Linear Algebra I MAT 2141 Fall 2018 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License Contents Preface 4 1 Vector spaces5 1.1 Fields.......................................5 1.2 Vector spaces...................................7 1.3 Some properties of vector spaces......................... 11 1.4 Linear combinations............................... 14 1.5 Subspaces..................................... 16 2 Linear maps 21 2.1 Definition and examples............................. 21 2.2 Kernel and image................................. 24 2.3 Vector spaces of linear maps........................... 27 2.4 Isomorphisms................................... 31 3 Structure of vector spaces 36 3.1 Spans and generating sets............................ 36 3.2 Linear dependence/independence........................ 38 3.3 Finitely generated vector spaces......................... 42 3.4 Basis and dimension............................... 45 3.5 The Dimension Theorem............................. 51 3.6 Dimensions of spaces of linear maps....................... 55 3.7 Dual spaces.................................... 56 4 Matrices 62 4.1 The matrix of a linear map............................ 62 4.2 Change of bases and similar matrices...................... 66 4.3 Gaussian elimination............................... 70 4.4 The rank of a matrix............................... 72 5 Determinants and multlinear maps 76 5.1 Multilinear maps................................. 76 5.2 The determinant................................. 79 5.3 Characterizing properties of the determinant.................. 82 5.4 Other properties of the determinant....................... 85 2 Contents 3 6 Inner product spaces 88 6.1 Definitions..................................... 88 6.2 Orthogonality................................... 91 6.3 Adjoints...................................... 100 7 Diagonalization 104 7.1 Eigenvectors, eigenvalues, and diagonalization................. 104 7.2 Criteria for diagonalization............................ 109 7.3 Self-adjoint operators and symmetric matrices................. 112 7.4 Diagonalization of self-adjoint operators.................... 115 7.5 Rigid motions................................... 121 A A taste of abstract algebra 125 A.1 Operations on sets................................ 125 A.2 Use of parentheses................................ 126 A.3 Identity elements................................. 128 A.4 Invertible elements................................ 129 A.5 Monoids...................................... 131 A.6 Fields....................................... 135 B Quotient spaces and the First Isomorphism Theorem 140 B.1 Equivalence relations and quotient sets..................... 140 B.2 Quotient vector spaces.............................. 145 B.3 The First Isomorphism Theorem......................... 147 B.4 Another proof of the Dimension Theorem.................... 150 Index 153 Preface These are lecture notes for the course Linear Algebra I (MAT 2141) at the University of Ottawa. In this course, we will take a more abstract approach to linear algebra than the one taken in MAT 1341 (the prerequisite for this course). Instead of working only with real or complex numbers, we will generalize to the setting where our coefficients lie in a field. The real numbers and complex numbers are both examples of fields, but there are others as well. We will revisit familiar topics such as matrices, linear maps, determinants, and diagonalization, but now in this more general setting and at a deeper level. We will also discuss more advanced topics such as dual spaces, multilinear maps, and inner product spaces. Compared to MAT 1341, this course will concentrate more on justification of results and mathematical rigour as opposed to computation. Almost all results will be accompanied by a proof and students will be expected to do proofs on assignments and exams. The appendices contains some introductory abstract algebra and discuss the topic of quotient vector spaces, including the important First Isomorphism Theorem. This material will not be covered in the course, and is included here only for the interested student who would like to explore delve further into the subject matter. Notation: In this course N = Z≥0 = f0; 1; 2; 3;::: g denotes the set of nonnegative integers. If Y is a subset of a set X, then X n Y = fx 2 X j x 62 Y g: Acknowledgement: Portions of these notes are based on lecture notes by Barry Jessup, Daniel Daigle, and Kirill Zainoulline. Alistair Savage Course website: https://alistairsavage.ca/mat2141 4 Chapter 1 Vector spaces In this chapter we introduce vector spaces, which form the central subject of the course. You have seen vector spaces in MAT 1341, and so much of the material in this chapter should be familiar. The topics in this chapter roughly correspond to [Tre, x1.1, x1.2, x1.7]. 1.1 Fields In MAT 1341, you did linear algebra over the real numbers and complex numbers. In fact, linear algebra can be done in a much more general context. For much of linear algebra, one can consider \scalars" from any mathematical object known as a field. The topic of fields is an interesting subject in its own right. We will not undertake a detailed study of fields in this course. The interested student is referred to Appendix A.6 for a discussion of this topic. For the purposes of this course, we will use the word field to mean a subfield of the complex numbers or a finite field, which we now explain. Definition 1.1.1 (Subfield of C). A subfield of the complex numbers C is a subset F ⊆ C that (a) contains 1, (b) is closed under addition (that is, x + y 2 F for all x; y 2 F ), (c) is closed under subtraction (that is, x − y 2 F for all x; y 2 F ), (d) is closed under multiplication (that is, xy 2 F for all x; y 2 F ), (e) is closed under taking the multiplicative inverse of a nonzero element (that is, x−1 2 F for all x 2 F , x 6= 0). Definition 1.1.2 (Finite fields). Let p be a prime number. The finite field with p elements is the set ¯ ¯ Fp = f0; 1;:::; n − 1g of integers modulo p, together with its usual addition and multiplication: x¯ +y ¯ = remainder after dividing x + y by p; x¯ · y¯ = remainder after dividing x · y by p: 5 6 Chapter 1. Vector spaces As long as the context is clear, we sometimes omit the bars and, for instance, write 2 instead of 2.¯ Example 1.1.3. (The field F2) We have F2 = f0; 1g, where 0 6= 1. The operations + and · on F are defined by: + 0 1 · 0 1 0 0 1 and 0 0 0 : 1 1 0 1 0 1 Example 1.1.4. In F5, we have 3¯ · 4¯ = 2¯; 2¯ + 4¯ = 1¯; −(2)¯ = 3¯; 2¯−1 = 3¯: Remark 1.1.5. We require p to be prime in Definition 1.1.2 since we want all nonzero elements of a field to have multiplicative inverses. See Exercise 1.1.3. For the purposes of this course, the term field will mean either a subfield of C or a finite field. (See Definition A.6.1 for the more general definition of a field.) Examples 1.1.6. (a) C, R, and Q are fields. (b) The integers Z are not a field since they are not closed under taking the inverse of a nonzero element. (c) The set R≥0 = fx 2 R j x ≥ 0g is not a field because it is not closed under subtraction. p Example 1.1.7 (The field Q( 2)). Let p p Q( 2) := fx + y 2 j x; y 2 Qg: Then p Q(Q( 2) (R: p p The factp that Q 6= Q( 2) follows from the fact that 2 isp not a rationalp number. The fact that Q( 2) 6= R follows, for example, from the fact that p3 62 Q( 2) (see Exercise 1.1.2). We leave it as an exercise (Exercise 1.1.1) to show that Q( 2) is a field. p Examples 1.1.8. (a) Q( 3) is also a subfield of C. p (b) Q(i) = Q( −1) is a subfield of C. We have Q(Q(i) (C and Q(i) 6⊆ R. We write F × for the set of nonzero elements of F , that is F × = F n f0g: We will see that most of the linear algebra that you saw in MAT 1341 can be done over any field. That is, fields can form the \scalars" in systems of equations and vector spaces. For example, we can solve the system x − 2x = 2 1 2 (1.1) x1 + x2 = 0 in R, C, Q, or F3 (Exercise 1.1.4). 1.2. Vector spaces 7 Exercises. p 1.1.1. Prove that Q( 2), as defined in Example 1.1.7 is a subfield of C. p p p p 1.1.2. Show that 3 62 Q( 2) and so Q( p2) and Q(p 3) are different fields. Hint: Prove the result by contradiction. Assume that 3p = a + bp 2 for some a; b 2 Q. Show that this leads to a contradiction, using the fact that 2 and 3 are irrational. 1.1.3. Consider the set F6 of integers modulo 6, together with the multiplication and addition of Definition 1.1.2. Show that 2¯ has no multiplicative inverse. 1.1.4. Find all the solutions to the system (1.1) over the fields R, C, Q, and F3. 1.2 Vector spaces For the remainder of the chapter, F is a field. Definition 1.2.1 (Vector space). A vector space over F is • a set V (whose objects are called vectors), • a binary operation + on V called vector addition, and • scalar multiplication: for each c 2 F and v 2 V , an element cv 2 V , called the scalar product of c and v, such that the following axioms are satisfied: (V1) For all u; v 2 V , we have u + v = v + u.(commutativity of vector addition) (V2) For all u; v; w 2 V , we have (u + v) + w = u + (v + w). (associativity of vector addition) (V3) There is an element 0 2 V such that, for all v 2 V , v + 0 = 0 + v = v.