Linear Algebra II (MAT 3141) Course notes written by Damien Roy with the assistance of Pierre Bel and financial support from the University of Ottawa for the development of pedagogical material in French (Translated into English by Alistair Savage) Fall 2012 Department of Mathematics and Statistics University of Ottawa Contents Preface v 1 Review of vector spaces1 1.1 Vector spaces...................................1 1.2 Vector subspaces.................................3 1.3 Bases........................................4 1.4 Direct sum.....................................7 2 Review of linear maps 13 2.1 Linear maps.................................... 13 2.2 The vector space LK (V; W )........................... 16 2.3 Composition.................................... 18 2.4 Matrices associated to linear maps....................... 21 2.5 Change of coordinates.............................. 24 2.6 Endomorphisms and invariant subspaces.................... 26 3 Review of diagonalization 33 3.1 Determinants and similar matrices....................... 33 3.2 Diagonalization of operators........................... 36 3.3 Diagonalization of matrices............................ 41 4 Polynomials, linear operators and matrices 47 4.1 The ring of linear operators........................... 47 4.2 Polynomial rings................................. 48 4.3 Evaluation at a linear operator or matrix.................... 54 5 Unique factorization in euclidean domains 59 5.1 Divisibility in integral domains......................... 59 5.2 Divisibility in terms of ideals........................... 62 5.3 Euclidean division of polynomials........................ 64 5.4 Euclidean domains................................ 65 5.5 The Unique Factorization Theorem....................... 69 5.6 The Fundamental Theorem of Algebra..................... 72 i ii CONTENTS 6 Modules 75 6.1 The notion of a module.............................. 75 6.2 Submodules.................................... 80 6.3 Free modules................................... 83 6.4 Direct sum..................................... 84 6.5 Homomorphisms of modules........................... 86 7 The structure theorem 91 7.1 Annihilators.................................... 91 7.2 Modules over a euclidean domain........................ 99 7.3 Primary decomposition.............................. 102 7.4 Jordan canonical form.............................. 108 8 The proof of the structure theorem 119 8.1 Submodules of free modules, Part I....................... 119 8.2 The Cayley-Hamilton Theorem......................... 124 8.3 Submodules of free modules, Part II....................... 127 8.4 The column module of a matrix......................... 130 8.5 Smith normal form................................ 134 8.6 Algorithms..................................... 141 9 Duality and the tensor product 155 9.1 Duality....................................... 155 9.2 Bilinear maps................................... 160 9.3 Tensor product.................................. 166 9.4 The Kronecker product.............................. 172 9.5 Multiple tensor products............................. 176 10 Inner product spaces 179 10.1 Review....................................... 179 10.2 Orthogonal operators............................... 186 10.3 The adjoint operator............................... 190 10.4 Spectral theorems................................. 193 10.5 Polar decomposition............................... 199 Appendices 205 A Review: groups, rings and fields 207 A.1 Monoids...................................... 207 A.2 Groups....................................... 209 A.3 Subgroups..................................... 212 A.4 Group homomorphisms.............................. 213 A.5 Rings........................................ 215 A.6 Subrings...................................... 219 A.7 Ring homomorphisms............................... 220 A.8 Fields....................................... 222 CONTENTS iii B The determinant 223 B.1 Multilinear maps................................. 223 B.2 The determinant................................. 228 B.3 The adjugate................................... 231 iv CONTENTS Preface These notes are aimed at students in the course Linear Algebra II (MAT 3141) at the University of Ottawa. The first three chapters contain a revision of basic notions covered in the prerequisite course Linear Algebra I (MAT 2141): vector spaces, linear maps and diagonalization. The rest of the course is divided into three parts. Chapters 4 to 8 make up the main part of the course. They culminate in the structure theorem for finite type modules over a euclidean ring, a result which leads to the Jordan canonical form of operators on a finite dimensional complex vector space, as well as to the structure theorem for abelian groups of finite type. Chapter 9 involves duality and the tensor product. Then Chapter 10 treats the spectral theorems for operators on an inner product space (after a review of some notions seen in the prerequisite course MAT 2141). With respect to the initial plan, they still lack an eleventh chapter on hermitian spaces, which will be added later. The notes are completed by two appendices. Appendix A provides a review of basic algebra that are used in the course: groups, rings and fields (and the morphisms betwen these objects). Appendix B covers the determinants of matrices with entries in an arbitrary commutative ring and their properties. Often the determinant is simply defined over a field and the goal of this appendix is to show that this notion easily generalizes to an arbitrary commutative ring. This course provides students the chance to learn the theory of modules that is not covered, due to lack of time, in other undergraduate algebra courses. To simplify matters and make the course easier to follow, I have avoided introducing quotient modules. This topic could be added through exercises (for more motivated students). According to comments I have received, students appreciate the richness of the concepts presented in this course and that they open new horizons to them. I thank Pierre Bel who greatly helped with the typing and updating of these notes. I also thank the Rectorat aux Etudes´ of the University of Ottawa for its financial support of this project via the funds for the development of French teaching materials. This project would not have been possible without their help. Finally, I thank my wife Laura Dutto for her help in the revision of preliminary versions of these notes and the students who took this course with me in the Winter and Fall of 2009 for their participation and their comments. Damien Roy (as translated by Alistair Savage) Ottawa, January 2010. v Chapter 1 Review of vector spaces This chapter, like the two that follow it, is devoted to a review of fundamental concepts of linear algebra from the prerequisite course MAT 2141. This first chapter concerns the main object of study in linear algebra: vector spaces. We recall here the notions of a vector space, vector subspace, basis, dimension, coordinates, and direct sum. The reader should pay special attention to the notion of direct sum, since it will play a vital role later in the course. 1.1 Vector spaces Definition 1.1.1 (Vector space). A vector space over a field K is a set V with operations of addition and scalar multiplication: V × V −! V K × V −! V and (u; v) 7−! u + v (a; v) 7−! av that satisfies the following axioms: VS1: u + (v + w) = (u + v) + w for all u; v; w 2 V: VS2: u + v = v + u VS3: There exists 0 2 V such that v + 0 = v for all v 2 V . VS4: For all v 2 V , there exists −v 2 V such that v + (−v) = 0. 9 VS5: 1v = v > VS6: a(bv) = (ab)v => for all a; b 2 K and u; v 2 V . VS7: (a + b)v = av + bv > VS8: a(u + v) = au + av ;> These eight axioms can be easily interpreted. The first four imply that (V; +) is an abelian group (cf. Appendix A). In particular, VS1 and VS2 imply that the order in which 1 2 CHAPTER 1. REVIEW OF VECTOR SPACES we add the elements v1;:::; vn of V does not affect their sum, denoted n X v1 + ··· + vn or vi: i=1 Condition VS3 uniquely determines the element 0 of V . We call it the zero!vector of V , the term vector being the generic name used to designate an element of a vector space. For each v 2 V , there exists one and only one vector −v 2 V that satisfies VS4. We call it the additive inverse of v. The existence of the additive inverse allows us to define subtraction in V by u − v := u + (−v): Axioms VS5 and VS6 involve only scalar multiplication, while the Axioms VS7 and VS8 link the two operations. These last axioms require that multiplication by a scalar be distributive over addition on the right and on the left (i.e. over addition in K and in V ). They imply general distributivity formulas: n ! n n n X X X X ai v = aiv and a vi = avi i=1 i=1 i=1 i=1 for all a; a1; : : : ; an 2 K and v; v1;:::; vn 2 V . We will return to these axioms later when we study the notion of a module over a commutative ring, which generalizes the idea of a vector space over a field. Example 1.1.2. Let n 2 N>0. The set 8 9 0 a 1 > 1 > <>B a C => n B 2 C K = B . C ; a1; a2; : : : ; an 2 K >@ . A > > > : an ; of n-tuples of elements of K is a vector space over K for the operations: 0 1 0 1 0 1 0 1 0 1 a1 b1 a1 + b1 a1 c a1 B a C B b C B a + b C B a C B c a C B 2 C B 2 C B 2 2 C B 2 C B 2 C B . C + B . C = B . C and c B . C = B . C : @ . A @ . A @ . A @ . A @ . A an bn an + bn an c an In particular, K1 = K is a vector space over K. Example 1.1.3. More generally, let m; n 2 N>0. The set 8 9 0 a ··· a 1 <> 11 1n => B . C Matm×n(K) = @ . A ; a11; : : : ; amn 2 K > > : am1 : : : amn ; 1.2. VECTOR SUBSPACES 3 of m × n matrices with entries in K is a vector space over K for the usual operations: 0 1 0 1 0 1 a11 ··· a1n b11 ··· b1n a11 + b11 ··· a1n + b1n B . C B . C B . C @ . A + @ . A = @ . A am1 : : : amn bm1 : : : bmn am1 + bm1 : : : amn + bmn 0 1 0 1 a11 ··· a1n c a11 ··· c a1n B .