Linear Transformations - Direct Sum Let $U, V$, and $W$ be finite dimensional vectors spaces over a field.
Suppose that $V\subset U$ is a subspace. Show that there is a subspace $W\subset U$ such that $U=V\oplus W$. 
only thing i know about this problem is that you have to use the null space. I'm pretty much lost! any help would be appreciated!

Hint: Start with a basis for $V$, $\{v_1,\ldots,v_k\}$, and extend it to a basis of $U$, $\{v_1,\ldots,v_k,u_{k+1},\ldots,u_n\}$. Now find a subspace of $U$ which has the properties that you want by using that extended basis.
