Lecture Notes on Linear and Multilinear Algebra 2301-610 Wicharn Lewkeeratiyutkul Department of Mathematics and Computer Science Faculty of Science Chulalongkorn University August 2014 Contents Preface iii 1 Vector Spaces 1 1.1 Vector Spaces and Subspaces . 2 1.2 Basis and Dimension . 10 1.3 Linear Maps . 18 1.4 Matrix Representation . 32 1.5 Change of Bases . 42 1.6 Sums and Direct Sums . 48 1.7 Quotient Spaces . 61 1.8 Dual Spaces . 65 2 Multilinear Algebra 73 2.1 Free Vector Spaces . 73 2.2 Multilinear Maps and Tensor Products . 78 2.3 Determinants . 95 2.4 Exterior Products . 101 3 Canonical Forms 107 3.1 Polynomials . 107 3.2 Diagonalization . 115 3.3 Minimal Polynomial . 128 3.4 Jordan Canonical Forms . 141 i ii CONTENTS 4 Inner Product Spaces 161 4.1 Bilinear and Sesquilinear Forms . 161 4.2 Inner Product Spaces . 167 4.3 Operators on Inner Product Spaces . 180 4.4 Spectral Theorem . 190 Bibliography 203 Preface This book grew out of the lecture notes for the course 2301-610 Linear and Multilinaer Algebra given at the Deparment of Mathematics, Faculty of Science, Chulalongkorn University that I have taught in the past 5 years. Linear Algebra is one of the most important subjects in Mathematics, with numerous applications in pure and applied sciences. A more theoretical linear algebra course will emphasize on linear maps between vector spaces, while an applied-oriented course will mainly work with matrices. Matrices have an ad- vantage of being easier to compute, while it is easier to establish the results by working with linear maps. This book tries to establish a close connection between the two aspects of the subject. I would like to thank my students who took this course with me and proof- read the early drafts. Special thanks go to Chao Kusollerschariya who provide technical help about latex and suggest several easier proofs, and Detchat Samart who supplied the proofs on polynomials. Please do not distribute. Wicharn Lewkeeratiyutkul iii Chapter 1 Vector Spaces In this chapter, we will study an abstract theory of vector spaces and linear maps between vector spaces. A vector space is a generalization of the space of vectors in the 2- or 3-dimensional Euclidean space. We can add two vectors and multiply a vector by a scalar. In a general framework, we still can add vectors, but the scalars don't have to be numbers; they are required to satisfy some algebraic properties which constitute a field. A vector space is defined to be a non-empty set that satisfies certain axioms that generalize the addition and scalar multiplication of 2 3 vectors in R and R . This will allow our theory to be applicable to a wide range of situations. Once we have some vector spaces, we can construct new vector spaces from existing ones by taking subspaces, direct sums and quotient spaces. We then introduce a basis for a vector space, which can be regarded as choosing a coor- dinate system. Once we fix a basis for the vector space, every other element can be written uniquely as a linear combination of elements in the basis. We also study a linear map between vector spaces. It is a function that preserves the vector space operations. If we fix bases for vector spaces V and W , a linear map from V into W can be represented by a matrix. This will allow the computational aspect of the theory. The set of all linear maps between two vector spaces is a vector space itself. The case when the target space is a scalar field is of particular importance, called the dual space of the vector space. 1 2 CHAPTER 1. VECTOR SPACES 1.1 Vector Spaces and Subspaces Definition 1.1.1. A field is a set F with two binary operations, + and ·, and two distinct elements 0 and 1, satisfying the following properties: (i) 8x; y; z 2 F ,(x + y) + z = x + (y + z); (ii) 8x 2 F , x + 0 = 0 + x = x; (iii) 8x 2 F 9y 2 F , x + y = y + x = 0; (iv) 8x; y 2 F , x + y = y + x; (v) 8x; y; z 2 F ,(x · y) · z = x · (y · z); (vi) 8x 2 F , x · 1 = 1 · x = x; (vii) 8x 2 F − f0g 9y 2 F , x · y = y · x = 1; (viii) 8x; y 2 F , x · y = y · x; (ix) 8x; y; z 2 F , x · (y + z) = x · y + x · z. Properties (i)-(iv) say that (F; +) is an abelian group. Properties (v)-(viii) say that (F − f0g; ·) is an abelian group. Property (ix) is the distributive law for the multiplication over addition. Example 1.1.2. Q, R, C, Zp, where p is a prime number, are fields. Definition 1.1.3. A vector space over a field F is a non-empty set V , together with an addition + : V × V ! V and a scalar multiplication · : F × V ! V , satisfying the following properties: (i) 8u; v; w 2 V ,(u + v) + w = u + (v + w); (ii) 90¯ 2 V 8v 2 V , v + 0¯ = 0¯ + v = v; (iii) 8v 2 V 9 − v 2 V , v + (−v) = (−v) + v = 0;¯ (iv) 8u; v 2 V , u + v = v + u; (v) 8m; n 2 F 8v 2 V ,(m + n) · v = m · v + n · v; 1.1. VECTOR SPACES AND SUBSPACES 3 (vi) 8m 2 F 8u; v 2 V , m · (u + v) = m · u + m · v; (vii) 8m; n 2 F 8v 2 V ,(m · n) · v = m · (n · v); (viii) 8v 2 V , 1 · v = v. Proposition 1.1.4. Let V be a vector space over a field F . Then (i) 8v 2 V , 0 · v = 0¯; (ii) 8k 2 F , k · 0¯ = 0¯; (iii) 8v 2 V 8k 2 F , k · v = 0¯ , k = 0 or v = 0¯; (iv) 8v 2 V , (−1) · v = −v. Proof. Let v 2 V and k 2 F . Then (i) 0 · v + v = 0 · v + 1 · v = (0 + 1) · v = 1 · v = v: Hence 0 · v = 0.¯ (ii) k · 0¯ = k · (0 · 0)¯ = (k · 0) · 0¯ = 0 · 0¯ = 0¯: (iii) If k · v = 0¯ and k 6= 0, then 1 1 v = 1 · v = · k · v = (k · v) = 0¯: k k (iv) (−1) · v + v = (−1) · v + 1 · v = (−1 + 1) · v = 0 · v = 0¯: Hence (−1) · v = −v. Remark. When there is no confusion, we will denote the additive identity 0¯ simply by 0. Example 1.1.5. The following sets with the given addition and scalar multipli- cation are vector spaces. (i) The set of n-tuples whose entries are in F : n F = f(x1; x2; : : : ; xn) j xi 2 F; i = 1; 2; : : : ; ng; with the addition and scalar multiplication given by (x1; : : : ; xn) + (y1; : : : ; yn) = (x1 + y1; : : : ; xn + yn); k(x1; : : : ; xn) = (kx1; : : : ; kxn): 4 CHAPTER 1. VECTOR SPACES (ii) The set of m × n matrices whose entries are in F : Mm×n(F ) = f[aij]m×n j aij 2 F; i = 1; 2; : : : ; m; j = 1; 2; : : : ; ng ; with the usual matrix addition and scalar multiplication. Note that if m = n, we write Mn(F ) for Mn×n(F ). (iii) The set of polynomials over F : n F [x] = fa0 + a1x + ··· + anx j n 2 N [ f0g; ai 2 F; i = 0; 1; : : : ; ng: with the usual polynomial addition and scalar multiplication. (iv) The set of sequences over F : S = f(xn) j xn 2 F for all n 2 Ng; with the addition and scalar multiplication given by (xn) + (yn) = (xn + yn); k(xn) = (kxn): Here we are not concerned with convergence of the sequences. (v) Let X be a non-empty set. The set of F -valued functions on X F(X) = ff : X ! F g with the following addition and scalar multiplication: (f + g)(x) = f(x) + g(x)(x 2 X); (kf)(x) = kf(x)(x 2 X): Once we have some vector spaces to begin with, there are several methods to construct new vector spaces from the old ones. We first consider a subset which is also a vector space under the same operations. Definition 1.1.6. Let V be a vector space over a field F .A subspace of V is a subset of V which is also a vector space over F under the same operations. We write W ≤ V to denote that W is a subspace of V . 1.1. VECTOR SPACES AND SUBSPACES 5 Proposition 1.1.7. Let W be a non-empty subset of a vector space V over a field F . Then the following statements are equivalent: (i) W is a subspace of V ; (ii) 8v 2 W 8w 2 W , v + w 2 W and 8v 2 W 8k 2 F; kv 2 W ; (iii) 8v 2 W 8w 2 W 8α 2 F 8β 2 F , αv + βw 2 W . Proof. We will establish (i) , (ii) and (ii) , (iii). (i) ) (ii). Assume W is a subspace of V . Then W is a vector space over a field F under the restriction of the addition and the scalar multiplication to W . Hence v + w 2 W and kv 2 W for any v; w 2 W and any k 2 F . (ii) ) (i). Assume (ii) holds. Since the axioms of a vector space hold for all elements in V , they also hold for elements in W as well. Since W is non-empty, we can choose an element v 2 W . Then 0¯ = 0 · v 2 W . Moreover, for any v 2 W , −v = (−1) · v 2 W .