164 APPENDIX A. MULTILINEAR ALGEBRA A.1 Vector spaces and linear maps We assume the reader is somewhat familiar with linear algebra, so at least most of this section should be review|its main purpose is to establish notation that is used in the rest of the notes, as well as to clarify the relationship between real and complex vector spaces. Appendix A Throughout this appendix, let F denote either of the fields R or C; we will refer to elements of this field as scalars. Recall that a vector space over F (or simply a real/complex vector space) is a set V together with two Multilinear algebra and index algebraic operations: (vector addition) V V V : (v; w) v + w notation • × ! 7! (scalar multiplication) F V V : (λ, v) λv • × ! 7! One should always keep in mind the standard examples Fn for n 0; as we will recall in a moment, every finite dimensional vector space≥ is Contents isomorphic to one of these. The operations are required to satisfy the A.1 Vector spaces . 164 following properties: A.2 Bases, indices and the summation convention 166 (associativity) (u + v) + w = u + (v + w). • A.3 Dual spaces . 169 (commutativity) v + w = w + v. • A.4 Inner products . 170 (additive identity) There exists a zero vector 0 V such that • 0 + v = v for all v V . 2 A.5 Direct sums . 174 2 (additive inverse) For each v V there is an inverse element A.6 Tensors and multilinear maps . 175 • v V such that v + ( v) = 0.2 (This is of course abbreviated − 2 − v v = 0.) A.7 The tensor product . 179 − (distributivity) For scalar multiplication, (λ + µ)v = λv + µv and A.8 Symmetric and exterior algebras . 182 • λ(v + w) = λv + λw. A.9 Duality and the Hodge star . 188 (scalar associativity) λ(µv) = (λµ)v. A.10 Tensors on manifolds . 190 • (scalar identity) 1v = v for all v V . • 2 Observe that every complex vector space can also be considered a real If linear algebra is the study of vector spaces and linear maps, then vector space, though the reverse is not true. That is, in a complex vector multilinear algebra is the study of tensor products and the natural gener- space, there is automatically a well defined notion of multiplication by real alizations of linear maps that arise from this construction. Such concepts scalars, but in real vector spaces, one has no notion of \multiplication by are extremely useful in differential geometry but are essentially algebraic i". As is also discussed in Chapter 2, such a notion can sometimes (though rather than geometric; we shall thus introduce them in this appendix us- not always) be defined as an extra piece of structure on a real vector space. ing only algebraic notions. We'll see finally in A.10 how to apply them For two vector spaces V and W over the same field F, a map to tangent spaces on manifolds and thus recoverx the usual formalism of tensor fields and differential forms. Along the way, we will explain the A : V W : v Av conventions of \upper" and \lower" index notation and the Einstein sum- ! 7! mation convention, which are standard among physicists but less familiar is called linear if it respects both vector addition and scalar multiplication, in general to mathematicians. meaning it satisfies the relations A(v + w) = Av + Aw and A(λv) = λ(Av) 163 A.1. VECTOR SPACES 165 166 APPENDIX A. MULTILINEAR ALGEBRA for all v; w V and λ F. Linear maps are also sometimes called vector with corresponding definitions for EndR(V ), EndC(V ) and EndC(V ). 2 2 space homomorphisms, and we therefore use the notation Observe that all these sets of linear maps are themselves also vector spaces in a natural way: simply define (A + B)v := Av + Bv and (λA)v := Hom(V; W ) := A : V W A is linear : f ! j g λ(Av). Given a vector space V , a subspace V 0 V is a subset which is closed The symbols L(V; W ) and (V; W ) are also quite common but are not used under both vector addition and scalar multiplication,⊂ i.e. v + w V 0 and F LC 2 in these notes. When = , we may sometimes want to specify that we λv V 0 for all v; w V 0 and λ F. Every linear map A Hom(V; W ) mean the set of real or complex linear maps by defining: gives2 rise to importan2t subspaces2of V and W : the kernel 2 HomR(V; W ) := A : V W A is real linear ker A = v V Av = 0 V f ! j g f 2 j g ⊂ HomC(V; W ) := Hom(V; W ): and image The first definition treats both V and W as real vector spaces, reducing im A = w W w = Av for some v V W: C R f 2 j 2 g ⊂ the set of scalars from to . The distinction is that a real linear map on We say that A Hom(V; W ) is injective (or one-to-one) if Av = Aw always C a complex vector space need not satisfy A(λv) = λ(Av) for all λ , but implies v = w,2and surjective (or onto) if every w W can be written as rather for λ R. Thus every complex linear map is also real linear,2 but 2 2 Av for some v V . It is useful to recall the basic algebraic fact that A is the reverse is not true: there are many more real linear maps in general. injective if and2only if its kernel is the trivial subspace 0 V . (Prove An example is the operation of complex conjugation it!) f g ⊂ An isomorphism between V and W is a linear map A Hom(V; W ) C C : x + iy x + iy = x iy: 2 ! 7! − that is both injective and surjective: in this case it is invertible, i.e. there is C another map A−1 Hom(W; V ) so that the compositions A−1A and AA−1 Indeed, we can consider as a real vector space via the one-to-one corre- 2 spondence are the identity map on V and W respectively. Two vector spaces are C R2 : x + iy (x; y): isomorphic if there exists an isomorphism between them. When V = W , ! 7! isomorphisms V V are also called automorphisms, and the space of Then the map z z¯ is equivalent to the linear map (x; y) (x; y) these is denoted b!y on R2; it is therefore7! real linear, but it does not respect multiplication7! −by complex scalars in general, e.g. iz = iz¯. It does however have another nice Aut(V ) = A End(V ) A is invertible : 6 f 2 j g property that deserves a name: for two complex vector spaces V and W , This is not a vector space since the sum of two invertible maps need not a map A : V W is called antilinear (or complex antilinear) if it is real be invertible. It is however a group, with the natural \multiplication" ! linear and also satisfies operation defined by composition of linear maps: A(iv) = i(Av): AB := A B: − ◦ Equivalently, such maps satisfy A(λv) = λ¯v for all λ C. The canonical Fn F 2 As a special case, for V = one has the general linear group GL(n; ) := example is complex conjugation in n dimensions: Aut(Fn). This and its subgroups are discussed in some detail in Ap- pendix B. Cn Cn : (z ; : : : ; z ) (z¯ ; : : : ; z¯ ); ! 1 n 7! 1 n and one obtains many more examples by composing this conjugation with A.2 Bases, indices and the summation con- any complex linear map. We denote the set of complex antilinear maps from V to W by vention HomC(V; W ): A basis of a vector space V is a set of vectors e(1); : : : ; e(n) V such that When the domain and target space are the same, a linear map V V 2 ! every v V can be expressed as is sometimes called a vector space endomorphism, and we therefore use the 2 n notation v = cje(j) End(V ) := Hom(V; V ); j=1 X A.2. BASES, INDICES AND THE SUMMATION CONVENTION 167 168 APPENDIX A. MULTILINEAR ALGEBRA for some unique set of scalars c ; : : : ; c F. If a basis of n vectors exists, f ; : : : ; f is a basis of W implies there exist unique scalars Ai F such 1 n 2 (1) (m) j 2 then the vector space V is called n-dimensional. Observe that the map that i n Ae(j) = A jf(i); n F V : (c1; : : : ; cn) cje(j) where again summation over i is implied on the right hand side. Then for ! 7! j j=1 any v = v e V , we exploit the properties of linearity and find1 X (j) 2 is then an isomorphism, so every n-dimensional vector space over F is i (j) i (j) k i k (j) (A ja i )v = (A ja i )v e(k) = A jv a i e(k) Fn ( ) ( ) ( ) (A.2) isomorphic to . Not every vector space is n-dimensional for some n 0: i j j i j j ≥ = (A v )f i = v A f i = v Ae j = A(v e j ) = Av: there are also infinite dimensional vector spaces, e.g. the set of continuous j ( ) j ( ) ( ) ( ) R (j) functions f : [0; 1] , with addition and scalar multiplication defined by i a ! Thus A = A j (i) , and we've also derived the standard formula for (f +g)(x) := f(x)+g(x) and (λf)(x) := λf(x).