let $\displaystyle T:U\rightarrow{V}$ be a linear mapping from vector spaces U to V and let X be a subspace of U. Show that

$\displaystyle T(X)=\{v\epsilonV\epsilonV|v=Tx\ \mbox{for some}\ x\epsilonX\}$ is a subspace of V

I can understand why this is, it seems pretty trivial but I am not sure how you would go about proving it.

Thanks for any help