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Linear Algebra II Linear Algebra II Peter Philip∗ Lecture Notes Originally Created for the Class of Spring Semester 2019 at LMU Munich Includes Subsequent Corrections and Revisions† September 19, 2021 Contents 1 Affine Subspaces and Geometry 4 1.1 AffineSubspaces ............................... 4 1.2 AffineHullandAffineIndependence . 6 1.3 AffineBases.................................. 10 1.4 BarycentricCoordinates and Convex Sets. ..... 12 1.5 AffineMaps .................................. 14 1.6 AffineGeometry................................ 20 2 Duality 22 2.1 LinearFormsandDualSpaces. 22 2.2 Annihilators.................................. 27 2.3 HyperplanesandLinearSystems . 31 2.4 DualMaps................................... 34 3 Symmetric Groups 38 ∗E-Mail: [email protected] †Resources used in the preparation of this text include [Bos13, For17, Lan05, Str08]. 1 CONTENTS 2 4 Multilinear Maps and Determinants 46 4.1 MultilinearMaps ............................... 46 4.2 Alternating Multilinear Maps and Determinants . ...... 49 4.3 DeterminantsofMatricesandLinearMaps . .... 57 5 Direct Sums and Projections 71 6 Eigenvalues 76 7 Commutative Rings, Polynomials 87 8 CharacteristicPolynomial,MinimalPolynomial 106 9 Jordan Normal Form 119 10 Vector Spaces with Inner Products 136 10.1 Definition,Examples . 136 10.2 PreservingNorm,Metric,InnerProduct . 138 10.3Orthogonality .................................145 10.4TheAdjointMap ...............................152 10.5 Hermitian,Unitary,andNormalMaps . 158 11 Definiteness of Quadratic Matrices over K 172 A Multilinear Maps 175 B Polynomials in Several Variables 176 C Quotient Rings 192 D Algebraic Field Extensions 199 D.1 BasicDefinitionsandProperties . 199 D.2 AlgebraicClosure...............................205 CONTENTS 3 References 212 1 AFFINE SUBSPACES AND GEOMETRY 4 1 Affine Subspaces and Geometry 1.1 Affine Subspaces Definition 1.1. Let V be a vector space over the field F . Then M V is called an affine subspace of V if, and only if, there exists a vector v V and a (vector)⊆ subspace U V such that M = v + U. We define dim M := dim U∈ to be the dimension of M (this⊆ notion of dimension is well-defined by the following Lem. 1.2(a)). — Thus, the affine subspaces of a vector space V are precisely the translations of vector subspaces U of V , i.e. the cosets of subspaces U, i.e. the elements of quotient spaces V/U. Lemma 1.2. Let V be a vector space over the field F . (a) If M is an affine subspace of V , then the vector subspace corresponding to M is unique, i.e. if M = v1 + U1 = v2 + U2 with v1,v2 V and vector subspaces U1, U2 V , then ∈ ⊆ U = U = u v : u,v M . (1.1) 1 2 { − ∈ } (b) If M = v + U is an affine subspace of V , then the vector v in this representation is unique if, and only if, U = 0 . { } Proof. (a): Let M = v + U with v V and vector subspace U V . Moreover, let 1 1 1 ∈ 1 ⊆ U := u v : u,v M . It suffices to show U1 = U. Suppose, u1 U1. Since v1 M and v{+−u M, we∈ have} u = v + u v U, showing U U.∈ If a U, then there∈ 1 1 ∈ 1 1 1 − 1 ∈ 1 ⊆ ∈ are u1,u2 U1 such that a = v1 + u1 (v1 + u2) = u1 u2 U1, showing U U1, as desired. ∈ − − ∈ ⊆ (b): If U = 0 , then M = v and v is unique. If M = v + U with 0 = u U, then M = v + U ={ v} + u + U with{ v}= v + u. 6 ∈ 6 Definition 1.3. In the situation of Def. 1.1, we call affine subspaces of dimension 0 points, of dimension 1 lines, and of dimension 2 planes – in R2 and R3, such objects are easily visualized and they then coincide with the points, lines, and planes with which one is already familiar. — Affine spaces and vector spaces share many structural properties. In consequence, one can develop a theory of affine spaces that is in many respects analogous to the theory 1 AFFINE SUBSPACES AND GEOMETRY 5 of vector spaces, as will be illustrated by some of the notions and results presented in the following. We start by defining so-called affine combinations, which are, for affine spaces, what linear combinations are for vector spaces: Definition 1.4. Let V be a vector space over the field F with v1,...,vn V and n ∈ λ1,...,λn F , n N. Then i=1 λi vi is called an affine combination of v1,...,vn if, ∈ n ∈ and only if, λi = 1. i=1 P Theorem 1.5.P Let V be a vector space over the field F , = M V . Then M is an affine subspace of V if, and only, if M is closed under∅ affine 6 combinations.⊆ More precisely, the following statements are equivalent: (i) M is an affine subspace of V . n n (ii) If n N, v1,...,vn M, and λ1,...,λn F with i=1 λi =1, then i=1 λi vi M. ∈ ∈ ∈ ∈ P P If char F =2, then (i) and (ii) are also equivalent to1 6 (iii) If v ,v M and λ ,λ F with λ + λ =1, then λ v + λ v M. 1 2 ∈ 1 2 ∈ 1 2 1 1 2 2 ∈ Proof. Exercise. The following Th. 1.6 is the analogon of [Phi19, Th. 5.7] for affine spaces: Theorem 1.6. Let V be a vector space over the field F . (a) Let I = be an index set and (M ) a family of affine subspaces of V . Then the 6 ∅ i i∈I intersection M := i∈I Mi is either empty or it is an affine subspace of V . (b) In contrast to intersections,T unions of affine subspaces are almost never affine sub- spaces. More precisely, if M1 and M2 are affine subspaces of V and char F =2 (i.e. 1 = 1 in F ), then 6 6 − M M is an affine subspace of V M M M M (1.2) 1 ∪ 2 ⇔ 1 ⊆ 2 ∨ 2 ⊆ 1 (where “ ” also holds for char F =2, but cf. Ex. 1.7 below). ⇐ 1 For char F = 2, (iii) does not imply (i) and (ii): Let F := Z2 = 0, 1 . Let V be a vector space over F with #V 4 (e.g. V = F 2). Let p,q,r V be distinct, M{:= }p,q,r (i.e. #M = 3). If ≥ ∈ { } λ1,λ2 F with λ1 +λ2 = 1, then (λ1,λ2) (0, 1), (1, 0) and (iii) is trivially true. On the other hand, v := p ∈+ q + r is an affine combination of p,q,r∈ { , since 1+1+1=1} in F ; but v / M: v = p + q + r = p implies q = r = r, and v = r, q likewise leads to a contradiction (this counterexample∈ was pointed out by Robin Mader).− 1 AFFINE SUBSPACES AND GEOMETRY 6 Proof. (a): Let M = . We use the characterization of Th. 1.5(ii) to show M is an 6 ∅ n affine subspace: If n N, v1,...,vn M, and λ1,...,λn F with k=1 λk = 1, then n ∈ ∈ ∈ v := k=1 λk vk Mi for each i I, implying v M. Thus, M is an affine subspace of V . ∈ ∈ ∈ P P (b): If M1 M2, then M1 M2 = M2, which is an affine subspace of V . If M2 M , then M⊆ M = M ,∪ which is an affine subspace of V . For the converse, we⊆ 1 1 ∪ 2 1 now assume char F = 2, M1 M2, and M1 M2 is an affine subspace of V . We have to show M 6 M . Let 6⊆m M M ∪and m M . Since M M is an 2 ⊆ 1 1 ∈ 1 \ 2 2 ∈ 2 1 ∪ 2 affine subspace, m2 + m2 m1 M1 M2 by Th. 1.5(ii). If m2 + m2 m1 M2, then m = m + m (m− + m∈ m∪) M , in contradiction to m /−M .∈ Thus, 1 2 2 − 2 2 − 1 ∈ 2 1 ∈ 2 m2 + m2 m1 M1. Since char F = 0, we have 2 := 1+1 = 0 in F , implying m = 1 (m−+ m ∈ m )+ 1 m M , i.e.6 M M . 6 2 2 2 2 − 1 2 1 ∈ 1 2 ⊆ 1 2 Example 1.7. Consider F := Z2 = 0, 1 and the vector space V := F over F . Then M := U := (0, 0), (1, 0) = (1, 0){ is} a vector subspace and, in particular, an affine 1 1 { } h{ }i subspace of V . The set M2 := (0, 1) + U1 = (0, 1), (1, 1) is also an affine subspace. Then M M = V is an affine subspace, even{ though neither} M M nor M M . 1 ∪ 2 1 ⊆ 2 2 ⊆ 1 1.2 Affine Hull and Affine Independence Next, we will define the affine hull of a subset A of a vector space, which is the affine analogon to the linear notion of the span of A (which is sometimes also called the linear hull of A): Definition 1.8. Let V be a vector space over the field F , = A V , and ∅ 6 ⊆ := M (V ): A M M is affine subspace of V , M ∈P ⊆ ∧ where we recall that (V ) denotes the power set of V . Then the set P aff A := M M∈M \ is called the affine hull of A. We call A a generating set of aff A. — The following Prop. 1.9 is the analogon of [Phi19, Prop. 5.9] for affine spaces: Proposition 1.9. Let V be a vector space over the field F and = A V . ∅ 6 ⊆ (a) aff A is an affine subspace of V , namely the smallest affine subspace of V containing A. 1 AFFINE SUBSPACES AND GEOMETRY 7 (b) aff A is the set of all affine combinations of elements from A, i.e. n n aff A = λ a : n N λ ,...,λ F a ,...,a A λ =1 .
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