Let V be an n-dimensional vector space, and assume for subspaces U and W. Suppose is a linear transformation such that and .

Show that the matrix of T with respect to a basis of V that contains bases of U and W, has block shape:

I am having trouble seeing how the matrix that represents T would be a 2x2 matrix.

So far in my working I have set the basis of U to be and the basis of W to be and I can see that under the T(u) (where ) can be written as a linear combination of the elements in the basis. But I can't see where to go from there. Help would really be appreciated.