Properties
A first immediate consequence of the definition is that B(x,y) = 0 whenever x = 0 or y = 0. (This is seen by writing the null vector 0 as 0·0 and moving the scalar 0 "outside", in front of B, by linearity.)
The set L(V,W;X) of all bilinear maps is a linear subspace of the space (viz. vector space, module) of all maps from V×W into X.
If V, W, X are finite-dimensional, then so is L(V,W;X). For X = F, i.e. bilinear forms, the dimension of this space is dim V × dim W (while the space L(V×W;F) of linear forms is of dimension dim V + dim W). To see this, choose a basis for V and W; then each bilinear map can be uniquely represented by the matrix B(ei,fj), and vice versa. Now, if X is a space of higher dimension, we obviously have dim L(V,W;X) = dim V × dim W × dim X.
Read more about this topic: Bilinear Map
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