# 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.