Multilinear Algebra a Bilinear Map in Chapter I
·~ ...,... .chapter II 1 60 MULTILINEAR ALGEBRA Here a:"~ E v,' a:J E VJ for j =F i, and X EF. Thus, a function rjl: v 1 X ••• X v n --+ v is a multilinear mapping if for all i e { 1, ... , n} and for all vectors a: 1 e V 1> ... , a:1_ 1 eV1_h a:1+ 1eV1+ 1 , ••. , a:.ev•• we have rjl(a: 1 , ... ,a:1_ 1, ··, a 1+ 1, ... , a:.)e HomJ>{V" V). Before proceeding further, let us give a few examples of multilinear maps. Example 1. .2: Ifn o:= 1, then a function cf>: V 1 ..... Vis a multilinear mapping if and only if r/1 is a linear transformation. Thus, linear tra~sformations are just special cases of multilinear maps. 0 Example 1.3: If n = 2, then a multilinear map r/1: V 1 x V 2 ..... V is what we called .Multilinear Algebra a bilinear map in Chapter I. For a concrete example, we have w: V x v• ..... F given by cv(a:, T) = T(a:) (equation 6.6 of Chapter 1). 0 '\1 Example 1.4: The determinant, det{A), of an n x n matrix A can be thought of .l as a multilinear mapping cf>: F .. x · · · x P--+ F in the following way: If 1. MULTILINEAR MAPS AND TENSOR PRODUCTS a: (a , ... , a .) e F" for i 1, ... , o, then set rjl(a: , ... , a:.) det(a J)· The fact "'I 1 = 11 1 = 1 = 1 that rJ> is multilinear is an easy computation, which we leave as an exercise at the In Chapter I, we dealt mainly with functions of one variable.
[Show full text]