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AAA Part IB of the Mathematical Tripos of the University of Cambridge Michaelmas 2012 Linear Algebra Lectured by: Notes by: Prof. I. Grojnowski Alex Chan Comments and corrections should be sent to [email protected]. This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. The following resources are not endorsed by the University of Cambridge. Printed Friday, 11 January 2013. Course schedule Definition of a vector space (over R or C), subspaces, the space spanned by a subset. Linear independence, bases, dimension. Direct sums and complementary subspaces. [3] Linear maps, isomorphisms. Relation between rank and nullity. The space of linear maps from U to V , representation by matrices. Change of basis. Row rank and column rank. [4] Determinant and trace of a square matrix. Determinant of a product of two matrices and of the inverse matrix. Determinant of an endomorphism. The adjugate matrix. [3] Eigenvalues and eigenvectors. Diagonal and triangular forms. Characteristic and min- imal polynomials. Cayley-Hamilton Theorem over C. Algebraic and geometric multi- plicity of eigenvalues. Statement and illustration of Jordan normal form. [4] Dual of a finite-dimensional vector space, dual bases and maps. Matrix representation, rank and determinant of dual map. [2] Bilinear forms. Matrix representation, change of basis. Symmetric forms and their link with quadratic forms. Diagonalisation of quadratic forms. Law of inertia, classification by rank and signature. Complex Hermitian forms. [4] Inner product spaces, orthonormal sets, orthogonal projection, V = W ⊕ W ?. Gram- Schmidt orthogonalisation. Adjoints. Diagonalisation of Hermitian matrices. Orthogo- nality of eigenvectors and properties of eigenvalues. [4] Contents 1 Vector spaces 3 1.1 Definitions ................................... 3 1.2 Subspaces ................................... 5 1.3 Bases ..................................... 6 1.4 Linear maps and matrices .......................... 10 1.5 Conservation of dimension: the Rank-nullity theorem ........... 16 1.6 Sums and intersections of subspaces .................... 21 2 Endomorphisms 25 2.1 Determinants ................................. 25 3 Jordan normal form 35 3.1 Eigenvectors and eigenvalues ........................ 35 3.2 Cayley-Hamilton theorem .......................... 41 3.3 Combinatorics of nilpotent matrices .................... 45 3.4 Applications of JNF ............................. 48 4 Duals 51 5 Bilinear forms 55 5.1 Symmetric forms ............................... 57 5.2 Anti-symmetric forms ............................ 62 6 Hermitian forms 67 6.1 Inner product spaces ............................. 69 6.2 Hermitian adjoints for inner products ................... 72 3 1 Vector spaces 1.1 Definitions 5 Oct We start by fixing a field, F. We say that F is a field if: • F is an abelian group under an operation called addition, (+), with additive iden- tity 0; • Fnf0g is an abelian group under an operation called multiplication, (·), with mul- tiplicative identity 1; • Multiplication is distributive over addition; that is, a (b + c) = ab + ac for all a; b; c 2 F. R C Fields we’ve encountered before include the reals , the complexp numbersn p , the ringo of integers modulo p, Z=p = Fp, the rationals Q, as well as Q( 3 ) = a + b 3 : a; b 2 Q , ... Everything we will discuss works over any field, but it’s best to have R and C in mind, since that’s what we’re most familiar with. Definition. A vector space over F is a tuple (V; +; ·) consisting of a set V , opera- tions + : V × V ! V (vector addition) and · : F × V ! V (scalar multiplication) such that (i) (V; +) is an abelian group, that is: • Associative: for all v1; v2; v3 2 V , (v1 + v2) + v3 = v1 + (v2 + v3); • Commutative: for all v1; v2 2 V , v1 + v2 = v2 + v1; • Identity: there is some (unique) 0 2 V such that, for all v 2 V , 0 + v = v = v + 0; • Inverse: for all v 2 V , there is some u 2 V with u + v = v + u = 0. This inverse is unique, and often denoted −v. (ii) Scalar multipication satisfies • Associative: for all λ1; λ2 2 F, v 2 V , λ1 · (λ2 · v) = (λ1λ2) · v; • Identity: for all v 2 V , the unit 1 2 F acts by 1 · v = v; • · distributes over +V : for all λ 2 F, v1; v2 2 V , λ·(v1 + v2) = λ·v1+λ·v2; • +F distributes over ·: for all λ1; λ2 2 F, v 2 V , (λ1 + λ2)·v = λ1·v+λ2·v; We usually say “the vector space V ” rather than (V; +; ·). 4 j Linear Algebra Let’s look at some examples: Examples 1.1. (i) f0g is a vector space. (ii) Vectors in the plane under vector addition form a vector{ space. } n (iii) The space of n-tuples with entries in F, denoted F = (a1; : : : ; an) : ai 2 F with component-wise addition (a1; : : : ; an) + (b1; : : : ; bn) = (a1 + b1; : : : ; an + bn) and scalar multiplication λ · (a1; : : : ; an) = (λa1; : : : ; λan) Proving that this is a vector space is an exercise. It is also a special case of the next example. (iv) Let X be any set, and FX = ff : X ! Fg be the set of all functions X ! F. This is a vector space, with addition defined pointwise: (f + g)(x) = f(x) + g(x) and multiplication also defined pointwise: (λ · f)(x) = λf(x) if λ 2 F, f; g 2 FX , x 2 X. If X = f1; : : : ; ng, then FX = Fn and we have the previous example. Proof that FX is a vector space. • As + in F is commutative, we have (f + g)(x) = f(x) + g(x) = g(x) + f(x) = (g + f)(x); so f + g = g + f. Similarly, f in F associative implies f + (g + h) = (f + g) + h, and that (−f)(x) = −f(x) and 0(x) = 0. • Axioms for scalar multiplication follow from the relationship between · and + in F. Check this yourself! (v) C is a vector space over R. Lemma 1.2. Let V be a vector space over F. (i) For all λ 2 F, λ · 0 = 0, and for all v 2 V , 0 · v = 0. (ii) Conversely, if λ · v = 0 and λ 2 F has λ =6 0, then v = 0. (iii) For all v 2 V , −1 · v = −v. Proof. (i) λ · 0 = λ · (0 + 0) = λ · 0 + λ · 0 =) λ · 0 = 0. 0 · v = (0 + 0) · v = 0 · v + 0 · v =) 0 · v = 0. (ii) As λ 2 F, λ =6 0, there exists λ−1 2 F such that λ−1λ = 1, so v = (λ−1λ) · v = λ−1 (λ · v), hence if λ · v = 0, we get v = λ−1 · 0 = 0 by (i). (iii) 0 = 0 · v = (1 + (−1)) · v = 1 · v + (−1 · v) = v + (−1 · v) =) −1 · v = −v. We will write λv rather than λ · v from now on, as the lemma means this will not cause any confusion. 5 1.2 Subspaces Definition. Let V be a vector space over F. A subset U ⊆ V is a vector subspace (or just a subspace), written U ≤ V , if the following holds: (i) 0 2 U; (ii) If u1; u2 2 U, then u1 + u2 2 U; (iii) If u 2 U, λ 2 F, then λu 2 U. Equivalently, U is a subspace if U ⊆ V , U =6 ; (U is non-empty) and for all u; v 2 U, λ, µ 2 F, λu + µv 2 U. Lemma 1.3. If V is a vector space over F and U ≤ V , then U is a vector space over F under the restriction of the operations + and · on V to U. (Proof is an exercise.) Examples 1.4. (i) f0g and V are always subspaces of V . n+m n+m (ii) f(r1; : : : ; rn; 0;:::; 0) : ri 2 Rg ⊆ R is a subspace of R . (iii) The following are all subspaces of sets of functions: { } C1(R) = f : R ! R j f continuous and differentiable { } ⊆ C(R) = f : R ! R j f continuous ⊆ RR = ff : R ! Rg : Proof. f; g continuous implies f + g is, and λf is, for λ 2 R; the zero function is continuous, so C(R) is a subspace of RR, similarly for C1(R). (iv) Let X be any set, and write { } X F[X] = (F )fin = f : X ! F j f(x) =6 0 for only finitely many x 2 X : This is the set of finitely supported functions, which is is a subspace of FX . X Proof that this is a subspace. f(x) = 0 =) λ f(x) = 0, so if f 2 (F )fin, then so is λf. Similarly, − (f + g) 1 (Fnf0g) ⊆ f −1(Fnf0g) [ g−1(Fnf0g) and if these two are finite, so is the LHS. Special case. Consider the case X = N, so { } N F[N] = (F )fin = (λ0; λ1;:::) j only finitely many λi are non-zero : We write xi for the function which sends i 7! 1, j 7! 0 if j =6 i; that is, for the tuple (0;:::; 0; 1; 0;:::) in the ith place. Thus {P } F[N] = λi j only finitely many λi non-zero : 6 j Linear Algebra Note that we can do better than a vector space here; we can define multipli- cation by (P )(P ) P i j i+j λi x µj x = λi µj · x : This is still in F[N]. It is more usual to denote this F[x], the polynomials in x over F (and this is a formal definition of the polynomial ring). 1.3 Bases 8 Oct Definition. Suppose V is a vector space over F, and S ⊆ V is a subset of V . Then v is a linear combination of elements of S if there is some n > 0 and λ1; : : : ; λn 2 F, v1; : : : ; vn 2 S such that v = λ1v1 + ··· + λnvn or if v = 0.
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