Vector Space Theory A course for second year students by Robert Howlett typesetting by TEX Contents Copyright University of Sydney Chapter 1: Preliminaries 1 §1a Logic and commonSchool sense of Mathematics and Statistics 1 §1b Sets and functions 3 §1c Relations 7 §1d Fields 10 Chapter 2: Matrices, row vectors and column vectors 18 §2a Matrix operations 18 §2b Simultaneous equations 24 §2c Partial pivoting 29 §2d Elementary matrices 32 §2e Determinants 35 §2f Introduction to eigenvalues 38 Chapter 3: Introduction to vector spaces 49 §3a Linearity 49 §3b Vector axioms 52 §3c Trivial consequences of the axioms 61 §3d Subspaces 63 §3e Linear combinations 71 Chapter 4: The structure of abstract vector spaces 81 §4a Preliminary lemmas 81 §4b Basis theorems 85 §4c The Replacement Lemma 86 §4d Two properties of linear transformations 91 §4e Coordinates relative to a basis 93 Chapter 5: Inner Product Spaces 99 §5a The inner product axioms 99 §5b Orthogonal projection 106 §5c Orthogonal and unitary transformations 116 §5d Quadratic forms 121 iii Chapter 6: Relationships between spaces 129 §6a Isomorphism 129 §6b Direct sums Copyright University of Sydney 134 §6c Quotient spaces 139 §6d The dual spaceSchool of Mathematics and Statistics 142 Chapter 7: Matrices and Linear Transformations 148 §7a The matrix of a linear transformation 148 §7b Multiplication of transformations and matrices 153 §7c The Main Theorem on Linear Transformations 157 §7d Rank and nullity of matrices 161 Chapter 8: Permutations and determinants 171 §8a Permutations 171 §8b Determinants 179 §8c Expansion along a row 188 Chapter 9: Classification of linear operators 194 §9a Similarity of matrices 194 §9b Invariant subspaces 200 §9c Algebraically closed fields 204 §9d Generalized eigenspaces 205 §9e Nilpotent operators 210 §9f The Jordan canonical form 216 §9g Polynomials 221 Index of notation 228 Index of examples 229 iv 1 Copyright University of Sydney Preliminaries School of Mathematics and Statistics The topics dealt with in this introductory chapter are of a general mathemat- ical nature, being just as relevant to other parts of mathematics as they are to vector space theory. In this course you will be expected to learn several things about vector spaces (of course!), but, perhaps even more importantly, you will be expected to acquire the ability to think clearly and express your- self clearly, for this is what mathematics is really all about. Accordingly, you are urged to read (or reread) Chapter 1 of “Proofs and Problems in Calculus” by G. P. Monro; some of the points made there are reiterated below. §1a Logic and common sense When reading or writing mathematics you should always remember that the mathematical symbols which are used are simply abbreviations for words. Mechanically replacing the symbols by the words they represent should result in grammatically correct and complete sentences. The meanings of a few commonly used symbols are given in the following table. Symbols To be read as { ... | ... } the set of all ... such that ... = is ∈ in or is in > greater than or is greater than Thus, for example, the following sequence of symbols { x ∈ X | x > a }= 6 ∅ is an abbreviated way of writing the sentence The set of all x in X such that x is greater than a is not the empty set. 1 2 Chapter One: Preliminaries When reading mathematics you should mentally translate all symbols in this fashion, and when writing mathematics you should make sure that what you write translates into meaningfulCopyright sentences. University of Sydney Next, you must learn how to write out proofs. The ability to construct proofs is probably the mostSchool important of Mathematics skill for the and student Statistics of pure mathe- matics to acquire. And it must be realized that this ability is nothing more than an extension of clear thinking. A proof is nothing more nor less than an explanation of why something is so. When asked to prove something, your first task is to be quite clear as to what is being asserted, then you must decide why it is true, then write the reason down, in plain language. There is never anything wrong with stating the obvious in a proof; likewise, people who leave out the “obvious” steps often make incorrect deductions. When trying to prove something, the logical structure of what you are trying to prove determines the logical structure of the proof; this observation may seem rather trite, but nonetheless it is often ignored. For instance, it frequently happens that students who are supposed to proving that a state- ment p is a consequence of statement q actually write out a proof that q is a consequence of p. To help you avoid such mistakes we list a few simple rules to aid in the construction of logically correct proofs, and you are urged, whenever you think you have successfully proved something, to always check that your proof has the correct logical structure. • To prove a statement of the form If p then q your first line should be Assume that p is true and your last line Therefore q is true. • The statement p if and only if q is logically equivalent to If p then q and if q then p. To prove it you must do two proofs of the kind described in the preced- ing paragraph. • Suppose that P (x) is some statement about x. Then to prove P (x) is true for all x in the set S Chapter One: Preliminaries 3 your first line should be Let x be an arbitrary element of the set S and your last line Copyright University of Sydney Therefore P (x) holds. School of Mathematics and Statistics • Statements of the form There exists an x such that P (x) is true are proved producing an example of such an x. Some of the above points are illustrated in the examples #1, #2, #3 and #4 at the end of the next section. §1b Sets and functions It has become traditional to base all mathematics on set theory, and we will assume that the reader has an intuitive familiarity with the basic concepts. For instance, we write S ⊆ A (S is a subset of A) if every element of S is an element of A. If S and T are two subsets of A then the union of S and T is the set S ∪ T = { x ∈ A | x ∈ S or x ∈ T } and the intersection of S and T is the set S ∩ T = { x ∈ A | x ∈ S and x ∈ T }. (Note that the ‘or’ above is the inclusive ‘or’—that which is sometimes written as ‘and/or’. In this book ‘or’ will always be used in this sense.) Given any two sets S and T the Cartesian product S × T of S and T is the set of all ordered pairs (s, t) with s ∈ S and t ∈ T ; that is, S × T = { (s, t) | s ∈ S, t ∈ T }. The Cartesian product of S and T always exists, for any two sets S and T . This is a fact which we ask the reader to take on trust. This course is not concerned with the foundations of mathematics, and to delve into formal treatments of such matters would sidetrack us too far from our main purpose. Similarly, we will not attempt to give formal definitions of the concepts of ‘function’ and ‘relation’. 4 Chapter One: Preliminaries Let A and B be sets. A function f from A to B is to be thought of as a rule which assigns to every element a of the set A an element f(a) of the set B. The set A is called the domainCopyrightof f and UniversityB the codomain of Sydney(or target) of f. We use the notation ‘f: A → B’ (read ‘f, from A to B’) to mean that f is a function with domain ASchooland codomain of MathematicsB. and Statistics A map is the same thing as a function. The terms mapping and trans- formation are also used. A function f: A → B is said to be injective (or one-to-one) if and only if no two distinct elements of A yield the same element of B. In other words, f is injective if and only if for all a1, a2 ∈ A, if f(a1) = f(a2) then a1 = a2. A function f: A → B is said to be surjective (or onto) if and only if for every element b of B there is an a in A such that f(a) = b. If a function is both injective and surjective we say that it is bijective (or a one-to-one correspondence). The image (or range) of a function f: A → B is the subset of B con- sisting of all elements obtained by applying f to elements of A. That is, im f = { f(a) | a ∈ A }. An alternative notation is ‘f(A)’ instead of ‘im f’. Clearly, f is surjective if and only if im f = B. The word ‘image’ is also used in a slightly different sense: if a ∈ A then the element f(a) ∈ B is sometimes called the image of a under the function f. The notation ‘a 7→ b’ means ‘a maps to b’; in other words, the function involved assigns the element b to the element a. Thus, ‘a 7→ b’ (under the function f) means exactly the same as ‘f(a) = b’. If f: A → B is a function and C a subset of B then the inverse image or preimage of C is the subset of A f −1(C) = { a ∈ A | f(a) ∈ C }. (The above sentence reads ‘f inverse of C is the set of all a in A such that f of a is in C.’ Alternatively, one could say ‘The inverse image of C under f’ instead of ‘f inverse of C’.) Let f: B → C and g: A → B be functions such that domain of f is the codomain of g.