DOCSLIB.ORG

Mathematics of Information Hilbert Spaces and Linear Operators

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematics of Information Hilbert Spaces and Linear Operators Chair for Mathematical Information Science Verner Vlačić Sternwartstrasse 7 CH-8092 Zürich Mathematics of Information Hilbert spaces and linear operators These notes are based on [1, Chap. 6, Chap. 8], [2, Chap. 1], [3, Chap. 4], [4, App. A] and [5]. 1 Vector spaces Let us consider a field F, which can be, for example, the field of real numbers R or the field of complex numbers C.A vector space over F is a set whose elements are called vectors and in which two operations, addition and multiplication by any of the elements of the field F (referred to as scalars), are defined with some algebraic properties. More precisely, we have Definition 1 (Vector space). A set X together with two operations (+; ·) is a vector space over F if the following properties are satisfied: (i) the first operation, called vector addition: X × X ! X denoted by + satisfies • (x + y) + z = x + (y + z) for all x; y; z 2 X (associativity of addition) • x + y = y + x for all x; y 2 X (commutativity of addition) (ii) there exists an element 0 2 X , called the zero vector, such that x + 0 = x for all x 2 X (iii) for every x 2 X , there exists an element in X , denoted by −x, such that x + (−x) = 0. (iv) the second operation, called scalar multiplication: F × X ! X denoted by · satisfies • 1 · x = x for all x 2 X • α · (β · x) = (αβ) · x for all α; β 2 F and x 2 X (associativity for scalar multiplication) • (α+β)·x = α·x+β ·x for all α; β 2 F and x 2 X (distributivity of scalar multiplication with respect to field addition) • α·(x+y) = α·x+α·y for all α 2 F and x; y 2 X (distributivity of scalar multiplication with respect to vector addition). We refer to X as a real vector space when F = R and as a complex vector space when F = C. Examples. 1. C is both a real and a complex vector space. N 2. The set F , f(x1; x2; : : : ; xN ): xk 2 Fg of all N-tuples forms a vector space over F. 3. The set F[x] of all polynomials with coefficients in F is a vector space over F. 4. The space FZ of all sequences of F is a vector space over F. M×N 5. The set F of all matrices of size M × N with entries in F forms a vector space over F under the laws of matrix addition and scalar multiplication. 6. If X is an arbitrary set and Y an arbitrary vector space over F, the set F(X ; Y) of all functions X!Y is a vector space over F under pointwise addition and multiplication. A subspace of a vector space X over F is a subset of X which is itself a vector space over F (with respect to the same operations). One can easily verify that a subset Y of X is a subspace using the following proposition. Proposition 1.1. Let X be a vector space over F and Y a nonempty subset of X . Then Y is a subspace of X if it contains 0 and if it is stable under linear combinations, that is, α · y + β · z 2 Y for all α; β 2 F and y; z 2 Y. Examples. 1. R and iR are subspaces of the real vector space C. N 2. The set of N-tuples (x1; x2; : : : ; xN−1; 0) with xk 2 R is a subspace of R . p 3. The set ` (Z), p 2 [1; 1], of all complex-valued sequence fukgk2Z which satisfy P+1 p k=−∞ jukj < 1; if p 2 [1; 1); supk2Z jukj < 1; if p = 1; is a subspace of CZ. 4. If X is an arbitrary set, Y a vector space over F, and Z a subspace of Y, then the set F(X ; Z) of functions X!Z is a subspace of F(X ; Y). 5. The space Cn[a; b] of all complex-valued functions with continuous derivatives of order 0 6 k 6 n on the closed, bounded interval [a; b] of the real line is a subspace of F([a; b]; C). 6. If (S; Σ; µ) is a measure space, we let Lp(S; Σ; µ) denote the space of all mea- surable functions mapping S to C whose absolute value raised to the p-power is µ-integrable, that is, Z p p L (S; Σ; µ) = f : S! C measurable: jfj dµ < 1 ; S two elements of Lp(S; Σ; µ) being considered as equivalent if they differ only on p a set whose measure is zero. L (S; Σ; µ) is a subspace of F(S; C). For simplicity, p N we often use the notation L (S) when S is a subset of R , Σ is the Borel σ-algebra over S, and µ the Lebesgue measure on S. In this case, two functions are equivalent when they are equal except possibly on a set of Lebesgue measure 1 N zero (e.g., a finite or countable set of points of R ). 2 Theorem 1 (Intersection of vector spaces). The intersection of any collection of subspaces of a vector space X over F is again a subspace of X . Definition 2. If S and T are linear subspaces of a vector space X with S\T = f0g, then we define the direct sum S ⊕ T by S ⊕ T = fx + y j x 2 S; y 2 T g: 2 Note that the union of two subspaces is in general not a subspace. Take for example X = R . 2 The lines xR and yR, with x = (1; 0) and y = (0; 1), are both subspaces of R , but xR [ yR is a not subspace, given that x + y = (1; 1) 2= xR [ yR. Definition 3 (Subspace spanned by S). Let S be a (possibly infinite) subset of a vector space X over F. The subspace spanned by S is the intersection of all subspaces containing S. It is the smallest subspace containing S. It is denoted by span(S) and may be written the set of all finite combinations of elements of S, that is, ( k ) X span(S) , λ`x` : k 2 N; x` 2 S; λ` 2 F : `=1 Examples. 1. If S is empty, then the subspace spanned by S is f0g. 2. In the real vector space C, we have the following: • the subspace spanned by f1g is R, • the subspace spanned by fig is iR, • the subspace spanned by f1; ig is C. N 3. In F[x], the subspace spanned by f1; x; : : : ; x g is the space FN [x] of all polyno- mials whose degree is less than N. Definition 4 (Linear independence). Let X be a vector space over F. A finite set fx1; x2; : : : ; xN g of vectors of X is said to be linearly independent if for every λ1; λ2; : : : ; λN 2 F, the equality λ1x1 + λ2x2 + ::: + λN xN = 0 2 implies that λ1 = λ2 = ::: = λN = 0. An infinite set S of vectors of X is linearly independent if every finite subset of S is linearly independent. Examples. 1. Any set of vectors containing the zero vector is linearly dependent. 2. The real vector space C, the set f1; ig is linearly independent. 2 N 3. The basic monomials f1; x; x ; : : : ; x g form a linearly independent set of F[x]. 4. The set of trigonometric functions f1; cos(x); sin(x); cos2(x); cos(x) sin(x); sin2(x)g is linearly dependent. 1Note, however, that there exist uncountable sets (e.g., the Cantor set) of Lebesgue measure zero. 3 Definition 5 (Dimension). A vector space X is N-dimensional if there exists N linearly independent vectors in X and any N + 1 vectors in X are linearly dependent. Definition 6 (Finite-dimensional space). A vector space X is finite-dimensional if X is N- dimensional for some integer N. Otherwise, X is infinite dimensional. N Examples. 1. The spaces F and FN [x] are finite-dimensional (their dimension is N). n p 2. The space F(X ; X ), F[x], C [a; b], ` (Z) are infinite-dimensional. 3 Definition 7 (Basis). A basis of a vector space X over F is a set of linearly independent and spanning vectors of X . 3 T T T Examples. 1. The set fekgk=1, where e1 = [1 0 0] , e2 = [0 1 0] , and e3 = [0 0 1] , 3 forms a basis for R . 2 N 2. The set f1; x; x ; : : : ; x g forms a basis for FN [x]. 2 3 3. The set f1; x; x ; x ;:::g forms a basis for F[x]. Theorem 2. Let X be a finite dimensional vector space. Any set of linearly independent vectors can be extended to a basis of X . 2 Inner products and norms From here, we will assume that F = R or F = C. Definition 8 (Norm). Let X be a vector space over F. A norm on X is a function which maps X to R and satisfies the following properties: (i) for all x 2 X , we have kxk > 0 and kxk = 0 implies x = 0 (positivity) (ii) for all x 2 X and for all α 2 F, we have kαxk = jαj kxk (homogeneity) (iii) for all x; y 2 X , we have kx + yk 6 kxk + kyk (triangle inequality). Definition 9. A vector space X over F is a normed vector space (or a pre-Banach space) if it is equipped with a norm k·k.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Spanning Sets the Only Algebraic Operations That Are Defined in a Vector Space V Are Those of Addition and Scalar Multiplication
    i i “main” 2007/2/16 page 258 i i 258 CHAPTER 4 Vector Spaces 23. Show that the set of all solutions to the nonhomoge- and let neous differential equation S1 + S2 = {v ∈ V : y + a1y + a2y = F(x), v = x + y for some x ∈ S1 and y ∈ S2} . where F(x) is nonzero on an interval I, is not a sub- 2 space of C (I). (a) Show that, in general, S1 ∪ S2 is not a subspace of V . 24. Let S1 and S2 be subspaces of a vector space V . Let (b) Show that S1 ∩ S2 is a subspace of V . S ∪ S ={v ∈ V : v ∈ S or v ∈ S }, 1 2 1 2 (c) Show that S1 + S2 is a subspace of V . S1 ∩ S2 ={v ∈ V : v ∈ S1 and v ∈ S2}, 4.4 Spanning Sets The only algebraic operations that are defined in a vector space V are those of addition and scalar multiplication. Consequently, the most general way in which we can combine the vectors v1, v2,...,vk in V is c1v1 + c2v2 +···+ckvk, (4.4.1) where c1,c2,...,ck are scalars. An expression of the form (4.4.1) is called a linear combination of v1, v2,...,vk. Since V is closed under addition and scalar multiplica- tion, it follows that the foregoing linear combination is itself a vector in V . One of the questions we wish to answer is whether every vector in a vector space can be obtained by taking linear combinations of a finite set of vectors. The following terminology is used in the case when the answer to this question is affirmative: DEFINITION 4.4.1 If every vector in a vector space V can be written as a linear combination of v1, v2, ..., vk, we say that V is spanned or generated by v1, v2, ..., vk and call the set of vectors {v1, v2,...,vk} a spanning set for V .
    [Show full text]
  • Worksheet 1, for the MATLAB Course by Hans G. Feichtinger, Edinburgh, Jan
    Worksheet 1, for the MATLAB course by Hans G. Feichtinger, Edinburgh, Jan. 9th Start MATLAB via: Start > Programs > School applications > Sci.+Eng. > Eng.+Electronics > MATLAB > R2007 (preferably). Generally speaking I suggest that you start your session by opening up a diary with the command: diary mydiary1.m and concluding the session with the command diary off. If you want to save your workspace you my want to call save today1.m in order to save all the current variable (names + values). Moreover, using the HELP command from MATLAB you can get help on more or less every MATLAB command. Try simply help inv or help plot. 1. Define an arbitrary (random) 3 × 3 matrix A, and check whether it is invertible. Calculate the inverse matrix. Then define an arbitrary right hand side vector b and determine the (unique) solution to the equation A ∗ x = b. Find in two different ways the solution to this problem, by either using the inverse matrix, or alternatively by applying Gauss elimination (i.e. the RREF command) to the extended system matrix [A; b]. In addition look at the output of the command rref([A,eye(3)]). What does it tell you? 2. Produce an \arbitrary" 7 × 7 matrix of rank 5. There are at east two simple ways to do this. Either by factorization, i.e. by obtaining it as a product of some 7 × 5 matrix with another random 5 × 7 matrix, or by purposely making two of the rows or columns linear dependent from the remaining ones. Maybe the second method is interesting (if you have time also try the first one): 3.
    [Show full text]
  • Math 308 Pop Quiz No. 1 Complete the Sentence/Definition
    Math 308 Pop Quiz No. 1 Complete the sentence/definition: (1) If a system of linear equations has ≥ 2 solutions, then solutions. (2) A system of linear equations can have either (a) a solution, or (b) solutions, or (c) solutions. (3) An m × n system of linear equations consists of (a) linear equations (b) in unknowns. (4) If an m × n system of linear equations is converted into a single matrix equation Ax = b, then (a) A is a × matrix (b) x is a × matrix (c) b is a × matrix (5) A system of linear equations is consistent if (6) A system of linear equations is inconsistent if (7) An m × n matrix has (a) columns and (b) rows. (8) Let A and B be sets. Then (a) fx j x 2 A or x 2 Bg is called the of A and B; (b) fx j x 2 A and x 2 Bg is called the of A and B; (9) Write down the symbols for the integers, the rational numbers, the real numbers, and the complex numbers, and indicate by using the notation for subset which is contained in which. (10) Let E be set of all even numbers, O the set of all odd numbers, and T the set of all integer multiples of 3. Then (a) E \ O = (use a single symbol for your answer) (b) E [ O = (use a single symbol for your answer) (c) E \ T = (use words for your answer) 1 2 Math 308 Answers to Pop Quiz No. 1 Complete the sentence/definition: (1) If a system of linear equations has ≥ 2 solutions, then it has infinitely many solutions.
    [Show full text]
  • Math 20480 – Example Set 03A Determinant of a Square Matrices
    Math 20480 { Example Set 03A Determinant of A Square Matrices. Case: 2 × 2 Matrices. a b The determinant of A = is c d a b det(A) = jAj = = c d For matrices with size larger than 2, we need to use cofactor expansion to obtain its value. Case: 3 × 3 Matrices. 0 1 a11 a12 a13 a11 a12 a13 The determinant of A = @a21 a22 a23A is det(A) = jAj = a21 a22 a23 a31 a32 a33 a31 a32 a33 (Cofactor expansion along the 1st row) = a11 − a12 + a13 (Cofactor expansion along the 2nd row) = (Cofactor expansion along the 1st column) = (Cofactor expansion along the 2nd column) = Example 1: 1 1 2 ? 1 −2 −1 = 1 −1 1 1 Case: 4 × 4 Matrices. a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44 (Cofactor expansion along the 1st row) = (Cofactor expansion along the 2nd column) = Example 2: 1 −2 −1 3 −1 2 0 −1 ? = 0 1 −2 2 3 −1 2 −3 2 Theorem 1: Let A be a square n × n matrix. Then the A has an inverse if and only if det(A) 6= 0. We call a matrix A with inverse a matrix. Theorem 2: Let A be a square n × n matrix and ~b be any n × 1 matrix. Then the equation A~x = ~b has a solution for ~x if and only if . Proof: Remark: If A is non-singular then the only solution of A~x = 0 is ~x = . 3 3. Without solving explicitly, determine if the following systems of equations have a unique solution.
    [Show full text]
  • Tutorial 3 Using MATLAB in Linear Algebra
    Edward Neuman Department of Mathematics Southern Illinois University at Carbondale [email protected] One of the nice features of MATLAB is its ease of computations with vectors and matrices. In this tutorial the following topics are discussed: vectors and matrices in MATLAB, solving systems of linear equations, the inverse of a matrix, determinants, vectors in n-dimensional Euclidean space, linear transformations, real vector spaces and the matrix eigenvalue problem. Applications of linear algebra to the curve fitting, message coding and computer graphics are also included. For the reader's convenience we include lists of special characters and MATLAB functions that are used in this tutorial. Special characters ; Semicolon operator ' Conjugated transpose .' Transpose * Times . Dot operator ^ Power operator [ ] Emty vector operator : Colon operator = Assignment == Equality \ Backslash or left division / Right division i, j Imaginary unit ~ Logical not ~= Logical not equal & Logical and | Logical or { } Cell 2 Function Description acos Inverse cosine axis Control axis scaling and appearance char Create character array chol Cholesky factorization cos Cosine function cross Vector cross product det Determinant diag Diagonal matrices and diagonals of a matrix double Convert to double precision eig Eigenvalues and eigenvectors eye Identity matrix fill Filled 2-D polygons fix Round towards zero fliplr Flip matrix in left/right direction flops Floating point operation count grid Grid lines hadamard Hadamard matrix hilb Hilbert
    [Show full text]
  • 3. Hilbert Spaces
    3. Hilbert spaces In this section we examine a special type of Banach spaces. We start with some algebraic preliminaries. Definition. Let K be either R or C, and let Let X and Y be vector spaces over K. A map φ : X × Y → K is said to be K-sesquilinear, if • for every x ∈ X, then map φx : Y 3 y 7−→ φ(x, y) ∈ K is linear; • for every y ∈ Y, then map φy : X 3 y 7−→ φ(x, y) ∈ K is conjugate linear, i.e. the map X 3 x 7−→ φy(x) ∈ K is linear. In the case K = R the above properties are equivalent to the fact that φ is bilinear. Remark 3.1. Let X be a vector space over C, and let φ : X × X → C be a C-sesquilinear map. Then φ is completely determined by the map Qφ : X 3 x 7−→ φ(x, x) ∈ C. This can be see by computing, for k ∈ {0, 1, 2, 3} the quantity k k k k k k k Qφ(x + i y) = φ(x + i y, x + i y) = φ(x, x) + φ(x, i y) + φ(i y, x) + φ(i y, i y) = = φ(x, x) + ikφ(x, y) + i−kφ(y, x) + φ(y, y), which then gives 3 1 X (1) φ(x, y) = i−kQ (x + iky), ∀ x, y ∈ X. 4 φ k=0 The map Qφ : X → C is called the quadratic form determined by φ. The identity (1) is referred to as the Polarization Identity.
    [Show full text]
  • Linear Algebra: a Field Guide
    Linear Algebra: A Field Guide David Grabovsky July 27, 2020 Contents I Basic Notions 2 1 Welcome to Vector Space 2 1.1 The Rules of the Game . .2 1.2 Basis and Dimension . .3 2 Linear Transformations 4 2.1 Definition and Properties . .4 2.2 Kernel, Image, and Isomorphism . .5 2.3 A Change of Perspective . .6 3 Two Extra Topics 7 3.1 Linear Systems . .7 3.2 Determinants . .8 II Diagonalizable Operators 9 4 Finding an Eigenbasis 9 5 Direct-Sum Decomposition 11 III The Spectral Theorem 12 6 Geometry and Structure 12 7 Linear Functionals and Duality 13 8 The Spectral Theorem 15 8.1 A Zoo of Operators . 15 8.2 The Stormy Finale . 17 A Textbooks and Further Reading 19 1 1 WELCOME TO VECTOR SPACE Part I Basic Notions Linear algebra is about structures called vector spaces whose elements, called vectors, may be added and scaled; and about maps between them, called linear transformations, that preserve their linear structure. In this sense, objects as different as numbers, pointed arrows in space, and even functions fall under the subject's purview. 1 Welcome to Vector Space 1.1 The Rules of the Game Definition 1.1. A field F is a set of objects called scalars that behave in a sufficiently number-like manner. In particular, for all x; y; z 2 F , the field must support: (a) Associative-commutative binary operations + and ·; (b) Additive and multiplicative identity elements 0 6= 1; (c) Unique additive and multiplicative inverses −x and x−1; (d) The distributive property: x(y + z) = xy + xz.
    [Show full text]
  • Vector Spaces Linear Independence, Bases and Dimension
    DM559 Linear and Integer Programming Lecture 7 Vector Spaces Linear Independence, Bases and Dimension Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Vector Spaces and Subspaces Linear independence Outline Bases and Dimension 1. Vector Spaces and Subspaces 2. Linear independence 3. Bases and Dimension 5 Vector Spaces and Subspaces Linear independence Outline Bases and Dimension 1. Vector Spaces and Subspaces 2. Linear independence 3. Bases and Dimension 6 Vector Spaces and Subspaces Linear independence Premise Bases and Dimension • We move to a higher level of abstraction • A vector space is a set with an addition and scalar multiplication that behave appropriately, that is, like Rn • Imagine a vector space as a class of a generic type (template) in object oriented programming, equipped with two operations. 7 Vector Spaces and Subspaces Linear independence Vector Spaces Bases and Dimension Definition (Vector Space) A (real) vector space V is a non-empty set equipped with an addition and a scalar multiplication operation such that for all α; β 2 R and all u; v; w 2 V : 1. u + v 2 V (closure under addition) 2. u + v = v + u (commutative law for addition) 3. u + (v + w) = (u + v) + w (associative law for addition) 4. there is a single member 0 of V , called the zero vector, such that for all v 2 V ; v + 0 = v 5. for every v 2 V there is an element w 2 V , written −v, called the negative of v, such that v + w = 0 6. αv 2 V (closure under scalar multiplication) 7.
    [Show full text]
  • NATURAL STRUCTURES in DIFFERENTIAL GEOMETRY Lucas
    NATURAL STRUCTURES IN DIFFERENTIAL GEOMETRY Lucas Mason-Brown A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in mathematics Trinity College Dublin March 2015 Declaration I declare that the thesis contained herein is entirely my own work, ex- cept where explicitly stated otherwise, and has not been submitted for a degree at this or any other university. I grant Trinity College Dublin per- mission to lend or copy this thesis upon request, with the understanding that this permission covers only single copies made for study purposes, subject to normal conditions of acknowledgement. Contents 1 Summary 3 2 Acknowledgment 6 I Natural Bundles and Operators 6 3 Jets 6 4 Natural Bundles 9 5 Natural Operators 11 6 Invariant-Theoretic Reduction 16 7 Classical Results 24 8 Natural Operators on Differential Forms 32 II Natural Operators on Alternating Multivector Fields 38 9 Twisted Algebras in VecZ 40 10 Ordered Multigraphs 48 11 Ordered Multigraphs and Natural Operators 52 12 Gerstenhaber Algebras 57 13 Pre-Gerstenhaber Algebras 58 irred 14 A Lie-admissable Structure on Graacyc 63 irred 15 A Co-algebra Structure on Graacyc 67 16 Loday's Rigidity Theorem 75 17 A Useful Consequence of K¨unneth'sTheorem 83 18 Chevalley-Eilenberg Cohomology 87 1 Summary Many of the most important constructions in differential geometry are functorial. Take, for example, the tangent bundle. The tangent bundle can be profitably viewed as a functor T : Manm ! Fib from the category of m-dimensional manifolds (with local diffeomorphisms) to 3 the category of fiber bundles (with locally invertible bundle maps).
    [Show full text]
  • Bases and Dimension 281
    i i “main” 2007/2/16 page 281 i i 4.6 Bases and Dimension 281 2 3 40. (a) Show that {1,x,x ,x } is linearly independent (a) If n>m, prove that {u1, u2,...,un} is linearly on every interval. dependent on V . k (b) If fk(x) = x for k = 0, 1,...,n, show that (b) If n = m, prove that {u1, u2,...,un} is linearly {f0,f1,...,fn} is linearly independent on every independent in V if and only if det[aij ] = 0. interval for all fixed n. (c) If n<m, prove that {u1, u2,...,un} is linearly 41. (a) Show that the functions independent in V if and only if rank(A) = n, r1x r2x r3x =[ ] f1(x) = e ,f2(x) = e ,f3(x) = e where A aij . have Wronskian (d) Which result from this section do these results generalize? 111 (r1+r2+r3)x W[f1,f2,f3](x) = e r1 r2 r3 r2 r2 r2 45. Prove from the definition of “linearly independent” 1 2 3 { } + + that if v1, v2,...,vn is linearly independent and = (r1 r2 r3)x − − − e (r3 r1)(r3 r2)(r2 r1), if A is an invertible n × n matrix, then the set {Av ,Av ,...,Av } is linearly independent. and hence determine the conditions on r1,r2,r3 1 2 n such that {f1,f2,f3} is linearly independent on { } every interval. 46. Prove that if v1, v2 is linearly independent and v3 { } { } (b) More generally, show that the set of functions is not in span v1, v2 , then v1, v2, v3 is linearly independent.
    [Show full text]