Dual Space from Wikipedia, the Free Encyclopedia Contents

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Dual space From Wikipedia, the free encyclopedia Contents

1 Defective matrix 1 1.1 Jordan block ...... 1 1.2 Example ...... 1 1.3 See also ...... 2 1.4 Notes ...... 2 1.5 References ...... 2

2 Deﬁnite quadratic form 3 2.1 Associated symmetric bilinear form ...... 3 2.2 Example ...... 3 2.3 See also ...... 3 2.4 References ...... 4

3 Delta operator 5 3.1 Examples ...... 5 3.2 Basic polynomials ...... 6 3.3 See also ...... 6 3.4 References ...... 6

4 Dependence relation 7 4.1 Examples ...... 7 4.2 See also ...... 7

5 Determinant 8 5.1 Deﬁnition ...... 8 5.1.1 2 × 2 matrices ...... 9 5.1.2 3 × 3 matrices ...... 11 5.1.3 n × n matrices ...... 11 5.2 Properties of the determinant ...... 13 5.2.1 Multiplicativity and matrix groups ...... 15 5.2.2 Laplace’s formula and the adjugate matrix ...... 15 5.2.3 Sylvester’s determinant theorem ...... 15 5.3 Properties of the determinant in relation to other notions ...... 16 5.3.1 Relation to eigenvalues and trace ...... 16

i ii CONTENTS

5.3.2 Cramer’s rule ...... 18 5.3.3 Block matrices ...... 18 5.3.4 Derivative ...... 19 5.4 Abstract algebraic aspects ...... 20 5.4.1 Determinant of an endomorphism ...... 20 5.4.2 Exterior algebra ...... 20 5.4.3 Square matrices over commutative rings and abstract properties ...... 21 5.5 Generalizations and related notions ...... 22 5.5.1 Inﬁnite matrices ...... 22 5.5.2 Related notions for non-commutative rings ...... 22 5.5.3 Further variants ...... 22 5.6 Calculation ...... 22 5.6.1 Decomposition methods ...... 22 5.6.2 Further methods ...... 23 5.7 History ...... 23 5.8 Applications ...... 24 5.8.1 Linear independence ...... 24 5.8.2 Orientation of a basis ...... 24 5.8.3 Volume and Jacobian determinant ...... 25 5.8.4 Vandermonde determinant (alternant) ...... 25 5.8.5 Circulants ...... 26 5.9 See also ...... 26 5.10 Notes ...... 26 5.11 References ...... 28 5.12 External links ...... 28

6 Dieudonné determinant 29 6.1 Properties ...... 29 6.2 Tannaka–Artin problem ...... 29 6.3 See also ...... 29 6.4 References ...... 30

7 Dimension (vector space) 31 7.1 Examples ...... 31 7.2 Facts ...... 31 7.3 Generalizations ...... 32 7.3.1 Trace ...... 32 7.4 See also ...... 32 7.5 Notes ...... 33 7.6 References ...... 33 7.7 External links ...... 33 CONTENTS iii

8 Direct sum of modules 34 8.1 Construction for vector spaces and abelian groups ...... 34 8.1.1 Construction for two vector spaces ...... 34 8.1.2 Construction for two abelian groups ...... 35 8.2 Construction for an arbitrary family of modules ...... 35 8.3 Properties ...... 36 8.4 Internal direct sum ...... 37 8.5 Universal property ...... 37 8.6 Grothendieck group ...... 37 8.7 Direct sum of modules with additional structure ...... 37 8.7.1 Direct sum of algebras ...... 37 8.7.2 Direct sum of Banach spaces ...... 38 8.7.3 Direct sum of modules with bilinear forms ...... 38 8.7.4 Direct sum of Hilbert spaces ...... 38 8.8 See also ...... 39 8.9 References ...... 39

9 Direction vector 41 9.1 Parametric equation for a line ...... 41 9.2 Generative versus predicate forms ...... 41 9.3 Predicate form of 2D line equation ...... 41 9.4 See also ...... 42 9.5 External links ...... 42

10 Dot product 43 10.1 Definition ...... 43 10.1.1 Algebraic definition ...... 43 10.1.2 Geometric definition ...... 44 10.1.3 Scalar projection and first properties ...... 44 10.1.4 Equivalence of the definitions ...... 45 10.2 Properties ...... 46 10.2.1 Application to the cosine law ...... 47 10.3 Triple product expansion ...... 47 10.4 Physics ...... 48 10.5 Generalizations ...... 49 10.5.1 Complex vectors ...... 49 10.5.2 Inner product ...... 49 10.5.3 Functions ...... 49 10.5.4 Weight function ...... 50 10.5.5 Dyadics and matrices ...... 50 10.5.6 Tensors ...... 50 10.6 See also ...... 50 iv CONTENTS

10.7 References ...... 50 10.8 External links ...... 50

11 Dual basis 51 11.1 A categorical and algebraic construction of the dual space ...... 51 11.2 Existence and uniqueness ...... 51 11.3 Introduction ...... 52 11.4 Examples ...... 52 11.5 See also ...... 53 11.6 References ...... 53

12 Dual basis in a ﬁeld extension 54

13 Dual norm 55 13.1 Examples ...... 55 13.2 Notes ...... 55 13.3 References ...... 56

14 Dual number 57 14.1 Linear representation ...... 57 14.2 Geometry ...... 57 14.2.1 Cycles ...... 58 14.3 Algebraic properties ...... 58 14.4 Generalization ...... 58 14.5 Diﬀerentiation ...... 58 14.6 Superspace ...... 59 14.7 Division ...... 59 14.8 Projective line ...... 60 14.9 See also ...... 60 14.10Notes and references ...... 60 14.11Further reading ...... 61

15 Dual space 62 15.1 Algebraic dual space ...... 62 15.1.1 Finite-dimensional case ...... 62 15.1.2 Inﬁnite-dimensional case ...... 63 15.1.3 Bilinear products and dual spaces ...... 64 15.1.4 Injection into the double-dual ...... 64 15.1.5 Transpose of a linear map ...... 65 15.1.6 Quotient spaces and annihilators ...... 65 15.2 Continuous dual space ...... 66 15.2.1 Examples ...... 67 15.2.2 Transpose of a continuous linear map ...... 67 CONTENTS v

15.2.3 Annihilators ...... 68 15.2.4 Further properties ...... 68 15.2.5 Topologies on the dual ...... 68 15.2.6 Double dual ...... 68 15.3 See also ...... 69 15.4 Notes ...... 70 15.5 References ...... 70 15.6 Text and image sources, contributors, and licenses ...... 71 15.6.1 Text ...... 71 15.6.2 Images ...... 72 15.6.3 Content license ...... 73 Chapter 1

Defective matrix

In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n × n matrix is defective if and only if it does not have n linearly independent eigenvectors.[1] A complete basis is formed by augmenting the eigenvectors with generalized eigenvectors, which are necessary for solving defective systems of ordinary diﬀerential equations and other problems. A defective matrix always has fewer than n distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues λ with algebraic multiplicity m > 1 (that is, they are multiple roots of the characteristic polynomial), but fewer than m linearly independent eigenvectors associated with λ.[1] However, every eigenvalue with algebraic multiplicity m always has m linearly independent generalized eigenvectors. A Hermitian matrix (or the special case of a real symmetric matrix) or a unitary matrix is never defective; more generally, a normal matrix (which includes Hermitian and unitary as special cases) is never defective.

1.1 Jordan block

Any Jordan block of size 2×2 or larger is defective. For example, the n × n Jordan block,

  λ 1  .   λ ..  J =  ,  .   .. 1 λ

has an eigenvalue, λ, with algebraic multiplicity n, but only one distinct eigenvector,

  1   0 v = .. . 0

1.2 Example

A simple example of a defective matrix is:

[ ] 3 1 0 3

1 2 CHAPTER 1. DEFECTIVE MATRIX which has a double eigenvalue of 3 but only one distinct eigenvector

[ ] 1 0

(and constant multiples thereof).

1.3 See also

• Jordan normal form

1.4 Notes

[1] Golub & Van Loan (1996, p. 316)

1.5 References

• Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins University Press, ISBN 0-8018-5414-8

• Strang, Gilbert (1988). Linear Algebra and Its Applications (3rd ed.). San Diego: Harcourt. ISBN 970-686- 609-4. Chapter 2

Deﬁnite quadratic form

In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every nonzero vector of V. According to that sign, the quadratic form is called positive definite or negative definite. A semidefinite (or semi-definite) quadratic form is defined in the same way, except that “positive” and “negative” are replaced by “not negative” and “not positive”, respectively. An indefinite quadratic form is one that takes on both positive and negative values. More generally, the definition applies to a vector space over an ordered field.[1]

2.1 Associated symmetric bilinear form

Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space.[2] A symmetric bilinear form is also described as deﬁnite, semideﬁnite, etc. according to its associated quadratic form. A quadratic form Q and its associated symmetric bilinear form B are related by the following equations:

Q(x) = B(x, x)

1 − − B(x, y) = B(y, x) = 2 (Q(x + y) Q(x) Q(y))

2.2 Example

As an example, let V = R2 , and consider the quadratic form

2 2 Q(x) = c1x1 + c2x2

where x = (x1, x2) and c1 and c2 are constants. If c1 > 0 and c2 > 0, the quadratic form Q is positive definite. If one of the constants is positive and the other is zero, then Q is positive semidefinite. If c1 > 0 and c2 < 0, then Q is indefinite.

2.3 See also

• Anisotropic quadratic form

• Positive deﬁnite function

• Positive deﬁnite matrix

3 4 CHAPTER 2. DEFINITE QUADRATIC FORM

2.4 References

[1] Milnor & Husemoller (1973) p.61

[2] This is true only over a field of characteristic different of 2, but here we consider only ordered fields which necessarily have characteristic 0.

• Nathanael Leedom Ackerman (2006) Lecture notes Math 371, Positive deﬁnite bilinear form is deﬁnition 0.5.0.7, weblink from University of California, Berkeley.

• Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics 106. Cambridge University Press. ISBN 0-521-40475-4. Zbl 0785.11021.

• Lang, Serge (2004), Algebra, Graduate Texts in Mathematics 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, p. 578, ISBN 978-0-387-95385-4

• Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzge- biete 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016. Chapter 3

Delta operator

In mathematics, a delta operator is a shift-equivariant linear operator Q: K[x] −→ K[x] on the vector space of polynomials in a variable x over a ﬁeld K that reduces degrees by one. To say that Q is shift-equivariant means that if g(x) = f(x + a) , then

(Qg)(x) = (Qf)(x + a).

In other words, if f is a "shift" of g , then Qf is also a shift of Qg , and has the same "shifting vector" a . To say that an operator reduces degree by one means that if f is a polynomial of degree n , then Qf is either a polynomial of degree n − 1 , or, in case n = 0 , Qf is 0. Sometimes a delta operator is defined to be a shift-equivariant linear transformation on polynomials in x that maps x to a nonzero constant. Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition, since shift-equivariance is a fairly strong condition.

3.1 Examples

• The forward diﬀerence operator

(∆f)(x) = f(x + 1) − f(x)

is a delta operator.

• Diﬀerentiation with respect to x, written as D, is also a delta operator.

• Any operator of the form

∑∞ k ckD k=1

n (n) th (where D (ƒ) = ƒ is the n derivative) with c1 ≠ 0 is a delta operator. It can be shown that all delta operators can be written in this form. For example, the diﬀerence operator given above can be expanded as ∞ ∑ Dk ∆ = eD − 1 = . k! k=1

5 6 CHAPTER 3. DELTA OPERATOR

• The generalized derivative of time scale calculus which uniﬁes the forward diﬀerence operator with the derivative of standard calculus is a delta operator.

• In computer science and cybernetics, the term “discrete-time delta operator” (δ) is generally taken to mean a diﬀerence operator

f(x + ∆t) − f(x) (δf)(x) = , ∆t

the Euler approximation of the usual derivative with a discrete sample time ∆t . The delta-formulation obtains a signiﬁcant number of numerical advantages compared to the shift-operator at fast sampling.

3.2 Basic polynomials

Every delta operator Q has a unique sequence of “basic polynomials”, a polynomial sequence deﬁned by three conditions:

• p0(x) = 1;

• pn(0) = 0;

• (Qpn)(x) = npn−1(x), ∀n ∈ N.

Such a sequence of basic polynomials is always of binomial type, and it can be shown that no other sequences of binomial type exist. If the ﬁrst two conditions above are dropped, then the third condition says this polynomial sequence is a Sheﬀer sequence—a more general concept.

3.3 See also

• Pincherle derivative

• Shift operator • Umbral calculus

3.4 References

• Nikol'Skii, Nikolai Kapitonovich (1986), Treatise on the shift operator: spectral function theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-15021-5 Chapter 4

Dependence relation

In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence. Let X be a set. A (binary) relation ◁ between an element a of X and a subset S of X is called a dependence relation, written a ◁ S , if it satisﬁes the following properties:

• if a ∈ S , then a ◁ S ;

• if a ◁ S , then there is a ﬁnite subset S0 of S , such that a ◁ S0 ; • if T is a subset of X such that b ∈ S implies b ◁ T , then a ◁ S implies a ◁ T ;

• if a ◁ S but a ̸◁S − {b} for some b ∈ S , then b ◁ (S − {b}) ∪ {a} .

Given a dependence relation ◁ on X , a subset S of X is said to be independent if a ̸◁S −{a} for all a ∈ S. If S ⊆ T , then S is said to span T if t ◁ S for every t ∈ T.S is said to be a basis of X if S is independent and S spans X. Remark. If X is a non-empty set with a dependence relation ◁ , then X always has a basis with respect to ◁. Furthermore, any two bases of X have the same cardinality.

4.1 Examples

• Let V be a vector space over a field F. The relation ◁ , defined by υ ◁ S if υ is in the subspace spanned by S , is a dependence relation. This is equivalent to the definition of linear dependence. • Let K be a field extension of F. Define ◁ by α◁S if α is algebraic over F (S). Then ◁ is a dependence relation. This is equivalent to the definition of algebraic dependence.

4.2 See also

• matroid

This article incorporates material from Dependence relation on PlanetMath, which is licensed under the Creative Com- mons Attribution/Share-Alike License.

7 Chapter 5

Determinant

This article is about determinants in mathematics. For determinants in epidemiology, see risk factor.

In linear algebra, the determinant is a useful value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. In the case of a 2 × 2 matrix, the speciﬁc formula for the determinant is simply the upper left element times the lower right element, minus the product of the other two elements. Similarly, suppose we have a 3 × 3 matrix A, and we want the speciﬁc formula for its determinant |A|:

a b c e f d f d e |A| = d e f = a − b + c = aei + bfg + cdh − ceg − bdi − afh. h i g i g h g h i

Each determinant of a 2 × 2 matrix in this equation is called a "minor" of the matrix A. The same sort of procedure can be used to find the determinant of a 4 × 4 matrix, the determinant of a 5 × 5 matrix, and so forth. Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and the determinant is used to solve those equations. The use of determinants in calculus includes the Jacobian determinant in the change of variables rule for integrals of functions of several variables. Determinants are also used to define the characteristic polynomial of a matrix, which is essential for eigenvalue problems in linear algebra. Sometimes, determinants are used merely as a compact notation for expressions that would otherwise be unwieldy to write down. It can be proven that any matrix has a unique inverse if its determinant is nonzero. Various other theorems can be proved as well, including that the determinant of a product of matrices is always equal to the product of determinants; and, the determinant of a Hermitian matrix is always real.

5.1 Deﬁnition

There are various ways to deﬁne the determinant of a square matrix A, i.e. one with the same number of rows and columns. Perhaps the simplest way to express the determinant is by considering the elements in the top row and the respective minors; starting at the left, multiply the element by the minor, then subtract the product of the next element and its minor, and alternate adding and subtracting such products until all elements in the top row have been exhausted. For example, here is the result for a 4 × 4 matrix:

a b c d f g h e g h e f h e f g e f g h = a j k l −b i k l +c i j l −d i j k . i j k l n o p m o p m n p m n o m n o p

Another way to deﬁne the determinant is expressed in terms of the columns of the matrix. If we write an n × n matrix A in terms of its column vectors

8 5.1. DEFINITION 9

[ ] A = a1, a2, . . . , an

where the aj are vectors of size n, then the determinant of A is deﬁned so that

[ ] [ ] det a , . . . , ba + cv, . . . , a = b det(A) + c det a , ..., v, ...,a [ 1 j n ] [ 1 n ] det a1, . . . , aj, aj+1, . . . , an = − det a1, . . . , aj+1, aj, . . . , an det(I) = 1

where b and c are scalars, v is any vector of size n and I is the identity matrix of size n. These equations say that the determinant is a linear function of each column, that interchanging adjacent columns reverses the sign of the determinant, and that the determinant of the identity matrix is 1. These properties mean that the determinant is an alternating multilinear function of the columns that maps the identity matrix to the underlying unit scalar. These suffice to uniquely calculate the determinant of any square matrix. Provided the underlying scalars form a field (more generally, a commutative ring with unity), the definition below shows that such a function exists, and it can be shown to be unique.[1] Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is −1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries. This expression grows rapidly with the size of the matrix (an n × n matrix contributes n! terms), so it will first be given explicitly for the case of 2 × 2 matrices and 3 × 3 matrices, followed by the rule for arbitrary size matrices, which subsumes these two cases. Assume A is a square matrix with n rows and n columns, so that it can be written as

  a1,1 a1,2 . . . a1,n   a2,1 a2,2 . . . a2,n  A =  . . . . .  . . .. .  an,1 an,2 . . . an,n

The entries can be numbers or expressions (as happens when the determinant is used to deﬁne a characteristic polynomial); the deﬁnition of the determinant depends only on the fact that they can be added and multiplied together in a commutative manner. The determinant of A is denoted as det(A), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:

a1,1 a1,2 . . . a1,n

a2,1 a2,2 . . . a2,n ......

an,1 an,2 . . . an,n

5.1.1 2 × 2 matrices

The determinant of a 2 × 2 matrix is deﬁned by

a b = ad − bc. c d

If the matrix entries are real numbers, the matrix A can be used to represent two linear maps: one that maps the standard basis vectors to the rows of A, and one that maps them to the columns of A. In either case, the images of the basis vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram deﬁned by the rows of the above matrix is the one with vertices at (0, 0), (a, b), (a + c, b + d), and (c, d), as shown in the accompanying diagram. 10 CHAPTER 5. DETERMINANT

(a+c,b+d)

(c,d)

ad−bc

(a,b)

(0,0)

The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram’s sides.

The absolute value of ad − bc is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by A. (The parallelogram formed by the columns of A is in general a different parallelogram, but since the determinant is symmetric with respect to rows and columns, the area will be the same.) The absolute value of the determinant together with the sign becomes the oriented area of the parallelogram. The oriented area is the same as the usual area, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the identity matrix). Thus the determinant gives the scaling factor and the orientation induced by the mapping represented by A. When the determinant is equal to one, the linear mapping defined by the matrix is equi-areal and orientation-preserving. The object known as the bivector is related to these ideas. In 2D, it can be interpreted as an oriented plane segment formed by imagining two vectors each with origin (0, 0), and coordinates (a, b) and (c, d). The bivector magnitude 5.1. DEFINITION 11

(denoted (a, b) ∧ (c, d)) is the signed area, which is also the determinant ad − bc.[2]

5.1.2 3 × 3 matrices

The volume of this parallelepiped is the absolute value of the determinant of the matrix formed by the rows constructed from the vectors r1, r2, and r3.

The determinant of a 3 × 3 matrix is deﬁned by

a b c e f d f d e d e f = a − b + c h i g i g h g h i = a(ei − fh) − b(di − fg) + c(dh − eg) = aei + bfg + cdh − ceg − bdi − afh.

The rule of Sarrus is a mnemonic for the 3 × 3 matrix determinant: the sum of the products of three diagonal north- west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements, when the copies of the ﬁrst two columns of the matrix are written beside it as in the illustration. This scheme for calculating the determinant of a 3 × 3 matrix does not carry over into higher dimensions.

5.1.3 n × n matrices

The determinant of a matrix of arbitrary size can be deﬁned by the Leibniz formula or the Laplace formula. The Leibniz formula for the determinant of an n × n matrix A is 12 CHAPTER 5. DETERMINANT

Sarrus’ rule: The determinant of the three columns on the left is the sum of the products along the solid diagonals minus the sum of the products along the dashed diagonals

∑ ∏n

det(A) = sgn(σ) ai,σi .

σ∈Sn i=1 Here the sum is computed over all permutations σ of the set {1, 2, ..., n}. A permutation is a function that reorders this set of integers. The value in the ith position after the reordering σ is denoted σi. For example, for n = 3, the original sequence 1, 2, 3 might be reordered to σ = [2, 3, 1], with σ1 = 2, σ2 = 3, and σ3 = 1. The set of all such permutations (also known as the symmetric group on n elements) is denoted Sn. For each permutation σ, sgn(σ) denotes the signature of σ, a value that is +1 whenever the reordering given by σ can be achieved by successively interchanging two entries an even number of times, and −1 whenever it can be achieved by an odd number of such interchanges. In any of the n! summands, the term

∏n

ai,σi i=1 is notation for the product of the entries at positions (i, σi), where i ranges from 1 to n:

· ··· a1,σ1 a2,σ2 an,σn . For example, the determinant of a 3 × 3 matrix A (n = 3) is

∑ ∏n ∏n ∏n ∏n

sgn(σ) ai,σi = sgn([1, 2, 3]) ai,[1,2,3]i + sgn([1, 3, 2]) ai,[1,3,2]i + sgn([2, 1, 3]) ai,[2,1,3]i

σ∈Sn i=1 i=1 i=1 i=1 ∏n ∏n ∏n

+ sgn([2, 3, 1]) ai,[2,3,1]i + sgn([3, 1, 2]) ai,[3,1,2]i + sgn([3, 2, 1]) ai,[3,2,1]i i=1 i=1 i=1 ∏n ∏n ∏n ∏n ∏n ∏n − − − = ai,[1,2,3]i ai,[1,3,2]i ai,[2,1,3]i + ai,[2,3,1]i + ai,[3,1,2]i ai,[3,2,1]i i=1 i=1 i=1 i=1 i=1 i=1

= a1,1a2,2a3,3 − a1,1a2,3a3,2 − a1,2a2,1a3,3 + a1,2a2,3a3,1

+ a1,3a2,1a3,2 − a1,3a2,2a3,1. 5.2. PROPERTIES OF THE DETERMINANT 13

Levi-Civita symbol

It is sometimes useful to extend the Leibniz formula to a summation in which not only permutations, but all sequences of n indices in the range 1, ..., n occur, ensuring that the contribution of a sequence will be zero unless it denotes a

permutation. Thus the totally antisymmetric Levi-Civita symbol εi1,··· ,in extends the signature of a permutation, by setting εσ(1),··· ,σ(n) = sgn(σ) for any permutation σ of n, and εi1,··· ,in = 0 when no permutation σ exists such that σ(j) = ij for j = 1, . . . , n (or equivalently, whenever some pair of indices are equal). The determinant for an n × n matrix can then be expressed using an n-fold summation as

∑n ··· det(A) = εi1···in a1,i1 an,in ,

i1,i2,...,in=1 or using two epsilon symbols as

1 ∑ det(A) = ε ··· ε ··· a ··· a , n! i1 in j1 jn i1j1 injn where now each ir and each jr should be summed over 1, ..., n.

5.2 Properties of the determinant

The determinant has many properties. Some basic properties of determinants are

1. det(In) = 1 where In is the n × n identity matrix. 2. det(AT) = det(A).

−1 1 −1 3. det(A ) = det(A) = det(A) . 4. For square matrices A and B of equal size,

det(AB) = det(A) det(B).

1. det(cA) = cn det(A) for an n × n matrix. 2. If A is a triangular matrix, i.e. ai,j = 0 whenever i > j or, alternatively, whenever i < j, then its determinant equals the product of the diagonal entries:

∏n det(A) = a1,1a2,2 ··· an,n = ai,i. i=1

This can be deduced from some of the properties below, but it follows most easily directly from the Leibniz formula (or from the Laplace expansion), in which the identity permutation is the only one that gives a non-zero contribution. A number of additional properties relate to the eﬀects on the determinant of changing particular rows or columns:

1. Viewing an n × n matrix as being composed of n columns, the determinant is an n-linear function. This means that if one column of a matrix A is written as a sum v + w of two column vectors, and all other columns are left unchanged, then the determinant of A is the sum of the determinants of the matrices obtained from A by replacing the column by v and then by w (and a similar relation holds when writing a column as a scalar multiple of a column vector). 2. If in a matrix, any row or column is 0, then the determinant of that particular matrix is 0. 14 CHAPTER 5. DETERMINANT

3. This n-linear function is an alternating form. This means that whenever two columns of a matrix are identical, or more generally some column can be expressed as a linear combination of the other columns (i.e. the columns of the matrix form a linearly dependent set), its determinant is 0.

Properties 1, 7 and 9 — which all follow from the Leibniz formula — completely characterize the determinant; in other words the determinant is the unique function from n × n matrices to scalars that is n-linear alternating in the columns, and takes the value 1 for the identity matrix (this characterization holds even if scalars are taken in any given commutative ring). To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a standard basis vector. These determinants are either 0 (by property 8) or else ±1 (by properties 1 and 11 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear. For matrices over non-commutative rings, properties 7 and 8 are incompatible for n ≥ 2,[3] so there is no good definition of the determinant in this setting. Property 2 above implies that properties for columns have their counterparts in terms of rows:

1. Viewing an n × n matrix as being composed of n rows, the determinant is an n-linear function.

2. This n-linear function is an alternating form: whenever two rows of a matrix are identical, its determinant is 0.

3. Interchanging any pair of columns or rows of a matrix multiplies its determinant by −1. This follows from properties 7 and 9 (it is a general property of multilinear alternating maps). More generally, any permutation of the rows or columns multiplies the determinant by the sign of the permutation. By permutation, it is meant viewing each row as a vector Ri (equivalently each column as Ci) and reordering the rows (or columns) by interchange of Rj and Rk (or Cj and Ck), where j,k are two indices chosen from 1 to n for an n × n square matrix.

4. Adding a scalar multiple of one column to another column does not change the value of the determinant. This is a consequence of properties 7 and 8: by property 7 the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0 by property 8. Similarly, adding a scalar multiple of one row to another row leaves the determinant unchanged.

Property 5 says that the determinant on n × n matrices is homogeneous of degree n. These properties can be used to facilitate the computation of determinants by simplifying the matrix to the point where the determinant can be determined immediately. Specifically, for matrices with coefficients in a field, properties 11 and 12 can be used to transform any matrix into a triangular matrix, whose determinant is given by property 6; this is essentially the method of Gaussian elimination. For example, the determinant of

  −2 2 −3 A = −1 1 3  2 0 −1 can be computed using the following matrices:

      −2 2 −3 −2 2 −3 −2 2 −3 B =  0 0 4.5,C =  0 0 4.5,D =  0 2 −4. 2 0 −1 0 2 −4 0 0 4.5

Here, B is obtained from A by adding −1/2×the ﬁrst row to the second, so that det(A) = det(B). C is obtained from B by adding the ﬁrst to the third row, so that det(C) = det(B). Finally, D is obtained from C by exchanging the second and third row, so that det(D) = −det(C). The determinant of the (upper) triangular matrix D is the product of its entries on the main diagonal: (−2) · 2 · 4.5 = −18. Therefore, det(A) = −det(D) = +18. 5.2. PROPERTIES OF THE DETERMINANT 15

5.2.1 Multiplicativity and matrix groups

The determinant of a matrix product of square matrices equals the product of their determinants:

det(AB) = det(A) det(B).

Thus the determinant is a multiplicative map. This property is a consequence of the characterization given above of the determinant as the unique n-linear alternating function of the columns with value 1 on the identity matrix, since the function Mn(K) → K that maps M ↦ det(AM) can easily be seen to be n-linear and alternating in the columns of M, and takes the value det(A) at the identity. The formula can be generalized to (square) products of rectangular matrices, giving the Cauchy–Binet formula, which also provides an independent proof of the multiplicative property. The determinant det(A) of a matrix A is non-zero if and only if A is invertible or, yet another equivalent statement, if its rank equals the size of the matrix. If so, the determinant of the inverse matrix is given by

1 det(A−1) = . det(A) In particular, products and inverses of matrices with determinant one still have this property. Thus, the set of such matrices (of ﬁxed size n) form a group known as the special linear group. More generally, the word “special” indicates the subgroup of another matrix group of matrices of determinant one. Examples include the special orthogonal group (which if n is 2 or 3 consists of all rotation matrices), and the special unitary group.

5.2.2 Laplace’s formula and the adjugate matrix

Laplace’s formula expresses the determinant of a matrix in terms of its minors. The minor Mi,j is deﬁned to be the determinant of the (n−1) × (n−1)-matrix that results from A by removing the ith row and the jth column. The expression (−1)i+jMi,j is known as cofactor. The determinant of A is given by

∑n ∑n i+j i+j det(A) = (−1) ai,jMi,j = (−1) ai,jMi,j. j=1 i=1 Calculating det(A) by means of that formula is referred to as expanding the determinant along a row or column. For the example 3 × 3 matrix

  −2 2 −3 A = −1 1 3  , 2 0 −1

Laplace expansion along the second column (j = 2, the sum runs over i) yields: However, Laplace expansion is eﬃcient for small matrices only. The adjugate matrix adj(A) is the transpose of the matrix consisting of the cofactors, i.e.,

i+j (adj(A))i,j = (−1) Mj,i.

5.2.3 Sylvester’s determinant theorem

Sylvester’s determinant theorem states that for A, an m × n matrix, and B, an n × m matrix (so that A and B have dimensions allowing them to be multiplied in either order):

det(Im + AB) = det(In + BA) 16 CHAPTER 5. DETERMINANT where Im and In are the m × m and n × n identity matrices, respectively. From this general result several consequences follow.

(a) For the case of column vector c and row vector r, each with m components, the formula allows quick calculation of the determinant of a matrix that diﬀers from the identity matrix by a matrix of rank 1:

det(Im + cr) = 1 + rc

(b) More generally,[4] for any invertible m × m matrix X,

−1 det(X + AB) = det(X) det(In + BX A) det(X + cr) = det(X)(1 + rX−1c) = det(X) + r adj(X) c

5.3 Properties of the determinant in relation to other notions

5.3.1 Relation to eigenvalues and trace

Main article: Eigenvalues and eigenvectors

Let A be an arbitrary n × n matrix of complex numbers with eigenvalues λ1 , λ2 , ... λn . (Here it is understood that an eigenvalue with algebraic multiplicities µ occurs µ times in this list.) Then the determinant of A is the product of all eigenvalues:

∏n det(A) = λi = λ1λ2 ··· λn i=1 The product of all non-zero eigenvalues is referred to as pseudo-determinant. Conversely, determinants can be used to ﬁnd the eigenvalues of the matrix A: they are the solutions of the characteristic equation det(A − xI) = 0, where I is the identity matrix of the same dimension as A. An Hermitian matrix is positive deﬁnite if all its eigenvalues are positive. Sylvester’s criterion asserts that this is equivalent to the determinants of the submatrices

  a1,1 a1,2 . . . a1,k   a2,1 a2,2 . . . a2,k Ak :=  . . . .   . . .. .  ak,1 ak,2 . . . ak,k being positive, for all k between 1 and n. The trace tr(A) is by deﬁnition the sum of the diagonal entries of A and also equals the sum of the eigenvalues. Thus, for complex matrices A, det(exp(A)) = exp(tr(A)) 5.3. PROPERTIES OF THE DETERMINANT IN RELATION TO OTHER NOTIONS 17

or, for real matrices A,

tr(A) = log(det(exp(A))).

Here exp(A) denotes the matrix exponential of A, because every eigenvalue λ of A corresponds to the eigenvalue exp(λ) of exp(A). In particular, given any logarithm of A, that is, any matrix L satisfying

exp(L) = A

the determinant of A is given by

det(A) = exp(tr(L)).

For example, for n = 2, n = 3, and n = 4, respectively,

( ) det(A) = (tr A)2 − tr(A2) /2, ( ) det(A) = (tr A)3 − 3 tr A tr(A2) + 2 tr(A3) /6, ( ) det(A) = (tr A)4 − 6 tr(A2)(tr A)2 + 3(tr(A2))2 + 8 tr(A3) tr A − 6 tr(A4) /24.

cf. Cayley-Hamilton theorem. Such expressions are deducible from Newton’s identities. In the general case,[5]

∑ ∏n (−1)kl+1 l kl det(A) = k tr(A ) , l l kl! k1,k2,...,kn l=1 where the sum is taken over the set of all integers kl ≥ 0 satisfying the equation

∑n lkl = n. l=1

I This formula can also be used to ﬁnd the determinant of a matrix A J with multidimensional indices I = (i1,i2,...,iᵣ) and J = (j1,j2,...,jᵣ). The product and trace of such matrices are deﬁned in a natural way as

∑ ∑ I I K I (AB)J = AK BJ , tr(A) = AI . K I An arbitrary dimension n identity can be obtained from the Mercator series expansion of the logarithm when A ∈ B(0, 1)

 k ∞ ∞ ∑ 1 ∑ (−1)j det(I + A) = − tr(Aj) , k! j k=0 j=1 where I is the identity matrix. The sum and the expansion of the exponential only need to go up to n instead of ∞, since the determinant cannot exceed O(An). See also: fr:Algorithme de Faddeev Leverrier 18 CHAPTER 5. DETERMINANT

Upper and lower bounds

For a positive deﬁnite matrix A, the trace operator gives the following tight lower and upper bounds on the log determinant tr(I − A−1) ≤ log det(A) ≤ tr(A − I) with equality if and only if A = I . This relationship can be derived via the formula for the KL-divergence between two multivariate normal distributions.

5.3.2 Cramer’s rule

For a matrix equation

Ax = b the solution is given by Cramer’s rule:

det(A ) x = i i = 1, 2, 3, . . . , n i det(A) where Ai is the matrix formed by replacing the ith column of A by the column vector b. This follows immediately by column expansion of the determinant, i.e.

[ ] ∑n [ ] det(Ai) = det a1, . . . , b, . . . , an = xj det a1, . . . , ai−1, aj, ai+1, . . . , an = xi det(A) j=1 where the vectors aj are the columns of A. The rule is also implied by the identity

A adj(A) = adj(A) A = det(A) In.

It has recently been shown that Cramer’s rule can be implemented in O(n3) time,[6] which is comparable to more common methods of solving systems of linear equations, such as LU, QR, or singular value decomposition.

5.3.3 Block matrices

Suppose A, B, C, and D are matrices of dimension n × n, n × m, m × n, and m × m, respectively. Then

( ) ( ) A 0 AB det = det(A) det(D) = det . CD 0 D

This can be seen from the Leibniz formula, or from a decomposition like (for the former case)

( ) ( )( ) A 0 A 0 I 0 = n . CD CIm 0 D When A is invertible, one has

( ) AB det = det(A) det(D − CA−1B). CD 5.3. PROPERTIES OF THE DETERMINANT IN RELATION TO OTHER NOTIONS 19

as can be seen by employing the decomposition

( ) ( )( ) −1 AB A 0 In A B = −1 . CD CIm 0 D − CA B

When D is invertible, a similar identity with det(D) factored out can be derived analogously,[7] that is,

( ) AB det = det(D) det(A − BD−1C). CD

When the blocks are square matrices of the same order further formulas hold. For example, if C and D commute (i.e., CD = DC), then the following formula comparable to the determinant of a 2 × 2 matrix holds:[8]

( ) AB det = det(AD − BC). CD

When A = D and B = C, the blocks are square matrices of the same order and the following formula holds (even if A and B do not commute)

( ) AB det = det(A − B) det(A + B). BA

When D is a 1×1 matrix, B is a column vector, and C is a row vector then

( ) AB det = (D − 1) det(A) + det(A − BC) = (D + 1) det A − det(A + BC) . CD

5.3.4 Derivative

By deﬁnition, e.g., using the Leibniz formula, the determinant of real (or analogously for complex) square matrices is a polynomial function from Rn × n to R. As such it is everywhere diﬀerentiable. Its derivative can be expressed using Jacobi’s formula:

( ) d det(A) dA = tr adj(A) . dα dα where adj(A) denotes the adjugate of A. In particular, if A is invertible, we have

( ) d det(A) dA = det(A) tr A−1 . dα dα Expressed in terms of the entries of A, these are

∂ det(A) −1 = adj(A)ji = det(A)(A )ji. ∂Aij Yet another equivalent formulation is det(A + ϵX) − det(A) = tr(adj(A)X)ϵ + O(ϵ2) = det(A) tr(A−1X)ϵ + O(ϵ2)

using big O notation. The special case where A = I , the identity matrix, yields 20 CHAPTER 5. DETERMINANT

det(I + ϵX) = 1 + tr(X)ϵ + O(ϵ2).

This identity is used in describing the tangent space of certain matrix Lie groups. [ ] If the matrix A is written as A = a b c where a, b, c are vectors, then the gradient over one of the three vectors may be written as the cross product of the other two:

∇a det(A) = b × c

∇b det(A) = c × a

∇c det(A) = a × b.

5.4 Abstract algebraic aspects

5.4.1 Determinant of an endomorphism

The above identities concerning the determinant of products and inverses of matrices imply that similar matrices have the same determinant: two matrices A and B are similar, if there exists an invertible matrix X such that A = X−1BX. Indeed, repeatedly applying the above identities yields det(A) = det(X)−1 det(B) det(X) = det(B) det(X)−1 det(X) = det(B).

The determinant is therefore also called a similarity invariant. The determinant of a linear transformation

T : V → V for some ﬁnite-dimensional vector space V is deﬁned to be the determinant of the matrix describing it, with respect to an arbitrary choice of basis in V. By the similarity invariance, this determinant is independent of the choice of the basis for V and therefore only depends on the endomorphism T.

5.4.2 Exterior algebra

The determinant of a linear transformation A : V → V of an n-dimensional vector space V can be formulated in a coordinate-free manner by considering the nth exterior power ΛnV of V. A induces a linear map

ΛnA :ΛnV → ΛnV v1 ∧ v2 ∧ · · · ∧ vn 7→ Av1 ∧ Av2 ∧ · · · ∧ Avn. As ΛnV is one-dimensional, the map ΛnA is given by multiplying with some scalar. This scalar coincides with the determinant of A, that is to say

n (Λ A)(v1 ∧ · · · ∧ vn) = det(A) · v1 ∧ · · · ∧ vn.

This deﬁnition agrees with the more concrete coordinate-dependent deﬁnition. This follows from the characterization of the determinant given above. For example, switching two columns changes the sign of the determinant; likewise, permuting the vectors in the exterior product v1 ∧ v2 ∧ v3 ∧ ... ∧ vn to v2 ∧ v1 ∧ v3 ∧ ... ∧ vn, say, also changes its sign. For this reason, the highest non-zero exterior power Λn(V) is sometimes also called the determinant of V and similarly for more involved objects such as vector bundles or chain complexes of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms ΛkV with k < n. 5.4. ABSTRACT ALGEBRAIC ASPECTS 21

Transformation on alternating multilinear n-forms

The vector space W of all alternating multilinear n-forms on an n-dimensional vector space V has dimension one. To each linear transformation T on V we associate a linear transformation T′ on W, where for each w in W we deﬁne (T′w)(x1, ..., xn) = w(Tx1, ..., Txn). As a linear transformation on a one-dimensional space, T′ is equivalent to a scalar multiple. We call this scalar the determinant of T.

5.4.3 Square matrices over commutative rings and abstract properties

The determinant can also be characterized as the unique function

D : Mn(K) → K

from the set of all n × n matrices with entries in a field K to this field satisfying the following three properties: first, D is an n-linear function: considering all but one column of A fixed, the determinant is linear in the remaining column, that is

D(v1, . . . , vi−1, avi+bw, vi+1, . . . , vn) = aD(v1, . . . , vi−1, vi, vi+1, . . . , vn)+bD(v1, . . . , vi−1, w, vi+1, . . . , vn)

for any column vectors v1, ..., vn, and w and any scalars (elements of K) a and b. Second, D is an alternating function: for any matrix A with two identical columns D(A) = 0. Finally, D(In) = 1. Here In is the identity matrix. This fact also implies that every other n-linear alternating function F:Mn(K) → K satisﬁes

F (M) = F (I)D(M).

This deﬁnition can also be extended where K is a commutative ring R, in which case a matrix is invertible if and only if its determinant is a invertible element in R. For example, a matrix A with entries in Z, the integers, is invertible (in the sense that there exists an inverse matrix with integer entries) if the determinant is +1 or −1. Such a matrix is called unimodular. The determinant deﬁnes a mapping

× GLn(R) → R , between the group of invertible n × n matrices with entries in R and the multiplicative group of units in R. Since it respects the multiplication in both groups, this map is a group homomorphism. Secondly, given a ring homomorphism f: R → S, there is a map GLn(R) → GLn(S) given by replacing all entries in R by their images under f. The determinant respects these maps, i.e., given a matrix A = (ai,j) with entries in R, the identity

f(det((ai,j ))) = det((f(ai,j)))

holds. For example, the determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo m of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo m (the latter determinant being computed using modular arithmetic). In the more high-brow parlance of category theory, the determinant is a natural transformation between the two functors GLn and (⋅)×.[9] Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of algebraic groups, from the general linear group to the multiplicative group,

det : GLn → Gm. 22 CHAPTER 5. DETERMINANT

5.5 Generalizations and related notions

5.5.1 Inﬁnite matrices

For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in the Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated. Functional analysis provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators. The Fredholm determinant defines the determinant for operators known as trace class operators by an appropriate generalization of the formula det(I + A) = exp(tr(log(I + A))). Another infinite-dimensional notion of determinant is the functional determinant.

5.5.2 Related notions for non-commutative rings

For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other ways to define the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, for instance the multiplicative property or the fact that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero bilinear form with a regular element of R as value on some pair of arguments implies that R is commutative). Nevertheless various notions of non-commutative determinant have been formulated, which preserve some of the properties of determinants, notably quasideterminants and the Dieudonné determinant. It may be noted that if one considers certain specific classes of matrices with non-commutative elements, then there are examples where one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include quantum groups and q-determinant, Capelli matrix and Capelli determinant, super-matrices and Berezinian; Manin matrices is the class of matrices which is most close to matrices with commutative elements.

5.5.3 Further variants

[10] Determinants of matrices in superrings (that is, Z2-graded rings) are known as Berezinians or superdeterminants. The permanent of a matrix is deﬁned as the determinant, except that the factors sgn(σ) occurring in Leibniz’s rule are omitted. The immanant generalizes both by introducing a character of the symmetric group Sn in Leibniz’s rule.

5.6 Calculation

Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in numerical linear algebra, where for applications like checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques.[11] Nonetheless, explicitly calculating determinants is required in some situations, and different methods are available to do so. Naive methods of implementing an algorithm to compute the determinant include using the Leibniz formula or Laplace’s formula. Both these approaches are extremely inefficient for large matrices, though, since the number of required operations grows very quickly: it is of order n!(n factorial) for an n × n matrix M. For example, Leibniz’s formula requires calculating n! products. Therefore, more involved techniques have been developed for calculating determinants.

5.6.1 Decomposition methods

Given a matrix A, some methods compute its determinant by writing A as a product of matrices whose determinants can be more easily computed. Such techniques are referred to as decomposition methods. Examples include the 5.7. HISTORY 23

LU decomposition, the QR decomposition or the Cholesky decomposition (for positive deﬁnite matrices). These methods are of order O(n3), which is a signiﬁcant improvement over O(n!) The LU decomposition expresses A in terms of a lower triangular matrix L, an upper triangular matrix U and a permutation matrix P:

A = P LU.

The determinants of L and U can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of P is just the sign ε of the corresponding permutation (which is +1 for an even number of permutations and is −1 for an uneven number of permutations). The determinant of A is then

det(A) = ε det(L) · det(U),

Moreover, the decomposition can be chosen such that L is a unitriangular matrix and therefore has determinant 1, in which case the formula further simpliﬁes to

det(A) = ε det(U).

5.6.2 Further methods

If the determinant of A and the inverse of A have already been computed, the matrix determinant lemma allows to quickly calculate the determinant of A + uvT, where u and v are column vectors. Since the definition of the determinant does not need divisions, a question arises: do fast algorithms exist that do not need divisions? This is especially interesting for matrices over rings. Indeed algorithms with run-time proportional to n4 exist. An algorithm of Mahajan and Vinay, and Berkowitz[12] is based on closed ordered walks (short clow). It computes more products than the determinant definition requires, but some of these products cancel and the sum of these products can be computed more efficiently. The final algorithm looks very much like an iterated product of triangular matrices. If two matrices of order n can be multiplied in time M(n), where M(n) ≥ na for some a > 2, then the determinant can be computed in time O(M(n)).[13] This means, for example, that an O(n2.376) algorithm exists based on the Coppersmith–Winograd algorithm. Algorithms can also be assessed according to their bit complexity, i.e., how many bits of accuracy are needed to store intermediate values occurring in the computation. For example, the Gaussian elimination (or LU decomposition) methods is of order O(n3), but the bit length of intermediate values can become exponentially long.[14] The Bareiss Algorithm, on the other hand, is an exact-division method based on Sylvester’s identity is also of order n3, but the bit complexity is roughly the bit size of the original entries in the matrix times n.[15]

5.7 History

Historically, determinants were used long before matrices: originally, a determinant was defined as a property of a system of linear equations. The determinant “determines” whether the system has a unique solution (which occurs precisely if the determinant is non-zero). In this sense, determinants were first used in the Chinese mathematics text- book The Nine Chapters on the Mathematical Art (, Chinese scholars, around the 3rd century BCE). In Europe, 2 × 2 determinants were considered by Cardano at the end of the 16th century and larger ones by Leibniz.[16][17][18][19] In Japan, Seki Takakazu ( ) is credited with the discovery of the resultant and the determinant (at first in 1683, the complete version no later than 1710). In Europe, Cramer (1750) added to the theory, treating the subject in relation to sets of equations. The recurrence law was first announced by Bézout (1764). It was Vandermonde (1771) who first recognized determinants as independent functions.[16] Laplace (1772) [20][21] gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case. Immediately following, Lagrange (1773) treated determinants of the second and third order. 24 CHAPTER 5. DETERMINANT

Lagrange was the first to apply determinants to questions of elimination theory; he proved many special cases of general identities. Gauss (1801) made the next advance. Like Lagrange, he made much use of determinants in the theory of numbers. He introduced the word determinant (Laplace had used resultant), though not in the present signification, but rather as applied to the discriminant of a quantic. Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem. The next contributor of importance is Binet (1811, 1812), who formally stated the theorem relating to the product of two matrices of m columns and n rows, which for the special case of m = n reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy, Cauchy also presented one on the subject. (See Cauchy–Binet formula.) In this he used the word determinant in its present sense,[22][23] summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet’s.[16][24] With him begins the theory in its generality. The next important figure was Jacobi[17] (from 1827). He early used the functional determinant which Sylvester later called the Jacobian, and in his memoirs in Crelle for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called alternants. About the time of Jacobi’s last memoirs, Sylvester (1839) and Cayley began their work.[25][26] The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by Lebesgue, Hesse, and Sylvester; persymmetric determinants by Sylvester and Hankel; circulants by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians (so called by Muir) by Christoffel and Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and Hessians by Sylvester; and symmetric gauche determinants by Trudi. Of the textbooks on the subject Spottiswoode’s was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.

5.8 Applications

5.8.1 Linear independence

As mentioned above, the determinant of a matrix (with real or complex entries, say) is zero if and only if the column vectors (or the row vectors) of the matrix are linearly dependent. Thus, determinants can be used to characterize 3 linearly dependent vectors. For example, given two linearly independent vectors v1, v2 in R , a third vector v3 lies in the plane spanned by the former two vectors exactly if the determinant of the 3 × 3 matrix consisting of the three vectors is zero. The same idea is also used in the theory of differential equations: given n functions f1(x), ..., fn(x) (supposed to be n − 1 times differentiable), the Wronskian is defined to be

··· f1(x) f2(x) fn(x) ′ ′ ··· ′ f1(x) f2(x) fn(x)

W (f1, . . . , fn)(x) = ...... (n−1) (n−1) ··· (n−1) f1 (x) f2 (x) fn (x)

It is non-zero (for some x) in a speciﬁed interval if and only if the given functions and all their derivatives up to order n−1 are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of analytic functions, this implies the given functions are linearly dependent. See the Wronskian and linear independence.

5.8.2 Orientation of a basis

Main article: Orientation (vector space)

The determinant can be thought of as assigning a number to every sequence of n vectors in Rn, by using the square matrix whose columns are the given vectors. For instance, an orthogonal matrix with entries in Rn represents an orthonormal basis in Euclidean space. The determinant of such a matrix determines whether the orientation of the 5.8. APPLICATIONS 25

basis is consistent with or opposite to the orientation of the standard basis. If the determinant is +1, the basis has the same orientation. If it is −1, the basis has the opposite orientation. More generally, if the determinant of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal 2 × 2 or 3 × 3 matrix, this is a rotation), while if it is negative, A switches the orientation of the basis.

5.8.3 Volume and Jacobian determinant

As pointed out above, the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if f: Rn → Rn is the linear map represented by the matrix A, and S is any measurable subset of Rn, then the volume of f(S) is given by |det(A)| times the volume of S. More generally, if the linear map f: Rn → Rm is represented by the m × n matrix A, then the n-dimensional volume of f(S) is given by:

√ volume(f(S)) = det(ATA) × volume(S). By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines. The volume of any tetrahedron, given its vertices a, b, c, and d, is (1/6)·|det(a − b, b − c, c − d)|, or any other combination of pairs of vertices that would form a spanning tree over the vertices. For a general diﬀerentiable function, much of the above carries over by considering the Jacobian matrix of f. For

f : Rn → Rn, the Jacobian is the n × n matrix whose entries are given by

( ) ∂f D(f) = i . ∂xj 1≤i,j≤n Its determinant, the Jacobian determinant appears in the higher-dimensional version of integration by substitution: for suitable functions f and an open subset U of R'n (the domain of f), the integral over f(U) of some other function φ: Rn → Rm is given by

∫ ∫ ϕ(v) dv = ϕ(f(u)) |det(D f)(u)| du. f(U) U The Jacobian also occurs in the inverse function theorem.

5.8.4 Vandermonde determinant (alternant)

Main article: Vandermonde matrix

Third order

1 1 1

x1 x2 x3 = (x3 − x2)(x3 − x1)(x2 − x1) . 2 2 2 x1 x2 x3 In general, the nth-order Vandermonde determinant is [27]

1 1 1 ··· 1 ··· x1 x2 x3 xn ∏ 2 2 2 ··· 2 x1 x2 x3 xn − = (xj xi) , ...... ≤ ≤ . . . . . 1 i

5.8.5 Circulants

Main article: Circulant matrix

Second order

x1 x2 = (x1 + x2)(x1 − x2) . x2 x1 Third order

x1 x2 x3 ( )( ) 2 2 x3 x1 x2 = (x1 + x2 + x3) x1 + ωx2 + ω x3 x1 + ω x2 + ωx3 ,

x2 x3 x1 where ω and ω2 are the complex cube roots of 1. In general, the nth-order circulant determinant is[27]

x1 x2 x3 ··· xn

x x x ··· x − n 1 2 n 1 ∏n ( ) ··· − xn−1 xn x1 xn−2 = x + x ω + x ω2 + ... + x ωn 1 , . . . . 1 2 j 3 j n j ...... j=1 . . . . .

x2 x3 x4 ··· x1 where ωj is an nth root of 1.

5.9 See also

• Dieudonné determinant • Functional determinant • Immanant • Matrix determinant lemma • Permanent • Pfaﬃan • Slater determinant

5.10 Notes

[1] Serge Lang, Linear Algebra, 2nd Edition, Addison-Wesley, 1971, pp 173, 191. [2] WildLinAlg episode 4, Norman J Wildberger, Univ. of New South Wales, 2010, lecture via youtube [3] In a non-commutative setting left-linearity (compatibility with left-multiplication by scalars) should be distinguished from right-linearity. Assuming linearity in the columns is taken to be left-linearity, one would have, for non-commuting scalars a, b:

1 0 1 0 a 0 a 0 1 0 ab = ab = a = = b = ba = ba, 0 1 0 b 0 b 0 1 0 1 a contradiction. There is no useful notion of multi-linear functions over a non-commutative ring. 5.10. NOTES 27

[4] Proofs can be found in http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/proof003.html

[5] A proof can be found in the Appendix B of Kondratyuk, L. A.; Krivoruchenko, M. I. (1992). “Superconducting quark matter in SU(2) color group”. Zeitschrift für Physik A 344: 99–115. doi:10.1007/BF01291027.

[6] Ken Habgood, Itamar Arel, A condensation-based application of Cramerʼs rule for solving large-scale linear systems, Journal of Discrete Algorithms, 10 (2012), pp. 98–109. Available online 1 July 2011, ISSN 1570–8667, 10.1016/j.jda.2011.06.007.

[7] These identities were taken from http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/proof003.html

[8] Proofs are given in J.R. Silvester, Determinants of Block Matrices, Math. Gazette, 84 (2000), pp. 460–467, available at http://www.jstor.org/stable/3620776 or freely at http://www.ee.iisc.ernet.in/new/people/faculty/prasantg/downloads/ blocks.pdf

[9] Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics 5 ((2nd ed.) ed.), Springer-Verlag, ISBN 0-387-98403-8

[10] Varadarajan, V. S (2004), Supersymmetry for mathematicians: An introduction, ISBN 978-0-8218-3574-6.

[11] L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, 1997). e.g. in Lecture 1: "... we mention that the determinant, though a convenient notion theoretically, rarely ﬁnds a useful role in numerical algorithms.”

[12] http://page.inf.fu-berlin.de/~{}rote/Papers/pdf/Division-free+algorithms.pdf

[13] Bunch, J. R.; Hopcroft, J. E. (1974). “Triangular Factorization and Inversion by Fast Matrix Multiplication”. Mathematics of Computation 28 (125): 231–236. doi:10.1090/S0025-5718-1974-0331751-8.

[14] Fang, Xin Gui; Havas, George (1997). “On the worst-case complexity of integer Gaussian elimination” (PDF). Proceedings of the 1997 international symposium on Symbolic and algebraic computation. ISSAC '97. Kihei, Maui, Hawaii, United States: ACM. pp. 28–31. doi:10.1145/258726.258740. ISBN 0-89791-875-4.

[15] Bareiss, Erwin (1968), “Sylvester’s Identity and Multistep Integer-Preserving Gaussian Elimination” (PDF), Mathematics of computation 22 (102): 565–578

[16] Campbell, H: “Linear Algebra With Applications”, pages 111–112. Appleton Century Crofts, 1971

[17] Eves, H: “An Introduction to the History of Mathematics”, pages 405, 493–494, Saunders College Publishing, 1990.

[18] A Brief History of Linear Algebra and Matrix Theory : http://darkwing.uoregon.edu/~{}vitulli/441.sp04/LinAlgHistory. html

[19] Cajori, F. A History of Mathematics p. 80

[20] Expansion of determinants in terms of minors: Laplace, Pierre-Simon (de) “Researches sur le calcul intégral et sur le systéme du monde,” Histoire de l'Académie Royale des Sciences (Paris), seconde partie, pages 267–376 (1772).

[21] Muir, Sir Thomas, The Theory of Determinants in the historical Order of Development [London, England: Macmillan and Co., Ltd., 1906]. JFM 37.0181.02

[22] The ﬁrst use of the word “determinant” in the modern sense appeared in: Cauchy, Augustin-Louis “Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et des signes contraires par suite des transpositions operées entre les variables qu'elles renferment,” which was ﬁrst read at the Institute de France in Paris on November 30, 1812, and which was subsequently published in the Journal de l'Ecole Polytechnique, Cahier 17, Tome 10, pages 29–112 (1815).

[23] Origins of mathematical terms: http://jeff560.tripod.com/d.html

[24] History of matrices and determinants: http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants. html

[25] The ﬁrst use of vertical lines to denote a determinant appeared in: Cayley, Arthur “On a theorem in the geometry of position,” Cambridge Mathematical Journal, vol. 2, pages 267–271 (1841).

[26] History of matrix notation: http://jeff560.tripod.com/matrices.html

[27] Gradshteyn, I. S., I. M. Ryzhik: “Table of Integrals, Series, and Products”, 14.31, Elsevier, 2007. 28 CHAPTER 5. DETERMINANT

5.11 References

See also: Linear algebra § Further reading

• Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0

• de Boor, Carl (1990), “An empty exercise” (PDF), ACM SIGNUM Newsletter 25 (2): 3–7, doi:10.1145/122272.122273. • Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978- 0-321-28713-7 • Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8 • Muir, Thomas (1960) [1933], A treatise on the theory of determinants, Revised and enlarged by William H. Metzler, New York, NY: Dover • Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3

• Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International • Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall

5.12 External links

• Hazewinkel, Michiel, ed. (2001), “Determinant”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608- 010-4 |ﬁrst1= missing |last1= in Authors list (help) • Weisstein, Eric W., “Determinant”, MathWorld.

• O'Connor, John J.; Robertson, Edmund F., “Matrices and determinants”, MacTutor History of Mathematics archive, University of St Andrews.

• WebApp to calculate determinants and descriptively solve systems of linear equations • Determinant Interactive Program and Tutorial

• Online Matrix Calculator • Linear algebra: determinants. Compute determinants of matrices up to order 6 using Laplace expansion you choose. • Matrices and Linear Algebra on the Earliest Uses Pages

• Determinants explained in an easy fashion in the 4th chapter as a part of a Linear Algebra course. • Instructional Video on taking the determinant of an nxn matrix (Khan Academy)

• Online matrix calculator (determinant, track, inverse, adjoint, transpose) Compute determinant of matrix up to order 8

• Derivation of Determinant of a Matrix Chapter 6

Dieudonné determinant

In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings and local rings. It was introduced by Dieudonné (1943). If K is a division ring, then the Dieudonné determinant is a homomorphism of groups from the group GLn(K) of invertible n by n matrices over K onto the abelianization K*/[K*, K*] of the multiplicative group K* of K. For example, the Dieudonné determinant for a 2-by-2 matrix is

( ) { a b −cb ifa = 0 det = . c d ad − aca−1b ifa ≠ 0

6.1 Properties

Let R be a local ring. There is a determinant map from the matrix ring GL(R) to the abelianised unit group R∗ₐ with the following properties:[1]

• The determinant is invariant under elementary row operations

• The determinant of the identity is 1

• If a row is left multiplied by a in R∗ then the determinant is left multiplied by a

• The determinant is multiplicative: det(AB) = det(A)det(B)

• If two rows are exchanged, the determinant is multiplied by −1

• The determinant is invariant under transposition

6.2 Tannaka–Artin problem

Assume that K is ﬁnite over its centre F. The reduced norm gives a homomorphism Nn from GLn(K) to F*. We also have a homomorphism from GLn(K) to F* obtained by composing the Dieudonné determinant from GLn(K) to * * * * * K /[K , K ] with the reduced norm N1 from GL1(K) = K to F via the abelianization. The Tannaka–Artin problem is whether these two maps have the same kernel SLn(K). This is true when F is locally compact[2] but false in general.[3]

6.3 See also

• Moore determinant over a division algebra

29 30 CHAPTER 6. DIEUDONNÉ DETERMINANT

6.4 References

[1] Rosenberg (1994) p.64

[2] Nakayama, Tadasi; Matsushima, Yozô (1943). "Über die multiplikative Gruppe einer p-adischen Divisionsalgebra”. Proc. Imp. Acad. Tokyo (in German) 19: 622–628. doi:10.3792/pia/1195573246. Zbl 0060.07901.

[3] Platonov, V.P. (1976). “The Tannaka-Artin problem and reduced K-theory”. Izv. Akad. Nauk SSSR, Ser. Mat. (in Russian) 40: 227–261. Zbl 0338.16005.

• Dieudonné, Jean (1943), “Les déterminants sur un corps non commutatif”, Bulletin de la Société Mathématique de France 71: 27–45, ISSN 0037-9484, MR 0012273, Zbl 0028.33904 • Rosenberg, Jonathan (1994), Algebraic K-theory and its applications, Graduate Texts in Mathematics 147, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94248-3, MR 1282290, Zbl 0801.19001. Errata • Serre, Jean-Pierre (2003), Trees, Springer, p. 74, ISBN 3-540-44237-5, Zbl 1013.20001

• Suprunenko, D.A. (2001), “Determinant”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Chapter 7

Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.[1][lower-alpha 1] For every vector space there exists a basis,[lower-alpha 2] and all bases of a vector space have equal cardinality;[lower-alpha 3] as a result, the dimension of a vector space is uniquely defined. We say V is finite-dimensional if the dimension of V is finite, and infinite-dimensional if its dimension is infinite. The dimension of the vector space V over the field F can be written as dimF(V) or as [V : F], read “dimension of V over F". When F can be inferred from context, dim(V) is typically written.

7.1 Examples

The vector space R3 has

       1 0 0         0 , 1 , 0  0 0 1

as a basis, and therefore we have dimR(R3) = 3. More generally, dimR(Rn) = n, and even more generally, dimF(Fn) = n for any ﬁeld F. The complex numbers C are both a real and complex vector space; we have dimR(C) = 2 and dimC(C) = 1. So the dimension depends on the base ﬁeld. The only vector space with dimension 0 is {0}, the vector space consisting only of its zero element.

7.2 Facts

If W is a linear subspace of V, then dim(W) ≤ dim(V). To show that two finite-dimensional vector spaces are equal, one often uses the following criterion: if V is a finite- dimensional vector space and W is a linear subspace of V with dim(W) = dim(V), then W = V. n R has the standard basis {e1, ..., en}, where ei is the i-th column of the corresponding identity matrix. Therefore Rn has dimension n. Any two vector spaces over F having the same dimension are isomorphic. Any bijective map between their bases can be uniquely extended to a bijective linear map between the vector spaces. If B is some set, a vector space with dimension |B| over F can be constructed as follows: take the set F(B) of all functions f : B → F such that f(b) = 0 for all but finitely many b in B. These functions can be added and multiplied with elements of F, and we obtain the desired F-vector space. An important result about dimensions is given by the rank–nullity theorem for linear maps.

31 32 CHAPTER 7. DIMENSION (VECTOR SPACE)

If F/K is a ﬁeld extension, then F is in particular a vector space over K. Furthermore, every F-vector space V is also a K-vector space. The dimensions are related by the formula

dimK(V) = dimK(F) dimF(V).

In particular, every complex vector space of dimension n is a real vector space of dimension 2n. Some simple formulae relate the dimension of a vector space with the cardinality of the base ﬁeld and the cardinality of the space itself. If V is a vector space over a ﬁeld F then, denoting the dimension of V by dim V, we have:

If dim V is ﬁnite, then |V| = |F|dim V . If dim V is inﬁnite, then |V| = max(|F|, dim V).

7.3 Generalizations

One can see a vector space as a particular case of a matroid, and in the latter there is a well-deﬁned notion of dimension. The length of a module and the rank of an abelian group both have several properties similar to the dimension of vector spaces. The Krull dimension of a commutative ring, named after Wolfgang Krull (1899–1971), is deﬁned to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.

7.3.1 Trace

See also: Internal direct product

Suppose M is some R-module, and Mi is a submodule of M for every i in I. If every x in M can be written in one and only one way as a sum of ﬁnitely many elements of the Mi, then we say that M is the internal direct sum of the submodules Mi (Halmos 1974, §18). In this case, M is naturally isomorphic to the (external) direct sum of the Mi as deﬁned above (Adamson 1972, p.61). A submodule N of M is a direct summand of M if there exists some other submodule N′ of M such that M is the internal direct sum of N and N′. In this case, N and N′ are complementary subspaces.

8.5 Universal property

In the language of category theory, the direct sum is a coproduct and hence a colimit in the category of left R- modules, which means that it is characterized by the following universal property. For every i in I, consider the natural embedding

⊕ ji : Mi → Mk k∈I which sends the elements of Mi to those functions which are zero for all arguments but i. If fi : Mi → M are arbitrary R-linear maps for every i, then there exists precisely one R-linear map

⊕ f : Mi → M i∈I such that f o ji = fi for all i. Dually, the direct product is the product.

8.6 Grothendieck group

The direct sum gives a collection of objects the structure of a commutative monoid, in that the addition of objects is defined, but not subtraction. In fact, subtraction can be defined, and every commutative monoid can be extended to an abelian group. This extension is known as the Grothendieck group. The extension is done by defining equivalence classes of pairs of objects, which allows certain pairs to be treated as inverses. The construction, detailed in the article on the Grothendieck group, is “universal”, in that it has the universal property of being unique, and homomorphic to any other embedding of an abelian monoid in an abelian group.

8.7 Direct sum of modules with additional structure

If the modules we are considering carry some additional structure (e.g. a norm or an inner product), then the direct sum of the modules can often be made to carry this additional structure, as well. In this case, we obtain the coproduct in the appropriate category of all objects carrying the additional structure. Two prominent examples occur for Banach spaces and Hilbert spaces. In some classical texts, the notion of direct sum of algebras over a ﬁeld is also introduced. This construction, however, does not provide a coproduct in the category of algebras, but a direct product (see note below and the remark on direct sums of rings).

8.7.1 Direct sum of algebras

A direct sum of algebras X and Y is the direct sum as vector spaces, with product 38 CHAPTER 8. DIRECT SUM OF MODULES

(x1 + y1)(x2 + y2) = (x1x2 + y1y2). Consider these classical examples:

R ⊕ R is ring isomorphic to split-complex numbers, also used in interval analysis. C ⊕ C is the algebra of tessarines introduced by James Cockle in 1848. H ⊕ H , called the split-biquaternions, was introduced by William Kingdon Cliﬀord in 1873.

Joseph Wedderburn exploited the concept of a direct sum of algebras in his classification of hypercomplex numbers. See his Lectures on Matrices (1934), page 151. Wedderburn makes clear the distinction between a direct sum and a direct product of algebras: For the direct sum the field of scalars acts jointly on both parts: λ(x⊕y) = λx⊕λy while for the direct product a scalar factor may be collected alternately with the parts, but not both: λ(x, y) = (λx, y) = (x, λy) . Ian R. Porteous uses the three direct sums above, denoting them 2R, 2C, 2H , as rings of scalars in his analysis of Clifford Algebras and the Classical Groups (1995). These direct sums also arise in the classification of composition algebras. It is worth mentioning that the construction described above, as well as Wedderburn’s use of the terms direct sum and direct product follow a different convention from the one in category theory. In categorical terms, Wedderburn’s direct sum is a categorical product, whilst Wedderburn’s direct product is a coproduct (or categorical sum), which (for commutative algebras) actually corresponds to the tensor product of algebras.

8.7.2 Direct sum of Banach spaces

The direct sum of two Banach spaces X and Y is the direct sum of X and Y considered as vector spaces, with the norm ||(x,y)|| = ||x||X + ||y||Y for all x in X and y in Y. Generally, if Xi is a collection of Banach spaces, where i traverses the index set I, then the direct sum ⨁i∈I Xi is a module consisting of all functions x deﬁned over I such that x(i) ∈ Xi for all i ∈ I and

∑ ∥ ∥ ∞ x(i) Xi < . i∈I The norm is given by the sum above. The direct sum with this norm is again a Banach space.

For example, if we take the index set I = N and Xi = R, then the direct sum ⨁i∈NXi is the space l1, which consists of all the sequences (ai) of reals with ﬁnite norm ||a|| = ∑i |ai|. A closed subspace A of a Banach space X is complemented if there is another closed subspace B of X such that X is equal to the internal direct sum A ⊕ B . Note that not every closed subspace is complemented, e.g. c0 is not complemented in ℓ∞ .

8.7.3 Direct sum of modules with bilinear forms

Let {(Mi,bi : i ∈ I} be a family indexed by I of modules equipped with bilinear forms. The orthogonal direct sum is the module direct sum with bilinear form B deﬁned by[1]

∑ B ((xi) , (yi)) = bi (xi, yi) i∈I in which the summation makes sense even for inﬁnite index sets I because only ﬁnitely many of the terms are non-zero.

8.7.4 Direct sum of Hilbert spaces

Further information: Positive deﬁnite kernel#Direct sum and tensor product 8.8. SEE ALSO 39

If ﬁnitely many Hilbert spaces H1,...,Hn are given, one can construct their orthogonal direct sum as above (since they are vector spaces), deﬁning the inner product as:

⟨(x1, ..., xn), (y1, ..., yn)⟩ = ⟨x1, y1⟩ + ... + ⟨xn, yn⟩.

The resulting direct sum is a Hilbert space which contains the given Hilbert spaces as mutually orthogonal subspaces. If infinitely many Hilbert spaces Hi for i in I are given, we can carry out the same construction; notice that when defining the inner product, only finitely many summands will be non-zero. However, the result will only be an inner product space and it will not necessarily be complete. We then define the direct sum of the Hilbert spaces Hi to be the completion of this inner product space. Alternatively and equivalently, one can define the direct sum of the Hilbert spaces Hi as the space of all functions α with domain I, such that α(i) is an element of Hi for every i in I and:

∑ 2 α(i) < ∞. i The inner product of two such function α and β is then deﬁned as:

∑ ⟨α, β⟩ = ⟨αi, βi⟩. i This space is complete and we get a Hilbert space. For example, if we take the index set I = N and Xi = R, then the direct sum ⨁i∈N Xi is the space l , which consists √∑ 2 ∥ ∥ ∥ ∥2 of all the sequences (ai) of reals with finite norm a = i ai . Comparing this with the example for Banach spaces, we see that the Banach space direct sum and the Hilbert space direct sum are not necessarily the same. But if there are only finitely many summands, then the Banach space direct sum is isomorphic to the Hilbert space direct sum, although the norm will be different. Every Hilbert space is isomorphic to a direct sum of sufficiently many copies of the base field (either R or C). This is equivalent to the assertion that every Hilbert space has an orthonormal basis. More generally, every closed subspace of a Hilbert space is complemented: it admits an orthogonal complement. Conversely, the Lindenstrauss–Tzafriri theorem asserts that if every closed subspace of a Banach space is complemented, then the Banach space is isomorphic (topologically) to a Hilbert space.

8.8 See also

• Biproduct

• Indecomposable module

• Jordan–Hölder theorem

• Krull–Schmidt theorem

• Split exact sequence

8.9 References

[1] Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete 73. Springer-Verlag. pp. 4–5. ISBN 3-540-06009-X. Zbl 0292.10016.

• Iain T. Adamson (1972), Elementary rings and modules, University Mathematical Texts, Oliver and Boyd, ISBN 0-05-002192-3 40 CHAPTER 8. DIRECT SUM OF MODULES

• Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9.

• Dummit, David S.; Foote, Richard M. (1991), Abstract algebra, Englewood Cliﬀs, NJ: Prentice Hall, Inc., ISBN 0-13-004771-6.

• Halmos, Paul (1974), Finite dimensional vector spaces, Springer, ISBN 0-387-90093-4 • Mac Lane, S.; Birkhoﬀ, G. (1999), Algebra, AMS Chelsea, ISBN 0-8218-1646-2. Chapter 9

Direction vector

In mathematics, a direction vector that describes a line D is any vector

−−→ AB where A and B are two distinct points on the line. If v is a direction vector for D, so is kv for any nonzero scalar k; and these are in fact all of the direction vectors for the line D. Under some deﬁnitions, the direction vector is required to be a unit vector, in which case each line has exactly two direction vectors, which are negatives of each other (equal in magnitude, opposite in direction).

9.1 Parametric equation for a line

In Euclidean space (any number of dimensions), given a point a and a nonzero vector v, a line is deﬁned parametrically by (a+tv), where the parameter t varies between -∞ and +∞. This line has v as a direction vector.

9.2 Generative versus predicate forms

The line equation a+tv is a generative form, but not a predicate form. Points may be generated along the line given values for a, t and v: p ← a +tv However, in order to function as a predicate, the representation must be suﬃcient to easily determine ( T / F ) whether any speciﬁed point p is on the line. If you substitute a known point into the above equation, it cannot be evaluated for equality because t was not supplied, only p .

9.3 Predicate form of 2D line equation

An example of a predicate form of the vector line equation in 2D is: p • o == L Here, the line is represented by two features: o and L. o is the line’s orientation, a normalized direction vector (unit vector) pointing perpendicular to its run direction. The orientation is computed using the same two quantities dx and dy that go into computing slope m: o ← ( dy , -dx ) norm = ( dy , -dx ) / || ( dy , -dx ) || (orientation of a 2D line) Orientation o has the advantage of not overcompressing the information vested in dx and dy into a single scalar as slope does, avoiding the need to appeal to inﬁnity as a value. Numerical algorithms beneﬁt by avoiding such ill-behaved exceptions (e.g. , slope of a vertical line).

41 42 CHAPTER 9. DIRECTION VECTOR

The 2nd feature of a 2D line represented this way is its location L. Intuitively and visually, L is the signed distance of the line from the origin (with positive distance increasing along direction o). Orientation must be solved before determining location. Once o is known, L can be computed given any known point p on the line: L ← p • o (location of a 2D line) Lines may be represented as feature pair ( o , L ) in all cases. Every line has an equivalent representation ( -o , -L ). To determine if a point p is on the line, plug the value of p into the vector line predicate and evaluate it: p • o == L (vector line predicate) In computation, the predicate must tolerate ﬁnite math error by incorporating an epsilon signifying acceptable equality: abs (p • o - L) < epsilon (ﬁnite precision vector line predicate) The predicate form for 2D lines can be extended to higher dimensions, using coordinate rotation (by matrix rotation). The general form is: R p == L where R is the matrix rotation that aligns the line with an axis, and L is the invariant vector Location of points on the line under this rotation.

9.4 See also

• Unit vector

9.5 External links

• Weisstein, Eric W., “Direction vector”, MathWorld. • Glossary, Nipissing University

• Finding the vector equation of a line • Lines in a plane - Orthogonality, Distances, MATH-tutorial

• Coordinate Systems, Points, Lines and Planes Chapter 10

Dot product

“Scalar product” redirects here. For the abstract scalar product, see Inner product space. For the product of a vector and a scalar, see Scalar multiplication.

In mathematics, the dot product, or scalar product (or sometimes inner product in the context of Euclidean space), is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation can be deﬁned either algebraically or geometrically. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. The name “dot product” is derived from the centered dot " · " that is often used to designate this operation; the alternative name “scalar product” emphasizes the scalar (rather than vectorial) nature of the result. In three-dimensional space, the dot product contrasts with the cross product of two vectors, which produces a pseudovector as the result. The dot product is directly related to the cosine of the angle between two vectors in Euclidean space of any number of dimensions.

10.1 Deﬁnition

The dot product is often defined in one of two ways: algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude of vectors). The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space. In modern presentations of Euclidean geometry, the points of space are defined in terms of their Cartesian coordinates, and Euclidean space itself is commonly identified with the real coordinate space Rn. In such a presentation, the notions of length and angles are not primitive. They are defined by means of the dot product: the length of a vector is defined as the square root of the dot product of the vector by itself, and the cosine of the (non oriented) angle of two vectors of length one is defined as their dot product. So the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations of Euclidean geometry.

10.1.1 Algebraic deﬁnition

[1] The dot product of two vectors A = [A1, A2, ..., An] and B = [B1, B2, ..., Bn] is deﬁned as:

∑n A · B = AiBi = A1B1 + A2B2 + ··· + AnBn i=1 where Σ denotes summation notation and n is the dimension of the vector space. For instance, in three-dimensional space, the dot product of vectors [1, 3, −5] and [4, −2, −1] is:

43 44 CHAPTER 10. DOT PRODUCT

[1, 3, −5] · [4, −2, −1] = (1)(4) + (3)(−2) + (−5)(−1) = 4 − 6 + 5 = 3.

10.1.2 Geometric deﬁnition

In Euclidean space, a Euclidean vector is a geometrical object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction that the arrow points. The magnitude of a vector A is denoted by ∥A∥ . The dot product of two Euclidean vectors A and B is deﬁned by[2]

A · B = ∥A∥ ∥B∥ cos θ, where θ is the angle between A and B. In particular, if A and B are orthogonal, then the angle between them is 90° and

A · B = 0.

At the other extreme, if they are codirectional, then the angle between them is 0° and

A · B = ∥A∥ ∥B∥

This implies that the dot product of a vector A by itself is

A · A = ∥A∥2 , which gives

√ ∥A∥ = A · A, the formula for the Euclidean length of the vector.

10.1.3 Scalar projection and ﬁrst properties

The scalar projection (or scalar component) of a Euclidean vector A in the direction of a Euclidean vector B is given by

AB = ∥A∥ cos θ, where θ is the angle between A and B. In terms of the geometric deﬁnition of the dot product, this can be rewritten

b AB = A · B, where Bb = B/ ∥B∥ is the unit vector in the direction of B. The dot product is thus characterized geometrically by[3]

A · B = AB ∥B∥ = BA ∥A∥ . 10.1. DEFINITION 45

Scalar projection

The dot product, deﬁned in this manner, is homogeneous under scaling in each variable, meaning that for any scalar α,

(αA) · B = α(A · B) = A · (αB).

It also satisﬁes a distributive law, meaning that

A · (B + C) = A · B + A · C.

These properties may be summarized by saying that the dot product is a bilinear form. Moreover, this bilinear form is positive deﬁnite, which means that A · A is never negative and is zero if and only if A = 0.

10.1.4 Equivalence of the deﬁnitions

n If e1, ..., en are the standard basis vectors in R , then we may write

∑ A = [A1,...,An] = Aiei ∑i B = [B1,...,Bn] = Biei. i The vectors ei are an orthonormal basis, which means that they have unit length and are at right angles to each other. Hence since these vectors have unit length 46 CHAPTER 10. DOT PRODUCT

Distributive law for the dot product

ei · ei = 1 and since they form right angles with each other, if i ≠ j,

ei · ej = 0. Also, by the geometric deﬁnition, for any vector ei and a vector A, we note

A · ei = ∥A∥ ∥ei∥ cos θ = ∥A∥ cos θ = Ai, where Ai is the component of vector A in the direction of ei. Now applying the distributivity of the geometric version of the dot product gives

∑ ∑ ∑ A · B = A · Biei = Bi(A · ei) = BiAi, i i i which is precisely the algebraic deﬁnition of the dot product. So the (geometric) dot product equals the (algebraic) dot product.

10.2 Properties

The dot product fulﬁlls the following properties if a, b, and c are real vectors and r is a scalar.[1][2] 10.3. TRIPLE PRODUCT EXPANSION 47

1. Commutative: a · b = b · a, which follows from the deﬁnition (θ is the angle between a and b): a · b = ∥a∥ ∥b∥ cos θ = ∥b∥ ∥a∥ cos θ = b · a. 2. Distributive over vector addition: a · (b + c) = a · b + a · c.

3. Bilinear: a · (rb + c) = r(a · b) + (a · c).

4. Scalar multiplication:

(c1a) · (c2b) = c1c2(a · b).

5. Orthogonal: Two non-zero vectors a and b are orthogonal if and only if a ⋅ b = 0. 6. No cancellation: Unlike multiplication of ordinary numbers, where if ab = ac, then b always equals c unless a is zero, the dot product does not obey the cancellation law: If a ⋅ b = a ⋅ c and a ≠ 0, then we can write: a ⋅ (b − c) = 0 by the distributive law; the result above says this just means that a is perpendicular to (b − c), which still allows (b − c) ≠ 0, and therefore b ≠ c. 7. Product Rule: If a and b are functions, then the derivative (denoted by a prime ′) of a ⋅ b is a′ ⋅ b + a ⋅ b′.

10.2.1 Application to the cosine law

Main article: law of cosines

Given two vectors a and b separated by angle θ (see image right), they form a triangle with a third side c = a − b. The dot product of this with itself is:

c · c = (a − b) · (a − b) = a · a − a · b − b · a + b · b = a2 − a · b − a · b + b2 = a2 − 2a · b + b2 c2 = a2 + b2 − 2ab cos θ which is the law of cosines.

10.3 Triple product expansion

Main article: Triple product

This is an identity (also known as Lagrange’s formula) involving the dot- and cross-products. It is written as:[1][2]

a × (b × c) = b(a · c) − c(a · b), which may be remembered as “BAC minus CAB”, keeping in mind which vectors are dotted together. This formula ﬁnds application in simplifying vector calculations in physics. 48 CHAPTER 10. DOT PRODUCT c b

a c = a − b Triangle with vector edges a and b, separated by angle θ.

10.4 Physics

In physics, vector magnitude is a scalar in the physical sense, i.e. a physical quantity independent of the coordinate system, expressed as the product of a numerical value and a physical unit, not just a number. The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system. Examples include:[4][5]

• Mechanical work is the dot product of force and displacement vectors.

• Magnetic ﬂux is the dot product of the magnetic ﬁeld and the vector area. 10.5. GENERALIZATIONS 49

10.5 Generalizations

10.5.1 Complex vectors

For vectors with complex entries, using the given definition of the dot product would lead to quite different properties. For instance the dot product of a vector with itself would be an arbitrary complex number, and could be zero without the vector being the zero vector (such vectors are called isotropic); this in turn would have consequences for notions like length and angle. Properties such as the positive-definite norm can be salvaged at the cost of giving up the symmetric and bilinear properties of the scalar product, through the alternative definition[1]

∑ a · b = aibi,

where bi is the complex conjugate of bi. Then the scalar product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However this scalar product is thus sesquilinear rather than bilinear: it is conjugate linear and not linear in b, and the scalar product is not symmetric, since

a · b = b · a. The angle between two complex vectors is then given by

Re(a · b) cos θ = . ∥a∥ ∥b∥ This type of scalar product is nevertheless useful, and leads to the notions of Hermitian form and of general inner product spaces.

10.5.2 Inner product

Main article: Inner product space

The inner product generalizes the dot product to abstract vector spaces over a field of scalars, being either the field of real numbers R or the field of complex numbers C . It is usually denoted by ⟨a , b⟩ . The inner product of two vectors over the field of complex numbers is, in general, a complex number, and is sesquilinear instead of bilinear. An inner product space is a normed vector space, and the inner product of a vector with itself is real and positive-definite.

10.5.3 Functions

The dot product is defined for vectors that have a finite number of entries. Thus these vectors can be regarded as discrete functions: a length-n vector u is, then, a function with domain {k ∈ ℕ ∣ 1 ≤ k ≤ n}, and ui is a notation for the image of i by the function/vector u. This notion can be generalized to continuous functions: just as the inner product on vectors uses a sum over corresponding components, the inner product on functions is defined as an integral over some interval a ≤ x ≤ b (also denoted [a, b]):[1]

∫ b ⟨u, v⟩ = u(x)v(x)dx a Generalized further to complex functions ψ(x) and χ(x), by analogy with the complex inner product above, gives[1]

∫ b ⟨ψ, χ⟩ = ψ(x)χ(x)dx. a 50 CHAPTER 10. DOT PRODUCT

10.5.4 Weight function

Inner products can have a weight function, i.e. a function which weight each term of the inner product with a value.

10.5.5 Dyadics and matrices

Matrices have the Frobenius inner product, which is analogous to the vector inner product. It is deﬁned as the sum of the products of the corresponding components of two matrices A and B having the same size:

∑ ∑ H H A : B = AijBij = tr(B A) = tr(AB ). i j ∑ ∑ T T T T A : B = AijBij = tr(B A) = tr(AB ) = tr(A B) = tr(BA ). i j Dyadics have a dot product and “double” dot product deﬁned on them, see Dyadics (Product of dyadic and dyadic) for their deﬁnitions.

10.5.6 Tensors

The inner product between a tensor of order n and a tensor of order m is a tensor of order n + m − 2, see tensor contraction for details.

10.6 See also

• Cauchy–Schwarz inequality • Cross product • Matrix multiplication

10.7 References

[1] S. Lipschutz, M. Lipson (2009). Linear Algebra (Schaum’s Outlines) (4th ed.). McGraw Hill. ISBN 978-0-07-154352-1.

[2] M.R. Spiegel, S. Lipschutz, D. Spellman (2009). Vector Analysis (Schaum’s Outlines) (2nd ed.). McGraw Hill. ISBN 978-0-07-161545-7.

[3] Arfken, G. B.; Weber, H. J. (2000). Mathematical Methods for Physicists (5th ed.). Boston, MA: Academic Press. pp. 14–15. ISBN 978-0-12-059825-0..

[4] K.F. Riley, M.P. Hobson, S.J. Bence (2010). Mathematical methods for physics and engineering (3rd ed.). Cambridge University Press. ISBN 978-0-521-86153-3.

[5] M. Mansﬁeld, C. O’Sullivan (2011). Understanding Physics (4th ed.). John Wiley & Sons. ISBN 978-0-47-0746370.

10.8 External links

• Hazewinkel, Michiel, ed. (2001), “Inner product”, Encyclopedia of Mathematics, Springer, ISBN 978-1- 55608-010-4 • Weisstein, Eric W., “Dot product”, MathWorld. • Explanation of dot product including with complex vectors • “Dot Product” by Bruce Torrence, Wolfram Demonstrations Project, 2007. Chapter 11

Dual basis

In linear algebra, given a vector space V with a basis B of vectors indexed by an index set I (the cardinality of I is the dimensionality of V), its dual set is a set B∗ of vectors in the dual space V∗ with the same index set I such that B and B∗ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span V∗. If it does span V∗, then B∗ is called the dual basis for the basis B. ∗ i Denoting the indexed vector sets as B = {vi}i∈I and B = {v }i∈I , being biorthogonal means that the elements pair to 1 if the indexes are equal, and to zero otherwise. Symbolically, evaluating a dual vector in V∗ on a vector in the original space V:

{

i i 1 ifi = j v (vj) = δ = j 0 ifi ≠ j,

i where δj is the Kronecker delta symbol.

11.1 A categorical and algebraic construction of the dual space

Another way to introduce the dual space of a vector space (module) is by introducing it in a categorical sense. To do this, let A be a module defined over the ring R (that is, A is an object in the category R-Mod ). Then we define ∗ the dual space of A , denoted A , to be HomR(A, R) , the module formed of all R -linear module homomorphisms from A into R . Note then that we may define a dual to the dual, referred to as the double dual of A , written as A∗∗ ∗ , and defined as HomR(A ,R) . To formally construct a basis for the dual space, we shall now restrict our view to the case where F is a finite- dimensional free (left) R -module, where R is a ring of unity. Then, we assume that the set X is a basis for F . From here, we define the Kronecker Delta function δxy over the basis X by δxy = 1 if x = y and δxy = 0 if x ≠ y . Then the set S = {fx : F → R | fx(y) = δxy} describes a linearly independent set with each fx ∈ HomR(F,R) . Since F is finite-dimensional, the basis X is of finite cardinality. Then, the set S is a basis to F ∗ and F ∗ is a free (right) R -module.

11.2 Existence and uniqueness

The dual set always exists and gives an injection from V into V∗, namely the mapping that sends vi to vi. This says, in particular, that the dual space has dimension greater or equal to that of V. However, the dual set of an infinite-dimensional V does not span its dual space V∗. For example, consider the map ∗ w in V from V into the underlying scalars F given by w(vi) = 1 for all∑i. This map is clearly nonzero on all vi. If w i i were a finite linear combination∑ of the dual basis vectors v , say w = i∈K αiv for a finite subset K of I, then for i any j not in K, w(vj) = ( i∈K αiv )(vj) = 0 , contradicting the definition of w. So, this w does not lie in the span of the dual set.

51 52 CHAPTER 11. DUAL BASIS

The dual of an infinite-dimensional space has greater dimensionality (this being a greater infinite cardinality) than the original space has, and thus these cannot have a basis with the same indexing set. However, a dual set of vectors exists, which defines a subspace of the dual isomorphic to the original space. Further, for topological vector spaces, a continuous dual space can be defined, in which case a dual basis may exist.

Finite-dimensional vector spaces

In the case of ﬁnite-dimensional vector spaces, the dual set is always a dual basis and it is unique. These bases are ∗ 1 n denoted by B = { e1, …, en } and B = { e , …, e }. If one denotes the evaluation of a covector on a vector as a pairing, the biorthogonality condition becomes:

⟨ ⟩ i i e , ej = δj.

The association of a dual basis with a basis gives a map from the space of bases of V to the space of bases of V∗, and this is also an isomorphism. For topological ﬁelds such as the real numbers, the space of duals is a topological space, and this gives a homeomorphism between the Stiefel manifolds of bases of these spaces.

11.3 Introduction

To perform operations with a vector, we must have a straightforward method of calculating its components. In a Cartesian frame the necessary operation is simple dot product by the base vector.[1] 1 2 3 e.g. x = x i1 + x i2 + x i3 where ik is the bases in Cartesian frame.The components of x can be found by k x = x · ik In non-Cartesian frame, we do not necessarily have eᵢ·e=0 for all i≠j. However, it may still be possible to ﬁnd a vector ei such that xi = x · ei (i = 1, 2, 3) . the equality holds when ei is the dual base of eᵢ k In Cartesian frame, we have e = ek = ik

11.4 Examples

For example, the standard basis vectors of R2 (the Cartesian plane) are

{( ) ( )} 1 0 {e , e } = , 1 2 0 1

and the standard basis vectors of its dual space R2* are

{( ) ( )} {e1, e2} = 1 0 , 0 1 .

1 2 In 3-dimensional Euclidean space, for a given basis {e1, e2, e3}, you can ﬁnd the biorthogonal (dual) basis {e , e , e3} by formulas below:

( ) ( ) ( ) e × e T e × e T e × e T e1 = 2 3 , e2 = 3 1 , e3 = 1 2 . V V V where T denotes the transpose and 11.5. SEE ALSO 53

V = (e1; e2; e3) = e1 · (e2 × e3) = e2 · (e3 × e1) = e3 · (e1 × e2) is the volume of the parallelepiped formed by the basis vectors e1, e2 and e3.

11.5 See also

• Reciprocal lattice • Miller index

• Zone axis

11.6 References

[1] Lebedev, Cloud & Eremeyev 2010, p. 12.

• Lebedev, Leonid P.; Cloud, Michael J.; Eremeyev, Victor A. (2010). Tensor Analysis With Applications to Mechanics. World Scientiﬁc. ISBN 978-981431312-4. Chapter 12

Dual basis in a ﬁeld extension

In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite extension L/K, by using the field trace. This requires the property that the field trace TrL/K provides a non-degenerate quadratic form over K. This can be guaranteed if the extension is separable; it is automatically true if K is a perfect field, and hence in the cases where K is finite, or of characteristic zero. A dual basis is not a concrete basis like the polynomial basis or the normal basis; rather it provides a way of using a second basis for computations. Consider two bases for elements in a finite field, GF(pm):

B1 = α0, α1, . . . , αm−1

and

B2 = γ0, γ1, . . . , γm−1 then B2 can be considered a dual basis of B1 provided

{ 0, if i ≠ j Tr(α · γ ) = i j 1, otherwise

Here the trace of a value in GF(pm) can be calculated as follows:

m∑−1 i Tr(β) = βp i=0 Using a dual basis can provide a way to easily communicate between devices that use diﬀerent bases, rather than having to explicitly convert between bases using the change of bases formula. Furthermore, if a dual basis is implemented then conversion from an element in the original basis to the dual basis can be accomplished with a multiplication by the multiplicative identity (usually 1).

54 Chapter 13

Dual norm

The concept of a dual norm arises in functional analysis, a branch of mathematics. Let X be a normed space (or, in a special case, a Banach space) over a number field F (i.e. F = C or F = R ) with norm ∥ · ∥ . Then the dual (or conjugate) normed space X′ (another notation X∗ ) is defined as the set of all continuous linear functionals from X into the base field F . If f : X → F is such a linear functional, then the dual norm[1] ∥ · ∥′ of f is defined by

{ } |f(x)| ∥f∥′ = sup{|f(x)| : x ∈ X, ∥x∥ ≤ 1} = sup : x ∈ X, x ≠ 0 . ∥x∥ With this norm, the dual space X′ is also a normed space, and moreover a Banach space, since X′ is always complete.[2]

13.1 Examples

1. Dual Norm of Vectors If p, q ∈ [1, ∞] satisfy 1/p + 1/q = 1 , then the ℓp and ℓq norms are dual to each other. In particular the Euclidean norm is self-dual (p = q = 2). Similarly, the Schatten p-norm on matrices is dual√ to the Schatten q-norm. √ For xTQx , the dual norm is yTQ−1y with Q positive deﬁnite. 2. Dual Norm of Matrices Frobenius norm v v u u u∑m ∑n √ umin∑{m, n} ∗ t ∥ ∥ t | |2 2 A F = aij = trace(A A) = σi i=1 j=1 i=1

Its dual norm is ∥B∥F Singular value norm

∥A∥2 = σmax(A) ∑ Dual norm i σi(B)

13.2 Notes

[1] A.N.Kolmogorov, S.V.Fomin (1957, III §23)

[2] http://www.seas.ucla.edu/~{}vandenbe/236C/lectures/proxop.pdf

55 56 CHAPTER 13. DUAL NORM

13.3 References

• Kolmogorov, A.N.; Fomin, S.V. (1957), Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces, Rochester: Graylock Press • Rudin, Walter (1991), Functional analysis, McGraw-Hill Science, ISBN 978-0-07-054236-5 Chapter 14

Dual number

For dual grammatical number found in some languages, see Dual (grammatical number).

In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 (ε is nilpotent). The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and b uniquely determined real numbers. Dual numbers can also be thought of as the Exterior algebra of a one dimensional vector space. The algebra of dual numbers is a ring that is a local ring since the principal ideal generated by ε is its only maximal ideal. Dual numbers form the coeﬃcients of dual quaternions.

14.1 Linear representation

Using matrices, dual numbers can be represented as

( ) ( ) 0 1 a b ε = and a + bε = 0 0 0 a

The sum and product of dual numbers are then calculated with ordinary matrix addition and matrix multiplication; both operations are commutative and associative within the algebra of dual numbers. This correspondence is analogous to the usual matrix representation of complex numbers. However, it is not the only representation with 2 × 2 real matrices, as is shown in the proﬁle of 2 × 2 real matrices. Like the complex plane and split-complex number plane, the dual numbers are one of the realizations of planar algebra.

14.2 Geometry

The “unit circle” of dual numbers consists of those with a = 1 or −1 since these satisfy z z* = 1 where z* = a − bε. However, note that

( ) ∑∞ exp(bε) = (bε)n/n! = 1 + bε n=0 so the exponential map applied to the ε-axis covers only half the “circle”. If a ≠ 0 and m = b /a, then z = a(1 + m ε) is the polar decomposition of the dual number z, and the slope m is its angular part. The concept of a rotation in the dual number plane is equivalent to a vertical shear mapping since (1 + p ε)(1 + q ε) = 1 + (p+q) ε. In absolute space and time the Galilean transformation

57 58 CHAPTER 14. DUAL NUMBER

( ) 1 v (t′, x′) = (t, x) , that is t′ = t, x′ = vt + x, 0 1 relates the resting coordinates system to a moving frame of reference of velocity v. With dual numbers t + x ε representing events along one space dimension and time, the same transformation is eﬀected with multiplication by (1 + v ε).

14.2.1 Cycles

Given two dual numbers p, and q, they determine the set of z such that the diﬀerence in slopes (“Galilean angle”) between the lines from z to p and q is constant. This set is a cycle in the dual number plane; since the equation setting the diﬀerence in slopes of the lines to a constant is a quadratic equation in the real part of z, a cycle is a parabola. The “cyclic rotation” of the dual number plane occurs as a motion of the projective line over dual numbers. According to Yaglom (pp. 92,3), the cycle Z = {z : y = α x2} is invariant under the composition of the shear

x1 = x, y1 = vx + y with the translation ′ ′ 2 x = x1 = v/2a, y = y1 + v /4a .

This composition is a cyclic rotation; the concept has been further developed by V. V. Kisil.[1]

14.3 Algebraic properties

In abstract algebra terms, the dual numbers can be described as the quotient of the polynomial ring R[X] by the ideal generated by the polynomial X2,

R[X]/(X2).

The image of X in the quotient is the unit ε. With this description, it is clear that the dual numbers form a commutative ring with characteristic 0. The inherited multiplication gives the dual numbers the structure of a commutative and associative algebra over the reals of dimension two. The algebra is not a division algebra or ﬁeld since the elements of the form 0 + bε are not invertible. All elements of this form are zero divisors (also see the section "Division"). The algebra of dual numbers is isomorphic to the exterior algebra of R1 .

14.4 Generalization

This construction can be carried out more generally: for a commutative ring R one can define the dual numbers over R as the quotient of the polynomial ring R[X] by the ideal (X2): the image of X then has square equal to zero and corresponds to the element ε from above. This ring and its generalisations play an important part in the algebraic theory of derivations and Kähler differentials (purely algebraic differential forms). Over any ring R, the dual number a + bε is a unit (i.e. multiplicatively invertible) if and only if a is a unit in R. In this case, the inverse of a + bε is a−1 − ba−2ε. As a consequence, we see that the dual numbers over any field (or any commutative local ring) form a local ring, its maximal ideal being the principal ideal generated by ε.

14.5 Diﬀerentiation

One application of dual numbers is automatic diﬀerentiation. Consider the real dual numbers above. Given any real 2 n polynomial P(x) = p0+p1x+p2x +...+pnx , it is straightforward to extend the domain of this polynomial from the reals to the dual numbers. Then we have this result: 14.6. SUPERSPACE 59

n P (a + bε) =p0 + p1(a + bε) + ... + pn(a + bε) 2 n =p0 + p1a + p2a + ... + pna n−1 + p1bε + 2p2abε + ... + npna bε =P (a) + bP ′(a)ε, where P ′ is the derivative of P .[2] By computing over the dual numbers, rather than over the reals, we can use this to compute derivatives of polynomials. More generally, we can extend any (smooth) real function to the dual numbers by looking at its Taylor series: f(a + ∑ ∞ f (n)(a)bnεn ′ bε) = n=0 n! = f(a) + bf (a)ε . By computing compositions of these functions over the dual numbers and examining the coeﬃcient of ε in the result we ﬁnd we have automatically computed the derivative of the composition. A similar method works for polynomials of n variables, using the exterior algebra of an n-dimensional vector space.

14.6 Superspace

Dual numbers ﬁnd applications in physics, where they constitute one of the simplest non-trivial examples of a superspace. The direction along ε is termed the “fermionic” direction, and the real component is termed the “bosonic” direction. The fermionic direction earns this name from the fact that fermions obey the Pauli exclusion principle: under the exchange of coordinates, the quantum mechanical wave function changes sign, and thus vanishes if two coordinates are brought together; this physical idea is captured by the algebraic relation ε2 = 0.

14.7 Division

Division of dual numbers is deﬁned when the real part of the denominator is non-zero. The division process is analogous to complex division in that the denominator is multiplied by its conjugate in order to cancel the non-real parts. Therefore, to divide an equation of the form:

a + bε c + dε We multiply the top and bottom by the conjugate of the denominator:

(a + bε)(c − dε) ac − adε + bcε − bdε2 ac − adε + bcε − 0 = = = (c + dε)(c − dε) (c2 + cdε − cdε − d2ε2) c2 − 0

ac + ε(bc − ad) = c2 a (bc − ad) = + ε c c2 Which is deﬁned when c is non-zero. If, on the other hand, c is zero while d is not, then the equation

a + bε = (x + yε)dε = xdε + 0

1. has no solution if a is nonzero 2. is otherwise solved by any dual number of the form 60 CHAPTER 14. DUAL NUMBER

b + yε d This means that the non-real part of the “quotient” is arbitrary and division is therefore not deﬁned for purely nonreal dual numbers. Indeed, they are (trivially) zero divisors and clearly form an ideal of the associative algebra (and thus ring) of the dual numbers.

14.8 Projective line

The idea of a projective line over dual numbers was advanced by Grünwald.[3] and Corrado Segre. [4] Just as the Riemann sphere needs a north pole point at infinity to close up the complex projective line, so a line at infinity succeeds in closing up the plane of dual numbers to a cylinder. [5] Suppose D is the ring of dual numbers x + y ε and U is the subset with x ≠ 0. Then U is the group of units of D. Let B = {(a,b) in D x D : a ∈ U or b ∈ U}. A relation is defined on B as follows: (a,b) ~ (c,d) when there is a u in U such that ua=c and ub=d. This relation is in fact an equivalence relation. The points of the projective line over D are equivalence classes in B under this relation: P(D) = B/ ~. Consider the embedding D → P(D) by z → U(z,1) where U(z,1) is the equivalence class of (z,1). Then points U(1,n), n2 = 0, are in P(D) but are not the image of any point under the embedding. P(D) is projected onto a cylinder by projection: Take a cylinder tangent to the double number plane on the line {y ε: y ∈ ℝ}, ε2 = 0. Now take the opposite line on the cylinder for the axis of a pencil of planes. The planes intersecting the dual number plane and cylinder provide a correspondence of points between these surfaces. The plane parallel to the dual number plane corresponds to points U(1,n), n2 = 0 in the projective line over dual numbers.

14.9 See also

• Perturbation theory

14.10 Notes and references

[1] V.V. Kisil (2007) “Inventing a Wheel, the Parabolic One” arXiv:0707.4024

[2] Berland, Håvard. “Automatic diﬀerentiation” (PDF). Retrieved 13 May 2013.

[3] Josef Grünwald (1906) "Über duale Zahlen und ihre Anwendung in der Geometrie”, Monatshefte für Mathematik 17: 81–136

[4] Corrado Segre (1912) “Le geometrie proiettive nei campi di numeri duali”, Paper XL of Opere, also Atti della R. Academia della Scienze di Torino, vol XLVII.

[5] I. M. Yaglom (1979) A Simple Non-Euclidean Geometry and its Physical Basis, pp 149–53, Springer, ISBN 0387-90332-1, MR 520230

• Bencivenga, Uldrico (1946) “Sulla rappresentazione geometrica della algebra doppie dotate di modulo”, Atti della real accademia della scienze e belle-lettre di Napoli, Ser (3) v.2 No7. MR 0021123.

• William Kingdon Cliﬀord (1873) Preliminary Sketch of Bi-quaternions, Proceedings of the London Mathe- matical Society 4:381–95

• Anthony A. Harkin & Joseph B. Harkin (2004) Geometry of Generalized Complex Numbers, Mathematics Magazine 77(2):118–29.

• Eduard Study (1903) Geometrie der Dynamen, page 196, from Cornell Historical Mathematical Monographs at Cornell University.

• Isaak Yaglom (1968) Complex Numbers in Geometry, pp 12–18, Academic Press. 14.11. FURTHER READING 61

14.11 Further reading

• Ian Fischer (1998). Dual-Number Methods in Kinematics, Statics and Dynamics. CRC Press. ISBN 978-0- 8493-9115-6. • E. Pennestri & R. Stefanelli (2007) Linear Algebra and Numerical Algorithms Using Dual Numbers, published in Multibody System Dynamics 18(3):323–49.

• D.P. Chevallier (1996) “On the transference principle in kinematics: its various forms and limitations”, Mech- anism and Machine Theory 31(1):57–76.

• M.A. Gungor (2009) “Dual Lorentzian spherical motions and dual Euler-Savary formuilas”, European Journal of Mechanics A Solids 28(4):820–6. Chapter 15

Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V together with a naturally induced linear structure. Dual vector spaces for finite-dimensional vector spaces show up in tensor analysis. When applied to vector spaces of functions (which are typically infinite- dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. There are two types of dual spaces: the algebraic dual space, and the continuous dual space. The algebraic dual space is defined for all vector spaces. When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.

15.1 Algebraic dual space

Given any vector space V over a ﬁeld F, the dual space V∗ is deﬁned as the set of all linear maps φ: V → F (linear functionals). The dual space V∗ itself becomes a vector space over F when equipped with an addition and scalar multiplication satisfying:

(φ + ψ)(x) = φ(x) + ψ(x) (aφ)(x) = a (φ(x))

for all φ and ψ ∈ V∗, x ∈ V, and a ∈ F. Elements of the algebraic dual space V∗ are sometimes called covectors or one-forms. The pairing of a functional φ in the dual space V∗ and an element x of V is sometimes denoted by a bracket: φ(x) = [φ,x] [1] or φ(x) = <φ,x>.[2] The pairing deﬁnes a nondegenerate bilinear mapping[3] [·,·] : V∗ × V → F.

15.1.1 Finite-dimensional case

∗ If V is finite-dimensional, then V has the same dimension as V. Given a basis {e1, ..., en} in V, it is possible to construct a specific basis in V∗, called the dual basis. This dual basis is a set {e1, ..., en} of linear functionals on V, defined by the relation

i 1 n i e (c e1 + ··· + c en) = c , i = 1, . . . , n

for any choice of coefficients ci ∈ F. In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations

i i e (ej) = δj

62 15.1. ALGEBRAIC DUAL SPACE 63

i 2 where δj is the Kronecker delta symbol. For example if V is R , and its basis chosen to be {e1 = (1, 0), e2 = (0, 1)}, 1 2 1 1 2 then e and e are one-forms (functions that map a vector to a scalar) such that e (e1) = 1, e (e2) = 0, e (e1) = 0, 2 and e (e2) = 1. (Note: The superscript here is the index, not an exponent). In particular, if we interpret Rn as the space of columns of n real numbers, its dual space is typically written as the space of rows of n real numbers. Such a row acts on Rn as a linear functional by ordinary matrix multiplication. One way to see this is that a functional maps every n-vector x into a real number y. Then, seeing this functional as a matrix M, and x, y as a n × 1 matrix and a 1 × 1 matrix (trivially, a real number) respectively, if we have Mx = y, then, by dimension reasons, M must be a 1 × n matrix, i.e., M must be a row vector. If V consists of the space of geometrical vectors in the plane, then the level curves of an element of V∗ form a family of parallel lines in V, because the range is 1-dimensional, so that every point in the range is a multiple of any one nonzero element. So an element of V∗ can be intuitively thought of as a particular family of parallel lines covering the plane. To compute the value of a functional on a given vector, one needs only to determine which of the lines the vector lies on. Or, informally, one “counts” how many lines the vector crosses. More generally, if V is a vector space of any dimension, then the level sets of a linear functional in V∗ are parallel hyperplanes in V, and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.[4]

15.1.2 Inﬁnite-dimensional case

If V is not finite-dimensional but has a basis[5] eα indexed by an infinite set A, then the same construction as in the finite-dimensional case yields linearly independent elements eα (α ∈ A) of the dual space, but they will not form a basis. Consider, for instance, the space R∞, whose elements are those sequences of real numbers that contain only finitely many non-zero entries, which has a basis indexed by the natural numbers N: for i ∈ N, ei is the sequence consisting of all zeroes except in the ith position, which is 1. The dual space of R∞ is RN, the space of all sequences of real numbers: such a sequence (an) is applied to an element (xn) of R∞ to give the number ∑anxn, which is a finite sum because there are only finitely many nonzero xn. The dimension of R∞ is countably infinite, whereas RN does not have a countable basis. This observation generalizes to any[5] infinite-dimensional vector space V over any field F: a choice of basis {eα : α A ∈ A} identifies V with the space (F )0 of functions f : A → F such that fα = f(α) is nonzero for only finitely many α ∈ A, where such a function f is identified with the vector

∑ fαeα α∈A in V (the sum is finite by the assumption on f, and any v ∈ V may be written in this way by the definition of the basis). The dual space of V may then be identified with the space FA of all functions from A to F: a linear functional T on V is uniquely determined by the values θα = T(eα) it takes on the basis of V, and any function θ : A → F (with θ(α) = θα) defines a linear functional T on V by

( ∑ ) ∑ ∑ T fαeα = fαT (eα) = fαθα. α∈A α∈A α∈A

Again the sum is finite because fα is nonzero for only finitely many α. A Note that (F )0 may be identified (essentially by definition) with the direct sum of infinitely many copies of F (viewed as a 1-dimensional vector space over itself) indexed by A, i.e., there are linear isomorphisms

⊕ ∼ A ∼ V = (F )0 = F. α∈A

On the other hand FA is (again by definition), the direct product of infinitely many copies of F indexed by A, and so the identification 64 CHAPTER 15. DUAL SPACE

( ⊕ )∗ ∏ ∏ ∗ ∼ ∼ ∗ ∼ ∼ A V = F = F = F = F α∈A α∈A α∈A is a special case of a general result relating direct sums (of modules) to direct products. Thus if the basis is inﬁnite, then the algebraic dual space is always of larger dimension (as a cardinal number) than the original vector space. This is in contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the original vector space even if the latter is inﬁnite-dimensional.

15.1.3 Bilinear products and dual spaces

If V is ﬁnite-dimensional, then V is isomorphic to V∗. But there is in general no natural isomorphism between these two spaces.[6] Any bilinear form ⟨·,·⟩ on V gives a mapping of V into its dual space via

v 7→ ⟨v, ·⟩

where the right hand side is deﬁned as the functional on V taking each w ∈ V to ⟨v,w⟩. In other words, the bilinear form determines a linear mapping

∗ Φ⟨·,·⟩ : V → V deﬁned by

[Φ⟨·,·⟩(v), w] = ⟨v, w⟩. If the bilinear form is nondegenerate, then this is an isomorphism onto a subspace of V∗. If V is ﬁnite-dimensional, then this is an isomorphism onto all of V∗. Conversely, any isomorphism Φ from V to a subspace of V∗ (resp., all of V∗) deﬁnes a unique nondegenerate bilinear form ⟨·,·⟩Φ on V by

⟨v, w⟩Φ = (Φ(v))(w) = [Φ(v), w]. Thus there is a one-to-one correspondence between isomorphisms of V to subspaces of (resp., all of) V∗ and nondegenerate bilinear forms on V. If the vector space V is over the complex ﬁeld, then sometimes it is more natural to consider sesquilinear forms instead of bilinear forms. In that case, a given sesquilinear form ⟨·,·⟩ determines an isomorphism of V with the complex conjugate of the dual space

∗ Φ⟨·,·⟩ : V → V . The conjugate space V∗ can be identiﬁed with the set of all additive complex-valued functionals f : V → C such that

f(αv) = αf(v).

15.1.4 Injection into the double-dual

There is a natural homomorphism Ψ from V into the double dual V∗∗, defined by (Ψ(v))(φ) = φ(v) for all v ∈ V, φ ∈ V∗. This map Ψ is always injective;[5] it is an isomorphism if and only if V is finite-dimensional. Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of a natural isomorphism. Note that infinite-dimensional Hilbert spaces are not a counterexample to this, as they are isomorphic to their continuous duals, not to their algebraic duals. 15.1. ALGEBRAIC DUAL SPACE 65

15.1.5 Transpose of a linear map

Main article: Transpose of a linear map

If f : V → W is a linear map, then the transpose (or dual) f ∗ : W∗ → V∗ is deﬁned by f ∗(φ) = φ ◦ f for every φ ∈ W∗. The resulting functional f ∗(φ) in V∗ is called the pullback of φ along f. The following identity holds for all φ ∈ W∗ and v ∈ V:

[f ∗(φ), v] = [φ, f(v)],

where the bracket [·,·] on the left is the duality pairing of V with its dual space, and that on the right is the duality pairing of W with its dual. This identity characterizes the transpose,[7] and is formally similar to the definition of the adjoint. The assignment f ↦ f ∗ produces an injective linear map between the space of linear operators from V to W and the space of linear operators from W∗ to V∗; this homomorphism is an isomorphism if and only if W is finite-dimensional. If V = W then the space of linear maps is actually an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that (fg)∗ = g∗f ∗. In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself. Note that one can identify (f ∗)∗ with f using the natural injection into the double dual. If the linear map f is represented by the matrix A with respect to two bases of V and W, then f ∗ is represented by the transpose matrix AT with respect to the dual bases of W∗ and V∗, hence the name. Alternatively, as f is represented by A acting on the left on column vectors, f ∗ is represented by the same matrix acting on the right on row vectors. These points of view are related by the canonical inner product on Rn, which identifies the space of column vectors with the dual space of row vectors.

15.1.6 Quotient spaces and annihilators

Let S be a subset of V. The annihilator of S in V∗, denoted here So, is the collection of linear functionals f ∈ V∗ such that [f, s] = 0 for all s ∈ S. That is, So consists of all linear functionals f : V → F such that the restriction to S vanishes: f|S = 0. The annihilator of a subset is itself a vector space. In particular, ∅o = V∗ is all of V∗ (vacuously), whereas Vo = 0 is the zero subspace. Furthermore, the assignment of an annihilator to a subset of V reverses inclusions, so that if S ⊂ T ⊂ V, then

0 ⊂ T o ⊂ So ⊂ V ∗.

Moreover, if A and B are two subsets of V, then

(A ∩ B)o ⊇ Ao + Bo,

and equality holds provided V is ﬁnite-dimensional. If Ai is any family of subsets of V indexed by i belonging to some index set I, then

( ) ∪ o ∩ o Ai = Ai . i∈I i∈I In particular if A and B are subspaces of V, it follows that 66 CHAPTER 15. DUAL SPACE

(A + B)o = Ao ∩ Bo.

If V is ﬁnite-dimensional, and W is a vector subspace, then

W oo = W

after identifying W with its image in the second dual space under the double duality isomorphism V ≈ V∗∗. Thus, in particular, forming the annihilator is a Galois connection on the lattice of subsets of a ﬁnite-dimensional vector space. If W is a subspace of V then the quotient space V/W is a vector space in its own right, and so has a dual. By the ﬁrst isomorphism theorem, a functional f : V → F factors through V/W if and only if W is in the kernel of f. There is thus an isomorphism

∗ ∼ o (V /W ) = W .

As a particular consequence, if V is a direct sum of two subspaces A and B, then V∗ is a direct sum of Ao and Bo.

15.2 Continuous dual space

When dealing with topological vector spaces, one is typically only interested in the continuous linear functionals from the space into the base field F = C (or R ). This gives rise to the notion of the “continuous dual space” or “topological dual” which is a linear subspace of the algebraic dual space V ∗ , denoted by V ′ . For any finite- dimensional normed vector space or topological vector space, such as Euclidean n-space, the continuous dual and the algebraic dual coincide. This is however false for any infinite-dimensional normed space, as shown by the example of discontinuous linear maps. Nevertheless in the theory of topological vector spaces the terms “continuous dual space” and “topological dual space” are rarely used, as a rule they are replaced by “dual space”, since there is no serious need to consider discontinuous maps in this field. For a topological vector space V its continuous dual space,[8] or topological dual space,[9] or just dual space[8][9][10][11] (in the sense of the theory of topological vector spaces) V ′ is defined as the space of all continuous linear functionals φ : V → F . There is a standard construction for introducing topology on the continuous dual V ′ of a topological vector space V : each given class A of bounded subsets in V defines a topology on V of uniform convergence on sets from A , or what is the same a topology generated by seminorms of the form

∥φ∥A = sup |φ(x)|, x∈A where φ is a continuous linear functional on V , and A runs over the class A . ′ This means that a net of functionals φi tends to a functional φ in V if and only if

∀A ∈ A ∥φ − φ∥ = sup |φ (x) − φ(x)| −→ 0. i A i →∞ x∈A i Usually (but not necessarily) the class A is supposed to satisfy the following conditions:

• each point x of V belongs to some set A ∈ A

∀x ∈ V ∃A ∈ A x ∈ A,

• each two sets A ∈ A and B ∈ A are contained in some set C ∈ A : 15.2. CONTINUOUS DUAL SPACE 67

∀A, B ∈ A ∃C ∈ A A ∪ B ⊆ C,

• A is closed under the operation of multiplication by scalars:

∀A ∈ A ∀λ ∈ F λ · A ∈ A, If these requirements are fulﬁlled then the corresponding topology on V ′ is Hausdorﬀ and the sets

′ UA = {φ ∈ V : ||φ||A < 1},A ∈ A, form its local base. Here are the three most important special cases.

• The strong topology on V ′ is the topology of uniform convergence on bounded subsets in V (so here A can be chosen as the class of all bounded subsets in V ). If V is a normed vector space (e.g., a Banach space or a Hilbert space) then the strong topology on V ′ is normed (in fact a Banach space if the ﬁeld of scalars is complete), with the norm

∥φ∥ = sup |φ(x)|. ∥x∥≤1

• The stereotype topology on V ′ is the topology of uniform convergence on totally bounded sets in V (so here A can be chosen as the class of all totally bounded subsets in V ).

• The weak topology on V ′ is the topology of uniform convergence on ﬁnite subsets in V (so here A can be chosen as the class of all ﬁnite subsets in V ).

Each of these three choices of topology on V ′ leads to a variant of reﬂexivity property for topological vector spaces.

15.2.1 Examples

Let 1 < p < ∞ be a real number and consider the Banach space ℓ p of all sequences a = (an) for which

( ) ∑∞ 1/p p ∥a∥p = |an| n=0 is finite. Define the number q by 1/p + 1/q = 1. Then the continuous dual of ℓ p is naturally identified with ℓ q: given an element φ ∈ (ℓ p)′, the corresponding element of ℓ q is the sequence (φ(en)) where en denotes the sequence whose n-th term is 1 and all others are zero. Conversely, given an element a = (an) ∈ ℓ q, the corresponding continuous linear functional φ on ℓ p is defined by φ(b) = ∑n anbn for all b = (bn) ∈ ℓ p (see Hölder’s inequality). In a similar manner, the continuous dual of ℓ 1 is naturally identified with ℓ ∞ (the space of bounded sequences). Furthermore, the continuous duals of the Banach spaces c (consisting of all convergent sequences, with the supremum 1 norm) and c0 (the sequences converging to zero) are both naturally identified with ℓ . By the Riesz representation theorem, the continuous dual of a Hilbert space is again a Hilbert space which is anti- isomorphic to the original space. This gives rise to the bra–ket notation used by physicists in the mathematical formulation of quantum mechanics.

15.2.2 Transpose of a continuous linear map

If T : V → W is a continuous linear map between two topological vector spaces, then the (continuous) transpose T′ : W′ → V′ is deﬁned by the same formula as before:

T ′(φ) = φ ◦ T, φ ∈ W ′. 68 CHAPTER 15. DUAL SPACE

The resulting functional T′(φ) is in V′. The assignment T → T′ produces a linear map between the space of continuous linear maps from V to W and the space of linear maps from W′ to V′. When T and U are composable continuous linear maps, then

(U ◦ T )′ = T ′ ◦ U ′.

When V and W are normed spaces, the norm of the transpose in L(W′, V′) is equal to that of T in L(V, W). Several properties of transposition depend upon the Hahn–Banach theorem. For example, the bounded linear map T has dense range if and only if the transpose T′ is injective. When T is a compact linear map between two Banach spaces V and W, then the transpose T′ is compact. This can be proved using the Arzelà–Ascoli theorem. When V is a Hilbert space, there is an antilinear isomorphism iV from V onto its continuous dual V′. For every bounded linear map T on V, the transpose and the adjoint operators are linked by

∗ ′ iV ◦ T = T ◦ iV .

When T is a continuous linear map between two topological vector spaces V and W, then the transpose T′ is continuous when W′ and V′ are equipped with"compatible” topologies: for example when, for X = V and X = W, both duals X′ have the strong topology β(X′, X) of uniform convergence on bounded sets of X, or both have the weak-∗ topology σ(X′, X) of pointwise convergence on X. The transpose T′ is continuous from β(W′, W) to β(V′, V), or from σ(W′, W) to σ(V′, V).

15.2.3 Annihilators

Assume that W is a closed linear subspace of a normed space V, and consider the annihilator of W in V′,

W ⊥ = {φ ∈ V ′ : W ⊂ ker φ}.

Then, the dual of the quotient V / W can be identiﬁed with W⊥, and the dual of W can be identiﬁed with the quotient V′ / W⊥.[12] Indeed, let P denote the canonical surjection from V onto the quotient V / W ; then, the transpose P′ is an isometric isomorphism from (V / W )′ into V′, with range equal to W⊥. If j denotes the injection map from W into V, then the kernel of the transpose j′ is the annihilator of W:

ker(j′) = W ⊥

and it follows from the Hahn–Banach theorem that j′ induces an isometric isomorphism V′ / W⊥ → W′.

15.2.4 Further properties

If the dual of a normed space V is separable, then so is the space V itself. The converse is not true: for example the space ℓ 1 is separable, but its dual ℓ ∞ is not.

15.2.5 Topologies on the dual

The topology of V and the topology of real or complex numbers can be used to induce on V′ a dual space topology.

15.2.6 Double dual

In analogy with the case of the algebraic double dual, there is always a naturally deﬁned continuous linear operator Ψ: V → V′′ from a normed space V into its continuous double dual V′′, deﬁned by 15.3. SEE ALSO 69

This is a natural transformation of vector addition from a vector space to its double dual. denotes the ordered pair of two vectors. The addition + sends x1 and x2 to x1 + x2. The addition +' induced by the transformation can be deﬁned as (Ψ(x1) +' Ψ(x2))(φ) = φ(x1 + x2) = φ(x) for any φ in the dual space.

Ψ(x)(φ) = φ(x), x ∈ V, φ ∈ V ′.

As a consequence of the Hahn–Banach theorem, this map is in fact an isometry, meaning ||Ψ(x)|| = ||x|| for all x in V. Normed spaces for which the map Ψ is a bijection are called reflexive. When V is a topological vector space, one can still define Ψ(x) by the same formula, for every x ∈ V, however several difficulties arise. First, when V is not locally convex, the continuous dual may be equal to {0} and the map Ψ trivial. However, if V is Hausdorff and locally convex, the map Ψ is injective from V to the algebraic dual V′∗ of the continuous dual, again as a consequence of the Hahn–Banach theorem.[13] Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual V′, so that the continuous double dual V′′ is not uniquely defined as a set. Saying that Ψ maps from V to V′′, or in other words, that Ψ(x) is continuous on V′ for every x ∈ V, is a reasonable minimal requirement on the topology of V′, namely that the evaluation mappings

φ ∈ V ′ 7→ φ(x), x ∈ V, be continuous for the chosen topology on V′. Further, there is still a choice of a topology on V′′, and continuity of Ψ depends upon this choice. As a consequence, deﬁning reﬂexivity in this framework is more involved than in the normed case.

15.3 See also

• Duality (mathematics)

• Duality (projective geometry)

• Pontryagin duality

• Reciprocal lattice – dual space basis, in crystallography

• Covariance and contravariance of vectors

• Dual norm 70 CHAPTER 15. DUAL SPACE

15.4 Notes

[1] Halmos (1974)

[2] Misner, Thorne & Wheeler (1973)

[3] In many areas, such as quantum mechanics, ⟨·,·⟩ is reserved for a sesquilinear form deﬁned on V × V.

[4] Misner, Thorne & Wheeler (1973, §2.5)

[5] Several assertions in this article require the axiom of choice for their justiﬁcation. The axiom of choice is needed to show that an arbitrary vector space has a basis: in particular it is needed to show that RN has a basis. It is also needed to show that the dual of an inﬁnite-dimensional vector space V is nonzero, and hence that the natural map from V to its double dual is injective.

[6] MacLane & Birkhoﬀ (1999, §VI.4)

[7] Halmos (1974, §44)

[8] A.P.Robertson, W.Robertson (1964, II.2)

[9] H.Schaefer (1966, II.4)

[10] W.Rudin (1973, 3.1)

[11] Nicolas Bourbaki (2003, II.42)

[12] Rudin (1991, chapter 4)

[13] If V is locally convex but not Hausdorﬀ, the kernel of Ψ is the smallest closed subspace containing {0}.

15.5 References

• Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9 • Bourbaki, Nicolas. (2003), Elements of mathematics, Topological vector spaces, Springer-Verlag

• Halmos, Paul (1974), Finite-dimensional Vector Spaces, Springer, ISBN 0-387-90093-4

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer- Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556

• MacLane, Saunders; Birkhoﬀ, Garrett (1999), Algebra (3rd ed.), AMS Chelsea Publishing, ISBN 0-8218- 1646-2.

• Misner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0-7167- 0344-0

• Rudin, Walter (1991). Functional analysis. McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236- 5.

• Robertson, A.P.; Robertson, W. (1964). Topological vector spaces. Cambridge University Press. • Schaefer, Helmuth H. (1971). Topological vector spaces. GTM 3. New York: Springer-Verlag. ISBN 0-387- 98726-6. 15.6. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 71

15.6 Text and image sources, contributors, and licenses

15.6.1 Text • Defective matrix Source: http://en.wikipedia.org/wiki/Defective_matrix?oldid=664022077 Contributors: Stevenj, Giftlite, Chnv, Cat- slash, Lantonov, Quantling, Partha lal, Plastikspork, Addbot, Luckas-bot, AManWithNoPlan, Anita5192, Helpful Pixie Bot and Anony- mous: 4 • Definite quadratic form Source: http://en.wikipedia.org/wiki/Definite_quadratic_form?oldid=652861601 Contributors: Jorge Stolfi, Malcolma, Crasshopper, Livingthingdan, Fabrictramp, David Eppstein, PixelBot, Jotterbot, Addbot, AnomieBOT, HRoestBot, Quondum, D.Lazard, KLBot2, Mark Arsten, Qetuth, Deltahedron and Sami.mkaddem • Delta operator Source: http://en.wikipedia.org/wiki/Delta_operator?oldid=607145000 Contributors: AxelBoldt, Tarquin, Patrick, Michael Hardy, Charles Matthews, Robbot, Oleg Alexandrov, Linas, Jeargle, AlainD, Jakob.scholbach, Daniele.tampieri, Tbone762, Marc van Leeuwen, Addbot, Luckas-bot, Yobot, Unara, SaschaWolff and Anonymous: 5 • Dependence relation Source: http://en.wikipedia.org/wiki/Dependence_relation?oldid=487280987 Contributors: Charles Matthews, Jitse Niesen, Josh Parris, Wavelength, Robbjedi, Keegan, Geometry guy and 777sms • Determinant Source: http://en.wikipedia.org/wiki/Determinant?oldid=666841511 Contributors: AxelBoldt, Bryan Derksen, Tarquin, Taw, Patrick, Michael Hardy, Wshun, Delirium, Ahoerstemeier, Stevenj, Snoyes, TheEternalVortex, AugPi, Andres, Wael Ellithy, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, Wik, Itai, McKay, Kwantus, Rogper~enwiki, Robbot, Fredrik, Benwing, Big Jim Fae Scot- land, MathMartin, Robinh, Tobias Bergemann, Tosha, Connelly, Giftlite, Christopher Parham, Arved, Gene Ward Smith, Recentchanges, BenFrantzDale, Tegla, Lethe, Fropuff, Bkonrad, TomViza, Alison, Ssd, Rpyle731, Quadell, Anythingyouwant, Icairns, Rpchase, Jew- bacca, Asqueella, Slady, Rich Farmbrough, TedPavlic, Ardonik, ArnoldReinhold, MuDavid, Paul August, Bender235, Zaslav, Gauge, Pt, Rgdboer, Nickj, Gershwinrb, Spoon!, Reinyday, Guiltyspark, Tgr, Obradovic Goran, Nsaa, Jumbuck, Wirawan0, Msa11usec, Nachiket- vartak, Burn, Danaman5, HenkvD, Jheald, Smithereens~enwiki, Oleg Alexandrov, Alkarex, Catfive, Joriki, N3vln, Shreevatsa, LOL, StradivariusTV, Siener, ^demon, Netdragon, Graham87, Zzedar, Rjwilmsi, Ae77, R.e.b., TheIncredibleEdibleOompaLoompa, Maro- zols, FlaBot, Anskas, Mathbot, RexNL, Chobot, Jersey Devil, Ztutz~enwiki, Gwernol, Algebraist, YurikBot, Wavelength, Personman, RussBot, Michael Slone, Josteinaj, Ino5hiro, Goffrie, Sangwine, NeilenMarais, Misza13, Crasshopper, Zwobot, Nethgirb, Entropeneur, Bota47, Jemebius, Woscafrench, Lt-wiki-bot, Cbogart2, Closedmouth, Jogers, Netrapt, Nekura, Vanka5, SmackBot, RDBury, Jagged 85, Alksub, Pedrose, Spireguy, Swerdnaneb, Shai-kun, Betacommand, Kurykh, Ian13, Oli Filth, Carbonrodney, Octahedron80, DHN- bot~enwiki, A. B., Tekhnofiend, Lhf, Trifon Triantafillidis, Adamp, Sayahoy, Kunal Bhalla, Merge, Stefano85, Balagen, Lambiam, J. Finkelstein, Dark Formal, Nat2, Cronholm144, Jim.belk, Kaarebrandt, Mets501, Cowanae, RekishiEJ, Alexandre Martins, AbsolutDan, Lavaka, Cthulhu.mythos, SkyWalker, Mrsaad31, CRGreathouse, Jackzhp, CBM, Mcstrother, Anakata, HenningThielemann, Leakeyjee, Doctormatt, Juansempere, Jrgetsin, Edwardrf, Thijs!bot, Epbr123, Headbomb, Marek69, Dmbrown00, Urdutext, Rfgeorge, AntiVandal- Bot, Salgueiro~enwiki, Rbb l181, Asmeurer, Thenub314, 01001, A-asadi, Tarif Ezaz, Coffee2theorems, Kokin, Jakob.scholbach, Sod- abottle, Pigei, Trioculite, JJ Harrison, Stdazi, Adam4445, Pensador82, Jondaman21, Heili.brenna, Cesarth~enwiki, Ulisse0, Numbo3, Rocchini, Jerry, Mikael Häggström, BrianOfRugby, Quantling, Cobi, Fylwind, Jasonevans, Cuzkatzimhut, VolkovBot, Camrn86, Jeff G., Hlevkin, Rei-bot, Anonymous Dissident, Josp-mathilde, MackSalmon, Elphion, MartinOtter, Kmhkmh, SwordSmurf, Wolfrock, Cam- puzano85, PhysPhD, Dmcq, HiDrNick, Demize, Priitliivak, Gerakibot, Icktoofay, Cwkmail, Rdmabry, Patamia, Paolo.dL, Lightmouse, JackSchmidt, AlanUS, Trashbird1240, Gejikeiji~enwiki, DonAByrd, ClueBot, Justin W Smith, The Thing That Should Not Be, Gre- walWiki, Barking Mad142, Mild Bill Hiccup, Timberframe, Sabri76, Bender2k14, EtudiantEco, Sun Creator, Kaspar.jan, EverettYou, Alexey Muranov, SchreiberBike, Thehelpfulone, Jshen6, Mcconnell3, Johnuniq, XLinkBot, Marc van Leeuwen, Nicolae Coman, San- dro.bosio, Truthnlove, Addbot, Roentgenium111, BjornPoonen, Samiswicked, Protonk, LaaknorBot, EconoPhysicist, SpBot, LinkFA- Bot, Mdnahas, Legobot, MariaDroujkova, Luckas-bot, Quadrescence, Yobot, RIBEYE special, Vanished user rt41as76lk, Mmxx, TestE- ditBot, VanishedUser sdu9aya9fasdsopa, Ipatrol, Kingpin13, Crystal whacker, Ffatossaajvazii, Materialscientist, LilHelpa, Xqbot, Dan- testyrael, J4lambert, Rayray28, Kstueve, Joejc, Logapragasan, Executive Outcomes, T8191, Spartan S58, Correnos, Lagelspeil, Rum- blethunder, Sławomir Biały, Greynose, Citation bot 1, Kd345205, SUL, AstaBOTh15, DrilBot, HRoestBot, LAncienne, Alexander Chervov, MastiBot, CarlBac, BPets, Istcol, Simon Sang, Kallikanzarid, FoxBot, Mobiusthefrost, Jordgette, Alex Sabaka, Datahaki, Cutelyaware, Unbitwise, Ijpulido, Nistra, EmausBot, Vincent Semeria, Wisapi, Bencbartlett, Alexandre Duret-Lutz, Hangitfresh, Ccan- dan, Luiscardona89, Ebrambot, Quondum, Eniagrom, Chocochipmuffin, Tolly4bolly, Gabriel10yf, Maschen, Donner60, Chewings72, JFB80, Euphrat1508, ClueBot NG, Wcherowi, Michael P. Barnett, Trivialsegfault, Chester Markel, Movses-bot, Snotbot, TreyGreer62, Khabgood, Helpful Pixie Bot, Supreme fascist, Cjkstephenson, Vibhor1997, Myshka spasayet lva, Cmmthf, René Vápeník, Thiagowfx, Dexbot, Deltahedron, Mkhan3189, Sebhofer, Joeinwiki, Trompedo, Colombomattia89, E290341, Babitaarora, Westwood0.137, Improb- able keeler, Hkmcszz, Abitslow, Leegrc, GeoffreyT2000, KH-1, Loraof, Premshah95, Sunnsambhi, Rywais, Ajbaumann and Anonymous: 387 • Dieudonné determinant Source: http://en.wikipedia.org/wiki/Dieudonn%C3%A9_determinant?oldid=644289855 Contributors: Takuya- Yobot, RjwilmsiBot, Quondum, Spectral ,דוד שי ,Murata, Charles Matthews, Giftlite, Rjwilmsi, R.e.b., Masnevets, Headbomb, Addbot sequence and Anonymous: 2 • Dimension (vector space) Source: http://en.wikipedia.org/wiki/Dimension_(vector_space)?oldid=651516115 Contributors: AxelBoldt, Zundark, Tarquin, Michael Hardy, Wshun, TakuyaMurata, AugPi, Charles Matthews, Gandalf61, MathMartin, Giftlite, Lethe, Bob.v.R, Krakhan, Army1987, Obradovic Goran, Jumbuck, Oleg Alexandrov, Mathbot, YurikBot, Tetracube, JahJah, Banus, SmackBot, Chris the speller, Bluebot, Oli Filth, Pierrecurie, Nbarth, Rludlow, Dreadstar, Tilin, SashatoBot, Jim.belk, Jbolden1517, Kanags, Epbr123, Faigl.ladislav, JAnDbot, Quentar~enwiki, David Eppstein, Geometry guy, Jmath666, Neparis, Rybu, Ivan Štambuk, Niceguyedc, Marc van Leeuwen, MystBot, Addbot, Kein Einstein, Legobot, Luckas-bot, Yobot, Star Flyer, Xqbot, Flavio Guitian, Edderso, EmausBot, Quondum, Zynwyx, Mesoderm, Helpful Pixie Bot, AvocatoBot, Grv87, Impsswoon and Anonymous: 23 • Direct sum of modules Source: http://en.wikipedia.org/wiki/Direct_sum_of_modules?oldid=636440060 Contributors: AxelBoldt, Zun- dark, Toby Bartels, Patrick, Chas zzz brown, Michael Hardy, Chinju, TakuyaMurata, AugPi, Ideyal, Revolver, Charles Matthews, Jitse Niesen, Robbot, Jaredwf, Tobias Bergemann, Giftlite, Lethe, Waltpohl, Daniel Brockman, Elwikipedista~enwiki, El C, Rgdboer, Mairi, C S, Domster, Oleg Alexandrov, Linas, Tertiary7, Salix alba, Mathbot, Chobot, Algebraist, YurikBot, KSmrq, Hakeem.gadi, Zvika, Smack- Bot, Silly rabbit, Nbarth, Dreadstar, Loodog, Mgiganteus1, Colonel Warden, James pic, ShelfSkewed, Myasuda, Mct mht, Thijs!bot, RobHar, Vanish2, Sullivan.t.j, Robin S, Alexwright, LokiClock, Rei-bot, Geometry guy, AlleborgoBot, MikeRumex, DesolateReality, Functor salad, He7d3r, Sun Creator, Leen Droogendijk, Addbot, Pilotkyber, Fyrael, Topology Expert, Tassedethe, Luckas-bot, Yobot, AnomieBOT, Sławomir Biały, Citation bot 1, MondalorBot, Set theorist, Quondum, Helpful Pixie Bot, Deltahedron and Anonymous: 27 72 CHAPTER 15. DUAL SPACE

• Direction vector Source: http://en.wikipedia.org/wiki/Direction_vector?oldid=627370231 Contributors: BenFrantzDale, D6, Gary, Oleg Alexandrov, Sbyrnes321, Dicklyon, Bruno321, Sherbrooke, JohnBlackburne, Neparis, Paolo.dL, Kiensvay, Addbot, Luckas-bot, Xqbot, J04n, Pbierre and Anonymous: 4 • Dot product Source: http://en.wikipedia.org/wiki/Dot_product?oldid=662544868 Contributors: Dan~enwiki, Tarquin, Mrwojo, Bde- sham, Patrick, Michael Hardy, Tim Starling, DIG~enwiki, SGBailey, Gbleem, SebastianHelm, Charles Matthews, Timwi, Dysprosia, Omegatron, Kk~enwiki, Robbot, Mazin07, Fredrik, Romanm, Naddy, Gandalf61, MathMartin, Wereon, Tosha, Giftlite, BenFrantzDale, MSGJ, Msm, Wwoods, Daniel Brockman, Matt Crypto, Tristanreid, Sreyan, Lockeownzj00, MarkSweep, Icairns, Frau Hitt~enwiki, Daviis~enwiki, Rich Farmbrough, Andros 1337, Dfan, Kwamikagami, Danshil, O18, Slicky, NanoTy, Kundor, Landroni, ArbiterOne, Davor~enwiki, Tony Sidaway, Oleg Alexandrov, LOL, StradivariusTV, GregorB, Isnow, Pfalstad, Qwertyus, NeoUrfahraner, JVz, Maro- zols, Yamamoto Ichiro, Windchaser, Fresheneesz, Kri, Chobot, Jersey Devil, Zath42, Wavelength, Hairy Dude, Hillman, Michael Slone, MrHumperdink, Markus Schmaus, KSmrq, Archelon, Vanished user 1029384756, Akulaalfa, Sangwine, Voidxor, Samuel Huang, Ospalh, Amadib, Tlevine, Heptazane, Naught101, HereToHelp, Ghazer~enwiki, Selfworm, InverseHypercube, Melchoir, Alksub, Mcherm, Mcld, GBL, Bluebot, Octahedron80, Audriusa, Cybercobra, Savidan, Hgilbert, Flip619, WhiteHatLurker, Luokehao, Hypnosifl, Dr.K., Emote, Harold f, CBM, Prlsmith, Ezrakilty, Arauzo, Shirulashem, Epbr123, Ryansca, Karlmeerbergen, LRG, Kborer, NocNokNeo, Icep, An- tiVandalBot, JAnDbot, Quentar~enwiki, Sanchom, Matt Kovacs, PhilKnight, Crunchy Numbers, M samadi, Redbaron486, Policron, Fylwind, VolkovBot, Pleasantville, JohnBlackburne, LokiClock, Slaunger, Anonymous Dissident, Victimofleisure, Howeworth, PaulTa- nenbaum, Smoo222, Neparis, SieBot, LeadSongDog, Interchange88, Paolo.dL, Oxymoron83, Rightisright4250, ClueBot, Justin W Smith, R000t, Niceguyedc, LizardJr8, Ecov, GoldenGoose100, Bender2k14, Muhandes, J.Gowers, Humanengr, XLinkBot, Marc van Leeuwen, Kal-El-Bot, Addbot, Narayansg, Econotechie, MrVanBot, AndersBot, Smartyepp, Zorrobot, TeH nOmInAtOr, Jarble, Luckas-bot, Yobot, Fflanner, Ehaussecker, AnomieBOT, Götz, Tucoxn, Ahmedo15, Raffamaiden, Aptronotus, WaysToEscape, Crispin.cheddarcock, Jose- manimala, Sławomir Biały, Khashishi, Elockid, Rosenlind, Chezz444, Nguoiviet979, Rausch, Duoduoduo, Harleymeyer, Limited Atone- ment, Jfmantis, TjBot, Mortees~enwiki, VACowboy41, EmausBot, Kpufferfish, Dcirovic, Slawekb, ZéroBot, Quondum, D.Lazard, AManWithNoPlan, Maschen, ChuispastonBot, DKPrestige, Locador, ClueBot NG, Gareth Griffith-Jones, ThomasMcLeod, AvocatoBot, Dmawxc, Matthew.p.korte, PsiEpsilon, Jdogzz, , ChrisGualtieri, Khazar2, Gundamlh, JoghurtGF, Ecce simia, Coobit, Couryt, 27394Days, Sci40002, Dlambert2499, Aqsa Idrees and Anonymous: 229 • Dual basis Source: http://en.wikipedia.org/wiki/Dual_basis?oldid=659506909 Contributors: Patrick, Charles Matthews, Giftlite, Sha- hab, Aschiller, Army1987, Ent, Woohookitty, Swamp Ig, Kri, Wavelength, Frigoris, SmackBot, Silly rabbit, Nbarth, Mhym, Rschwieb, Jbolden1517, Thermochap, TFighterPilot, Scholarus, VolkovBot, Wikiwide, LokiClock, TXiKiBoT, Drorata, AlleborgoBot, Drew- ster1829, Jshen6, Addbot, Luckas-bot, Yobot, Calle, AnomieBOT, Erik9bot, BenzolBot, Quondum, Highestbottoms, J824h, Martin Mok, , MeTend, Alakzi and Anonymous: 9 • Dual basis in a field extension Source: http://en.wikipedia.org/wiki/Dual_basis_in_a_field_extension?oldid=644686663 Contributors: Charles Matthews, Gene Ward Smith, CryptoDerk, ArnoldReinhold, SmackBot, Ideal gas equation, Niceguyedc, Erik9bot, Brad7777, LimeyCinema1960 and Anonymous: 1 • Dual norm Source: http://en.wikipedia.org/wiki/Dual_norm?oldid=665008541 Contributors: Toby Bartels, Michael Hardy, Nandhp, SmackBot, Stifle, Ensign beedrill, Grubb257, Furankid, Quondum, U+003F, Eozhik and Anonymous: 6 • Dual number Source: http://en.wikipedia.org/wiki/Dual_number?oldid=649634397 Contributors: AxelBoldt, Zundark, Tarquin, Karl E. V. Palmen, Benwbrum, XJaM, PierreAbbat, Peterlin~enwiki, Michael Hardy, Dominus, Gabbe, Barak~enwiki, Charles Matthews, Tosha, Giftlite, BenFrantzDale, Macrakis, Vivacissamamente, Mormegil, Rgdboer, Keenan Pepper, Oleg Alexandrov, Linas, Jeff02, R160K, KSmrq, Yahya Abdal-Aziz, Reyk, Paul White, Gesslein, Jaysbro, SmackBot, InverseHypercube, BiT, Bluebot, Tsca.bot, Tesseran, Rschwieb, AlainD, Thijs!bot, Tercer, Iida-yosiaki, Krtg, TXiKiBoT, VVVBot, Strilanc, DFRussia, V.V.Kisil, Luca Antonelli, Addbot, LaaknorBot, PV=nRT, Luckas-bot, Yobot, Ptbotgourou, Nallimbot, Slow Phil, Chatul, MathFacts, Tkuvho, Dude1818, Hexadecachoron, Diannaa, EmausBot, ZéroBot, Quondum, JmountZedZed, , Citizentoad, Anixx1, CsDix, Cyrapas, Tentinator and Anonymous: 25 • Dual space Source: http://en.wikipedia.org/wiki/Dual_space?oldid=665420273 Contributors: Damian Yerrick, AxelBoldt, Zundark, The Anome, Fcueto, Michael Hardy, Snoyes, AugPi, Ideyal, Barak~enwiki, Loren Rosen, Charles Matthews, Dysprosia, Jitse Niesen, Jens- Mueller, Mattblack82, Wikibot, Tobias Bergemann, Giftlite, BenFrantzDale, Lethe, Fropuff, Wwoods, Dratman, Vadmium, Sreyan, ChickenMerengo, Sam nead, Frau Hitt~enwiki, TheObtuseAngleOfDoom, Guanabot, Sligocki, Jheald, Trieu, Oleg Alexandrov, Joriki, Arneth, LOL, Juan Marquez, Small potato, Chobot, Algebraist, Michael Slone, KSmrq, Archelon, Gaius Cornelius, Crasshopper, Nfm, Marc Harper, Evilbu, GrinBot~enwiki, KnightRider~enwiki, SmackBot, KYN, Optikos, Kurykh, Silly rabbit, Pierrecurie, Nbarth, Colonies Chris, Texas Dervish, UncleDouggie, Jbolden1517, JRSpriggs, Anakata, Thijs!bot, RobHar, LachlanA, Smartcat, Magioladitis, Albmont, David Eppstein, Lantonov, Daniel5Ko, JohnBlackburne, LokiClock, TXiKiBoT, Cgwaldman, Torsten Nahm~enwiki, Gagelman, Ge- ometry guy, Morestuff, Billy2007, Alexander.fairley, Paolo.dL, Ddxc, Anchor Link Bot, Hadas583, Yasmar, Pointillist blur, Addbot, Download, CarsracBot, TeH nOmInAtOr, Legobot, Luckas-bot, Yobot, Ht686rg90, ArthurBot, LilHelpa, Bdmy, Anschmid13, TechBot, Omnipaedista, RibotBOT, Kaoru Itou, Lisp21, Sławomir Biały, HRoestBot, Stpasha, Filip Albert, Alph Bot, Slawekb, ZéroBot, Sgoder, Quondum, SporkBot, Zfeinst, Zueignung, ChuispastonBot, Anita5192, Helpful Pixie Bot, BG19bot, Papadim.G, IkamusumeFan, Stausifr, ChrisGualtieri, CarrieVS, Eozhik, Katterjohn, Abitslow, Austrartsua, Mgkrupa and Anonymous: 96

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