Let T, U: V to W be linear transformations. Prove that:
a. R(T+U) is a subset of R(T)+R(U).
b. If W is finite-dimensional, then rank(T+U)=< rank(T)+rank(U).
c. Deduct from b that rank(A+B)=< rank(A)+rank(B) for any m x n matrices
A and B.