V and W are finite dimensional vector spaces over F a field. F is a subfield of K. T:V->W is a linear transformation.
First I have to prove that 1 tensor T is a K-linear transformation from the vector spaces K tensor V to K tensor W over K. (1 is the identity map from K to itself).
Then, let B={v_1, v_2,....v_n} and E={w_1,...,w_m} be bases of V and W respectively.
I have to prove that the matrix of 1 tensor T with the bases {1 tensor v_1, 1 tensor v_2,...,1 tensor v_n} and {1 tensor w_1, 1 tensor w_2,...,1 tensor w_m} is the same as the matrix of T with respect to B and E.