I need to prove the following proposition:
Let $V$ be a finite dimentional vector space and $T \in End(V)$ a non-zero linear operator, then $T$ is invertible iff for all $U$ and $W$ subspaces of $V$ it is the case that $V=U \oplus W \implies T(V)=T(U)\oplus T(W)$
I have already proved that if $T$ is invertible then given any two subspaces $U$ and $W$ s.t $V=U \oplus W$ it is the case that $T(V)=T(U)\oplus T(W)$ but I'm stuck with the converse. If I'm not wrong it should be enough to prove that $T$ is injective or surjective (no need to prove both) since domain and codomain have the same dimention, so my intention was to prove injectivity by taking some vector $v \in Ker(T)$ and concluding that $v=0$ but i couldn't get to that result. Any suggestions on how to prove surjectivity will also be welcome. Thanks