If " V " and " W " are finite-dimensional, then the space of all linear transformations from " W " to " V ", denoted, is generated by such outer products; in fact, the rank of a matrix is the minimal number of such outer products needed to express it as a sum ( this is the "'tensor rank "'of a matrix ).
42.
If " V " and " W " are finite-dimensional, then the space of all linear transformations from " W " to " V ", denoted, is generated by such outer products; in fact, the rank of a matrix is the minimal number of such outer products needed to express it as a sum ( this is the "'tensor rank "'of a matrix ).
43.
More abstractly, the outer product is the bilinear map W \ times V ^ * \ to \ operatorname { Hom } ( V, W ) sending a vector and a covector to a rank 1 linear transformation ( simple tensor of type ( 1, 1 ) ), while the inner product is the bilinear evaluation map V ^ * \ times V \ to F given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector / vector distinction.
