Alexey L. Gorodentsev - Algebra II: Textbook for Students of Mathematics

Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia

Let V 1, V 2,, V n and W be free modules of finite ranks d 1, d 2,, d n and d respectively. Then the module of multilinear maps () is uniquely determined by its values on all collections of the basis vectors because for an arbitrary collection of vectors v 1, v 2,, v n , where each v V is linearly expressed through the basis as it follows from the multilinearity of that Every vector () is uniquely expanded as Thus, the multilinear maps () are in bijection with the ( n + 1)-dimensional matrices of size d d 1 d 2 d n with elements a ij 1 j 2 jn K . For n =1, such a matrix is the usual 2-dimensional d d 1 matrix of a linear map V W , where d 1=dim V , d =dim W . For n =2, a bilinear map V 1 V 2 W is encoded by the three-dimensional matrix of size d d 1 d 2 formed by the constants , etc. A map is recovered from its matrix by the formula The addition and multiplication by constants in has the effect on matrices of componentwise addition and multiplication by constants. Therefore, is isomorphic to the K -module of ( n + 1)-dimensional matrices of size d d 1 d 2 d n with elements from K . The latter module is free with a basis formed by the matrices E ij 1 j 2 jn having 1 in the position ( ij 1 j 2 jn ) and 0 everywhere else. The corresponding basis of consists of the multilinear maps An arbitrary collection of vectors () is mapped to In particular, if is a field and V 1, V 2,, V n , W are finite-dimensional vector spaces over , then .

Given a multilinear map of K - modules and an arbitrary K - module W , composing with the linear maps F : U W assigns the map which is obviously linear in F .