Prove that $V=V_0oplus V_1$ where $T(v_0)=0$ for all $v_0in V_0$ and $T(v_1)=v_1$ for all $v_1in V_1$, where $T$ is an idempotent linear trans.

Let $T:Vrightarrow V$ be an idempotent linear transformation. Prove that $V=V_0oplus V_1$ where $T(v_0)=0$ for all $v_0in V_0$ and $T(v_1)=v_1$ for all $v_1in V_1$.

My thoughts: Since $T$ is idempotent, we know that $T^2=T$. So, I first need to show that $V=V_0+V_1$, then I need to show that $V_0cap V_1={0}$. So, if we let $vin T$, then $T^2v=Tvimplies T(Tv-v)=0$ and so $Tv-v=thetain ker T$… So would it help to try and show that $V_0=ker T$ and $V_1=Im T$ and try to go from there..? Any help is greatly appreciated! Thank you.