Help with elementary Linear Algebra proof?

I've been brushing up on some linear algebra this summer, and here's a problem I found in a book that I can't seem to solve. Any help is appreciated!

Let V and W be vector spaces, and let T and U be nonzero linear transformations from V into W. If the intersection of R(T) and R(U)* is 0, prove that {T, U} is a linearly independent subset of L(V, W).**

*R(T) and R(U) denote the range of the linear transformation of T and U.

** L(V, W) denotes the space of linear transformations from V to W.