Linear Algebra Subspace regarding linear map

Hi, i'm having trouble starting this proof and how to prove the ideas necessary.

Let V and W be finite-dimensional vector spaces over F. Given T is in L(V,W), show that there is a subspace U of V such that the following are true:

U(intersection)null(T)= {0} and range(T)={Tu:u in U}.

Thank you

Re: Linear Algebra Subspace regarding linear map

How about the linear map which takes every element of V to the 0 element of W. Then if you let U be the subspace with only the 0 element of V. then $\displaystyle U \cap Ker(T) = U = {0}$ and $\displaystyle Range(T) = 0 = {T*0, 0 \in {0} = U}$