Linear Independence, Basis, and Dimensions Making the Abstraction Concrete
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10 7 General Vector Spaces Proposition & Definition 7.14
10 7 General vector spaces Proposition & Definition 7.14. Monomial basis of n(R) P Letn N 0. The particular polynomialsm 0,m 1,...,m n n(R) defined by ∈ ∈P n 1 n m0(x) = 1,m 1(x) = x, . ,m n 1(x) =x − ,m n(x) =x for allx R − ∈ are called monomials. The family =(m 0,m 1,...,m n) forms a basis of n(R) B P and is called the monomial basis. Hence dim( n(R)) =n+1. P Corollary 7.15. The method of equating the coefficients Letp andq be two real polynomials with degreen N, which means ∈ n n p(x) =a nx +...+a 1x+a 0 andq(x) =b nx +...+b 1x+b 0 for some coefficientsa n, . , a1, a0, bn, . , b1, b0 R. ∈ If we have the equalityp=q, which means n n anx +...+a 1x+a 0 =b nx +...+b 1x+b 0, (7.3) for allx R, then we can concludea n =b n, . , a1 =b 1 anda 0 =b 0. ∈ VL18 ↓ Remark: Since dim( n(R)) =n+1 and we have the inclusions P 0(R) 1(R) 2(R) (R) (R), P ⊂P ⊂P ⊂···⊂P ⊂F we conclude that dim( (R)) and dim( (R)) cannot befinite natural numbers. Sym- P F bolically, we write dim( (R)) = in such a case. P ∞ 7.4 Coordinates with respect to a basis 11 7.4 Coordinates with respect to a basis 7.4.1 Basis implies coordinates Again, we deal with the caseF=R andF=C simultaneously. Therefore, letV be an F-vector space with the two operations+ and . -
HILBERT SPACE GEOMETRY Definition: a Vector Space Over Is a Set V (Whose Elements Are Called Vectors) Together with a Binary Operation
HILBERT SPACE GEOMETRY Definition: A vector space over is a set V (whose elements are called vectors) together with a binary operation +:V×V→V, which is called vector addition, and an external binary operation ⋅: ×V→V, which is called scalar multiplication, such that (i) (V,+) is a commutative group (whose neutral element is called zero vector) and (ii) for all λ,µ∈ , x,y∈V: λ(µx)=(λµ)x, 1 x=x, λ(x+y)=(λx)+(λy), (λ+µ)x=(λx)+(µx), where the image of (x,y)∈V×V under + is written as x+y and the image of (λ,x)∈ ×V under ⋅ is written as λx or as λ⋅x. Exercise: Show that the set 2 together with vector addition and scalar multiplication defined by x y x + y 1 + 1 = 1 1 + x 2 y2 x 2 y2 x λx and λ 1 = 1 , λ x 2 x 2 respectively, is a vector space. 1 Remark: Usually we do not distinguish strictly between a vector space (V,+,⋅) and the set of its vectors V. For example, in the next definition V will first denote the vector space and then the set of its vectors. Definition: If V is a vector space and M⊆V, then the set of all linear combinations of elements of M is called linear hull or linear span of M. It is denoted by span(M). By convention, span(∅)={0}. Proposition: If V is a vector space, then the linear hull of any subset M of V (together with the restriction of the vector addition to M×M and the restriction of the scalar multiplication to ×M) is also a vector space. -
21. Orthonormal Bases
21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else. -
Selecting a Monomial Basis for Sums of Squares Programming Over a Quotient Ring
Selecting a Monomial Basis for Sums of Squares Programming over a Quotient Ring Frank Permenter1 and Pablo A. Parrilo2 Abstract— In this paper we describe a method for choosing for λi(x) and s(x), this feasibility problem is equivalent to a “good” monomial basis for a sums of squares (SOS) program an SDP. This follows because the sum of squares constraint formulated over a quotient ring. It is known that the monomial is equivalent to a semidefinite constraint on the coefficients basis need only include standard monomials with respect to a Groebner basis. We show that in many cases it is possible of s(x), and the equality constraint is equivalent to linear to use a reduced subset of standard monomials by combining equations on the coefficients of s(x) and λi(x). Groebner basis techniques with the well-known Newton poly- Naturally, the complexity of this sums of squares program tope reduction. This reduced subset of standard monomials grows with the complexity of the underlying variety, as yields a smaller semidefinite program for obtaining a certificate does the size of the corresponding SDP. It is therefore of non-negativity of a polynomial on an algebraic variety. natural to explore how algebraic structure can be exploited I. INTRODUCTION to simplify this sums of squares program. Consider the Many practical engineering problems require demonstrat- following reformulation [10], which is feasible if and only ing non-negativity of a multivariate polynomial over an if (2) is feasible: algebraic variety, i.e. over the solution set of polynomial Find s(x) equations. -
Multivector Differentiation and Linear Algebra 0.5Cm 17Th Santaló
Multivector differentiation and Linear Algebra 17th Santalo´ Summer School 2016, Santander Joan Lasenby Signal Processing Group, Engineering Department, Cambridge, UK and Trinity College Cambridge [email protected], www-sigproc.eng.cam.ac.uk/ s jl 23 August 2016 1 / 78 Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. 2 / 78 Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. 3 / 78 Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. 4 / 78 Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... 5 / 78 Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary 6 / 78 We now want to generalise this idea to enable us to find the derivative of F(X), in the A ‘direction’ – where X is a general mixed grade multivector (so F(X) is a general multivector valued function of X). Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. Then, provided A has same grades as X, it makes sense to define: F(X + tA) − F(X) A ∗ ¶XF(X) = lim t!0 t The Multivector Derivative Recall our definition of the directional derivative in the a direction F(x + ea) − F(x) a·r F(x) = lim e!0 e 7 / 78 Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. -
Linear Independence, Span, and Basis of a Set of Vectors What Is Linear Independence?
LECTURENOTES · purdue university MA 26500 Kyle Kloster Linear Algebra October 22, 2014 Linear Independence, Span, and Basis of a Set of Vectors What is linear independence? A set of vectors S = fv1; ··· ; vkg is linearly independent if none of the vectors vi can be written as a linear combination of the other vectors, i.e. vj = α1v1 + ··· + αkvk. Suppose the vector vj can be written as a linear combination of the other vectors, i.e. there exist scalars αi such that vj = α1v1 + ··· + αkvk holds. (This is equivalent to saying that the vectors v1; ··· ; vk are linearly dependent). We can subtract vj to move it over to the other side to get an expression 0 = α1v1 + ··· αkvk (where the term vj now appears on the right hand side. In other words, the condition that \the set of vectors S = fv1; ··· ; vkg is linearly dependent" is equivalent to the condition that there exists αi not all of which are zero such that 2 3 α1 6α27 0 = v v ··· v 6 7 : 1 2 k 6 . 7 4 . 5 αk More concisely, form the matrix V whose columns are the vectors vi. Then the set S of vectors vi is a linearly dependent set if there is a nonzero solution x such that V x = 0. This means that the condition that \the set of vectors S = fv1; ··· ; vkg is linearly independent" is equivalent to the condition that \the only solution x to the equation V x = 0 is the zero vector, i.e. x = 0. How do you determine if a set is lin. -
MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam
MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 2006, 09:30am - 10:50am Problem 1 (15 points). Let the matrix A be given by 1 −2 −1 2 −1 5 6 3 : 5 −4 5 4 5 (a) Find the inverse A−1 of A, if it exists. (b) Based on your answer in (a), determine whether the columns of A span R3. (Justify your answer!) Solution. (a) To check whether A is invertible, we row reduce the augmented matrix [A I3]: 1 −2 −1 1 0 0 1 −2 −1 1 0 0 2 −1 5 6 0 1 0 3 ∼ : : : ∼ 2 0 3 5 1 1 0 3 : 5 −4 5 0 0 1 0 0 0 −7 −2 1 4 5 4 5 Since the last row in the echelon form of A contains only zeros, A is not row equivalent to I3. Hence, A is not invertible, and A−1 does not exist. (b) Since A is not invertible by (a), the Invertible Matrix Theorem says that the columns of A cannot span R3. Problem 2 (15 points). Let the vectors b1; : : : ; b4 be defined by 3 2 −1 0 0 5 1 0 −1 1 0 1 1 0 0 1 b1 = ; b2 = ; b3 = ; and b4 = : −2 −5 3 0 B C B C B C B C B 4 C B 7 C B 0 C B −3 C @ A @ A @ A @ A (a) Determine if the set B = fb1; b2; b3; b4g is linearly independent by computing the determi- nant of the matrix B = [b1 b2 b3 b4]. -
Does Geometric Algebra Provide a Loophole to Bell's Theorem?
Discussion Does Geometric Algebra provide a loophole to Bell’s Theorem? Richard David Gill 1 1 Leiden University, Faculty of Science, Mathematical Institute; [email protected] Version October 30, 2019 submitted to Entropy Abstract: Geometric Algebra, championed by David Hestenes as a universal language for physics, was used as a framework for the quantum mechanics of interacting qubits by Chris Doran, Anthony Lasenby and others. Independently of this, Joy Christian in 2007 claimed to have refuted Bell’s theorem with a local realistic model of the singlet correlations by taking account of the geometry of space as expressed through Geometric Algebra. A series of papers culminated in a book Christian (2014). The present paper first explores Geometric Algebra as a tool for quantum information and explains why it did not live up to its early promise. In summary, whereas the mapping between 3D geometry and the mathematics of one qubit is already familiar, Doran and Lasenby’s ingenious extension to a system of entangled qubits does not yield new insight but just reproduces standard QI computations in a clumsy way. The tensor product of two Clifford algebras is not a Clifford algebra. The dimension is too large, an ad hoc fix is needed, several are possible. I further analyse two of Christian’s earliest, shortest, least technical, and most accessible works (Christian 2007, 2011), exposing conceptual and algebraic errors. Since 2015, when the first version of this paper was posted to arXiv, Christian has published ambitious extensions of his theory in RSOS (Royal Society - Open Source), arXiv:1806.02392, and in IEEE Access, arXiv:1405.2355. -
Discover Linear Algebra Incomplete Preliminary Draft
Discover Linear Algebra Incomplete Preliminary Draft Date: November 28, 2017 L´aszl´oBabai in collaboration with Noah Halford All rights reserved. Approved for instructional use only. Commercial distribution prohibited. c 2016 L´aszl´oBabai. Last updated: November 10, 2016 Preface TO BE WRITTEN. Babai: Discover Linear Algebra. ii This chapter last updated August 21, 2016 c 2016 L´aszl´oBabai. Contents Notation ix I Matrix Theory 1 Introduction to Part I 2 1 (F, R) Column Vectors 3 1.1 (F) Column vector basics . 3 1.1.1 The domain of scalars . 3 1.2 (F) Subspaces and span . 6 1.3 (F) Linear independence and the First Miracle of Linear Algebra . 8 1.4 (F) Dot product . 12 1.5 (R) Dot product over R ................................. 14 1.6 (F) Additional exercises . 14 2 (F) Matrices 15 2.1 Matrix basics . 15 2.2 Matrix multiplication . 18 2.3 Arithmetic of diagonal and triangular matrices . 22 2.4 Permutation Matrices . 24 2.5 Additional exercises . 26 3 (F) Matrix Rank 28 3.1 Column and row rank . 28 iii iv CONTENTS 3.2 Elementary operations and Gaussian elimination . 29 3.3 Invariance of column and row rank, the Second Miracle of Linear Algebra . 31 3.4 Matrix rank and invertibility . 33 3.5 Codimension (optional) . 34 3.6 Additional exercises . 35 4 (F) Theory of Systems of Linear Equations I: Qualitative Theory 38 4.1 Homogeneous systems of linear equations . 38 4.2 General systems of linear equations . 40 5 (F, R) Affine and Convex Combinations (optional) 42 5.1 (F) Affine combinations . -
28. Exterior Powers
28. Exterior powers 28.1 Desiderata 28.2 Definitions, uniqueness, existence 28.3 Some elementary facts 28.4 Exterior powers Vif of maps 28.5 Exterior powers of free modules 28.6 Determinants revisited 28.7 Minors of matrices 28.8 Uniqueness in the structure theorem 28.9 Cartan's lemma 28.10 Cayley-Hamilton Theorem 28.11 Worked examples While many of the arguments here have analogues for tensor products, it is worthwhile to repeat these arguments with the relevant variations, both for practice, and to be sensitive to the differences. 1. Desiderata Again, we review missing items in our development of linear algebra. We are missing a development of determinants of matrices whose entries may be in commutative rings, rather than fields. We would like an intrinsic definition of determinants of endomorphisms, rather than one that depends upon a choice of coordinates, even if we eventually prove that the determinant is independent of the coordinates. We anticipate that Artin's axiomatization of determinants of matrices should be mirrored in much of what we do here. We want a direct and natural proof of the Cayley-Hamilton theorem. Linear algebra over fields is insufficient, since the introduction of the indeterminate x in the definition of the characteristic polynomial takes us outside the class of vector spaces over fields. We want to give a conceptual proof for the uniqueness part of the structure theorem for finitely-generated modules over principal ideal domains. Multi-linear algebra over fields is surely insufficient for this. 417 418 Exterior powers 2. Definitions, uniqueness, existence Let R be a commutative ring with 1. -
1 Expressing Vectors in Coordinates
Math 416 - Abstract Linear Algebra Fall 2011, section E1 Working in coordinates In these notes, we explain the idea of working \in coordinates" or coordinate-free, and how the two are related. 1 Expressing vectors in coordinates Let V be an n-dimensional vector space. Recall that a choice of basis fv1; : : : ; vng of V is n the same data as an isomorphism ': V ' R , which sends the basis fv1; : : : ; vng of V to the n standard basis fe1; : : : ; eng of R . In other words, we have ' n ': V −! R vi 7! ei 2 3 c1 6 . 7 v = c1v1 + ::: + cnvn 7! 4 . 5 : cn This allows us to manipulate abstract vectors v = c v + ::: + c v simply as lists of numbers, 2 3 1 1 n n c1 6 . 7 n the coordinate vectors 4 . 5 2 R with respect to the basis fv1; : : : ; vng. Note that the cn coordinates of v 2 V depend on the choice of basis. 2 3 c1 Notation: Write [v] := 6 . 7 2 n for the coordinates of v 2 V with respect to the basis fvig 4 . 5 R cn fv1; : : : ; vng. For shorthand notation, let us name the basis A := fv1; : : : ; vng and then write [v]A for the coordinates of v with respect to the basis A. 2 2 Example: Using the monomial basis f1; x; x g of P2 = fa0 + a1x + a2x j ai 2 Rg, we obtain an isomorphism ' 3 ': P2 −! R 2 3 a0 2 a0 + a1x + a2x 7! 4a15 : a2 2 3 a0 2 In the notation above, we have [a0 + a1x + a2x ]fxig = 4a15. -
Linear Independence, the Wronskian, and Variation of Parameters
LINEAR INDEPENDENCE, THE WRONSKIAN, AND VARIATION OF PARAMETERS JAMES KEESLING In this post we determine when a set of solutions of a linear differential equation are linearly independent. We first discuss the linear space of solutions for a homogeneous differential equation. 1. Homogeneous Linear Differential Equations We start with homogeneous linear nth-order ordinary differential equations with general coefficients. The form for the nth-order type of equation is the following. dnx dn−1x (1) a (t) + a (t) + ··· + a (t)x = 0 n dtn n−1 dtn−1 0 It is straightforward to solve such an equation if the functions ai(t) are all constants. However, for general functions as above, it may not be so easy. However, we do have a principle that is useful. Because the equation is linear and homogeneous, if we have a set of solutions fx1(t); : : : ; xn(t)g, then any linear combination of the solutions is also a solution. That is (2) x(t) = C1x1(t) + C2x2(t) + ··· + Cnxn(t) is also a solution for any choice of constants fC1;C2;:::;Cng. Now if the solutions fx1(t); : : : ; xn(t)g are linearly independent, then (2) is the general solution of the differential equation. We will explain why later. What does it mean for the functions, fx1(t); : : : ; xn(t)g, to be linearly independent? The simple straightforward answer is that (3) C1x1(t) + C2x2(t) + ··· + Cnxn(t) = 0 implies that C1 = 0, C2 = 0, ::: , and Cn = 0 where the Ci's are arbitrary constants. This is the definition, but it is not so easy to determine from it just when the condition holds to show that a given set of functions, fx1(t); x2(t); : : : ; xng, is linearly independent.