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Linear Algebra II Andrew Kobin Fall 2014 Contents Contents Contents 0 Introduction 1 1 The Algebra of Vector Spaces 2 1.1 Scalar Fields . .2 1.2 Vector Spaces and Linear Transformations . .3 1.3 Constructing Vector Spaces . .7 1.4 Linear Independence, Span and Bases . 10 1.5 More on Linear Transformations . 17 1.6 Similar and Conjugate Matrices . 23 1.7 Kernel and Image . 24 1.8 The Isomorphism Theorems . 27 2 Duality 29 2.1 Functionals . 29 2.2 Self Duality . 31 2.3 Perpendicular Subspaces . 31 3 Multilinearity 34 3.1 Multilinear Functions . 34 3.2 Tensor Products . 35 3.3 Tensoring Transformations . 38 3.4 Alternating Functions and Exterior Products . 39 3.5 The Determinant . 43 4 Diagonalizability 46 4.1 Eigenvalues and Eigenvectors . 46 4.2 Minimal Polynomials . 50 4.3 Cyclic Linear Systems . 53 4.4 Jordan Canonical Form . 59 5 Inner Product Spaces 65 5.1 Inner Products and Norms . 65 5.2 Orthogonality . 68 5.3 The Adjoint . 71 5.4 Normal Operators . 73 5.5 Spectral Decomposition . 74 5.6 Unitary Transformations . 75 i 0 Introduction 0 Introduction These notes were taken from a second-semester linear algebra course taught by Dr. Frank Moore at Wake Forest University in the fall of 2014. The main reference for the course is A (Terse) Introduction to Linear Algebra by Katznelson. The goal of the course is to expand upon the concepts introduced in a first-semester linear algebra course by generalizing real vector spaces to vector spaces over an arbitrary field, and describing linear operators, canonical forms, inner products and linear groups on such spaces. The main topics and theorems that are covered here are: A rigorous treatment of the algebraic theory of vector spaces and their transformations Duality, dual bases, the adjoint transformation Bilinear maps and the tensor product Alternating functions, the exterior product and the determinant Eigenvalues, eigenvectors, the Cayley-Hamiliton Theorem Jordan canonical form and the theory of diagonalizability Inner product spaces Orthonormal bases and the Gram-Schmidt process Spectral theory 1 1 The Algebra of Vector Spaces 1 The Algebra of Vector Spaces 1.1 Scalar Fields The first question we must answer is: what should the scalars be in a general treatment of linear algebra? The answer is fields: Definition. A field is a set F with two binary operations 00+00 : F × F −! F 00·00 : F × F −! F (a; b) 7−! a + b (a; b) 7−! ab satisfying the following axioms: (1) There is an element 0 2 F such that a + 0 = 0 + a = a for all a 2 F. (2) For all a; b 2 F, a + b = b + a. (3) For each a 2 F, there is an element −a 2 F such that a + (−a) = 0. (4) For all a; b 2 F, ab = ba. (5) For all nonzero a 2 F (the set of such elements is denoted F×) there exists a−1 2 F such that aa−1 = 1. (6) For any a; b; c 2 F, a(bc) = (ab)c. (7) For any a; b; c 2 F, a(b + c) = ab + ac. Note that axioms (1) { (4) say that (F; +) is an abelian group, (5) { (8) say that (F×; ·) is an abelian group, and axiom (9) serves to describe the interaction between the operations. We sometimes say that + and · \play nicely". Examples. 1 The real numbers R, the rational numbers Q and the complex numbers C are all common examples of fields. 2 Z2 = f0; 1g is the smallest possible field, sometimes called the finite field of character- istic 2. This example is important in computer science, particularly coding theory. ¯ ¯ ¯ 3 For any p a prime integer, Zp = f0; 1; 2;:::; p − 1g is called the finite field of char- acteristic p. In contrast, Q, R and C are called infinite fields. 4 The integers Z are not a field. Z has + and · but no multiplicative inverses (i.e. it fails axiom (7)). 5 Let n be a composite number. Then Zn is not a field. For example, consider Z4 = f0¯; 1¯; 2¯; 3¯g. In this set 2¯ does not have an inverse. 2 1.2 Vector Spaces and Linear Transformations 1 The Algebra of Vector Spaces 1.2 Vector Spaces and Linear Transformations Definition. A vector space over a field F is a set V with two operations 00+00 : V × V −! V 00·00 : F × V −! V (u; v) 7−! u + v (a; v) 7−! av called vector addition and scaling, respectively, such that (1) (V; +) satisfies axioms (1) { (4), i.e. V is an additive abelian group. (2) For all a; b 2 F and v 2 V , a(bv) = (ab)v. (3) For all v 2 V , 1v = v. (4) For all a; b 2 F and v 2 V , (a + b)v = av + bv. (5) For all a 2 F and u; v 2 V , a(u + v) = au + av. Examples. n 1 F = f(a1; : : : ; an) j ai 2 Fg is the canonical n-dimensional vector space over F. The operations are the usual componentwise addition and scaling. In particular, a field is a (one-dimensional) vector space over itself. 2 f0g is the trivial vector space over any field F. 3 Mat(n; m; F) denotes the set of all n × m matrices with coefficients in F. It is an F-vector space under componentwise addition and scaling, and is isomorphic to (which will be described later) Fnm. 2 n 4 F[x] = fa0 + a1x + a2x + ::: + anx j ai 2 Fg is called the polynomial algebra over F. It is a vector space via coefficient addition and polynomial multiplication (extended FOIL). This is an example of an infinite dimensional vector space. 5 Let X be a set. Then FX = ff : X ! F j f is a functiong is a vector space over F. For f; g 2 FX and for all x 2 X, addition and scaling are defined by (f + g)(x) = f(x) + g(x) and f(cx) = cf(x): The zero function 0 : X ! F sends every element x 7! 0. It is easy to verify that FX satisfies the properties of a vector space. 6 Let V = ff : R ! C j f satisfies the ODE 3f 000(x) − sin x f 00(x) + 2f(x) = 0g. Then V is a vector space under pointwise addition and scaling. For example, if f; g 2 V and x 2 R then 3(f + g)000(x) − sin x(f + g)00(x) + 2(f + g)(x) = (3f 000(x) − sin x f 00(x) + 2f(x)) + (3g000(x) − sin x g00(x) + 2g(x)) = 0 + 0 = 0: Thus f + g 2 V . The proof for scalar multiplication is similar. 3 1.2 Vector Spaces and Linear Transformations 1 The Algebra of Vector Spaces Next we seek a way of comparing two vector spaces. Definition. A linear transformation is a function T : V ! W , where V and W are F-vector spaces, satisfying (1) T (u + v) = T (u) + T (v) (2) T (cv) = cT (u) where u; v 2 V and c 2 F. Recall the definitions of one-to-one, onto and bijective. Definition. A function f : X ! Y is (a) One-to-one if f(x1) = f(x2) implies x1 = x2. (b) Onto if for all y 2 Y there is an x 2 X such that f(x) = y. (c) Bijective if it is both one-to-one and onto. Remark. Let f : X ! Y be a function. (1) f is 1-1 () there is a function g : im(f) ! X such that gf = idX . (2) f is onto () there is a function h : Y ! X such that fh = idY . Definition. A bijective linear transformation is called an isomorphism. Examples. 1 Recall the vector space Mat(n; m; F). Define a map nm T : Mat(n; m; F) −! F 0 1 a11 B a21 C 2 3 B . C a ··· a B . C 11 1m B . C 6 . .. 7 B a C A = 4 . 5 7−! B n1 C B a C an1 ··· anm B 12 C B . C @ . A anm There is a clear candidate for an inverse function to T : nm S : F −! Mat(n; m; F) 0 1 a11 B a21 C B . C 2 3 B . C a ··· a B . C 11 1m B a C 6 . .. 7 B n1 C 7−! 4 . 5 B a C B 12 C an1 ··· anm B . C @ . A anm and checking T is linear is trivial, so T is an isomorphism. This confirms our earlier ∼ nm comment that Mat(n; m; F) = F . 4 1.2 Vector Spaces and Linear Transformations 1 The Algebra of Vector Spaces 2 Let X be a finite set, say X = fx1; : : : ; xng. Define n X T : F −! F 0 1 a1 . f : X ! F B . C 7−! @ A xi 7! ai an One can verify that this gives an isomorphism. 3 The function Int : C[0; 1] −! R Z 1 f 7−! f(x) dx 0 is a linear transformation since we know from calculus that integrals are linear. However Int is definitely not an isomorphism since there are many continuous functions on [0,1] that have integral 0. 4 Let A 2 Mat(n; m; F) be given by 2 3 a11 ··· a1m 6 . .. 7 A = 4 . 5 an1 ··· anm Then we can define a linear transformation m n TA : F −! F v 7−! Av where Av denotes the usual matrix-vector multiplication. In many branches of algebra there is a notion of subsets of an algebraic object which have the same properties as the larger object.
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