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LINEAR ALGEBRA Contents 1. Vector Spaces 2 1.1. Definitions LINEAR ALGEBRA SIMON WADSLEY Contents 1. Vector spaces2 1.1. Definitions and examples2 1.2. Linear independence, bases and the Steinitz exchange lemma4 1.3. Direct sum7 2. Linear maps9 2.1. Definitions and examples9 2.2. Linear maps and matrices 11 2.3. The first isomorphism theorem and the rank-nullity theorem 13 2.4. Change of basis 15 2.5. Elementary matrix operations 17 3. Determinants of matrices 18 4. Endomorphisms 23 4.1. Invariants 23 4.2. The Cayley-Hamilton Theorem 28 4.3. Multiplicities of eigenvalues and Jordan Normal Form 32 5. Duality 38 5.1. Dual spaces 38 5.2. Dual maps 40 6. Bilinear Forms 42 6.1. Definitions and Examples 42 6.2. Symmetric bilinear forms and quadratic forms 44 6.3. Hermitian forms 48 7. Inner product spaces 50 7.1. Definitions and basic properties 50 7.2. Gram{Schmidt orthogonalisation 51 7.3. Adjoints 53 7.4. Spectral theory 55 8. Alternating forms 57 1 2 SIMON WADSLEY Lecture 1 1. Vector spaces Linear algebra can be summarised as the study of vector spaces and linear maps between them. This is a second ‘first course' in Linear Algebra. That is to say, we will define everything we use but will assume some familiarity with the concepts (picked up from the IA course Vectors and Matrices for example). 1.1. Definitions and examples. Examples. (1) For each non-negative integer n, the set Rn of column vectors of length n with real entries is a vector space (over R). An (m × n)-matrix A with real entries can be viewed as a linear map Rn ! Rm via v 7! Av. In fact, as we will see, every linear map from Rn ! Rm is of this form. (2) Let X be a set and RX := ff : X ! Rg be equipped with an addition given by (f + g)(x) := f(x) + g(x) and a multiplication by scalars (in R) given by (λf)(x) = λ(f(x)). Then RX is a vector space (over R). (3) If [a; b] is a closed interval in R then C([a; b]; R) := ff 2 R[a;b] j f is continuousg is an R-vector space by restricting the operations on R[a;b]. Similarly C1([a; b]; R) := ff 2 C([a; b]; R) j f is infinitely differentiableg is an R-vector space. (4) The set of (m × n)-matrices with real entries is a vector space over R. Convention. In this course we will use F to denote either R or C. Most of the results will be true for any field F; but since general fields are not officially defined until Groups, Rings and Modules next term we follow the schedules in not addressing that. What do our examples of vector spaces above have in common? In each case we have a notion of addition of `vectors' and scalar multiplication of `vectors' by elements in R. Definition. An F-vector space is an abelian group (V; +) equipped with a function F × V ! V ;(λ, v) 7! λv such that (a) λ(µv) = (λµ)v for all λ, µ 2 F and v 2 V ; (b) λ(u + v) = λu + λv for all λ 2 F and u; v 2 V ; (c)( λ + µ)v = λv + µv for all λ, µ 2 F and v 2 V ; (d)1 v = v for all v 2 V . Note that this means that we can add, subtract and rescale elements in a vector space and these operations behave in the ways that we are used to. Note also that in general a vector space does not come equipped with notions of length or of angle. We will discuss how to recover these at the end of the course. At that point particular properties of the field F will be important. Convention. We will always write 0 to denote the additive identity of a vector space V . By slight abuse of notation we will also write 0 to denote the vector space f0g. LINEAR ALGEBRA 3 Exercise. (1) Convince yourself that all the vector spaces mentioned thus far do indeed satisfy the axioms for a vector space. (2) Show that for any v in any vector space V , 0v = 0 and (−1)v = −v Definition. Suppose that V is a vector space over F. A subset U ⊂ V is a (linear) subspace if (a) for all u1; u2 2 U, u1 + u2 2 U; (b) for all λ 2 F and u 2 U, λu 2 U; (c)0 2 U. Remarks. (1) It is straightforward to see that U ⊂ V is a subspace if and only if U 6= ; and λu1 + µu2 2 U for all u1; u2 2 U and λ, µ 2 F . (2) If U is a subspace of V then U is a vector space under the inherited operations. Examples. 80 1 9 x1 < 3 = 3 (1) @x2A 2 R : x1 + x2 + x3 = t is a subspace of R if and only if t = 0. : x3 ; (2) Let X be a set. We define the support of a function f : X ! F to be suppf := fx 2 X : f(x) 6= 0g: Then ff 2 FX : jsuppfj < 1g is a subspace of FX . Definition. Let V be a vector space over F and S ⊂ V a subset of V . Then the span of S in V , ( n ) X hSi := λisi : λi 2 F; si 2 S; n > 0 i=1 Remark. For any subset S ⊂ V , hSi is the smallest subspace of V containing S. Example. Suppose that V is R3. 80 1 0 1 0 19 80 1 9 < 1 0 1 = < a = If S = @0A ; @1A ; @2A then hSi = @bA : a; b 2 R : : 0 1 2 ; : b ; Note also that every subset of S of order 2 has the same span as S. Example. Let X be a set and for each x 2 X, define δx : X ! F by ( 1 if y = x δx(y) = 0 if y 6= x: X Then hδx : x 2 Xi = ff 2 F : jsuppfj < 1g: Definition. Suppose that U and W are subspaces of a vector space V over F. Then the sum of U and W is the set U + W := fu + w : u 2 U; w 2 W g: Proposition. If U and W are subspaces of a vector space V over F then U \ W and U + W are also subspaces of V . 4 SIMON WADSLEY Proof. Certainly both U \ W and U + W contain 0. Suppose that v1; v2 2 U \ W , u1; u2 2 U, w1; w2 2 W ,and λ, µ 2 F. Then λv1 + µv2 2 U \ W and λ(u1 + w1) + µ(u2 + w2) = (λu1 + µu2) + (λw1 + µw2) 2 U + W: So U \ W and U + W are subspaces of V . *Quotient spaces*. Suppose that V is a vector space over F and U is a subspace of V . Then the quotient group V=U can be made into a vector space over F by definining λ(v + U) = (λv) + U for λ 2 F and v 2 V . Exercise. Justify the claim that this makes V=U into a vector space over F. Lecture 2 1.2. Linear independence, bases and the Steinitz exchange lemma. Definition. Let V be a vector space over F and S ⊂ V . (a) We say that S spans V if V = hSi. (b) We say that S is linearly independent (LI) if, whenever n X λisi = 0 i=1 with λi 2 F, and si distinct elements of S, it follows that λi = 0 for all i. If S is not linearly independent then we say that S is linearly dependent (LD). (c) We say that S is a basis for V if S spans and is linearly independent. 80 1 0 1 0 19 < 1 0 1 = Example. Suppose that V is R3 and S = @0A ; @1A ; @2A . Then S is linearly : 0 1 2 ; 011 001 011 dependent since 1 @0A+2 @1A+(−1) @2A = 0. Moreover S does not span V since 0 1 2 001 @0A is not in hSi. However, every subset of S of order 2 is linearly independent 1 and forms a basis for hSi. Remark. Note that no linearly independent set can contain the zero vector since 1 · 0 = 0. Convention. The span of the empty set h;i is the zero subspace 0. Thus the empty set is a basis of 0. One may consider this to not be so much a convention as the only reasonable interpretation of the definitions of span, linearly independent and basis in this case. Lemma. A subset S of a vector space V over F is linearly dependent if and only if Pn there exist s0; s1; : : : ; sn 2 S distinct and λ1; : : : ; λn 2 F such that s0 = i=1 λisi. LINEAR ALGEBRA 5 P Proof. Suppose that S is linearly dependent so that λisi = 0 for some si 2 S distinct and λi 2 F with λj 6= 0 say. Then X −λi sj = si: λj i6=j Pn Pn Conversely, if s0 = i=1 λisi then (−1)s0 + i=1 λisi = 0. Proposition. Let V be a vector space over F. Then fe1; : : : ; eng is a basis for V Pn if and only if every element v 2 V can be written uniquely as v = i=1 λiei with λi 2 F. Proof. First we observe that by definition fe1; : : : ; eng spans V if and only if every P element v of V can be written in at least one way as v = λiei with λi 2 F.
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