Let $V_0$ and $V$ be subspaces of $\Omega$. For $V_1=V\cap V_0,$ show that $P_{V_1}=P_VP_{V_0}.$
I know that the above statement is not true but can't think of a counterexample. Also, if $V\perp V_0$ then the above statement is true, isn't it?
Here $\Omega$ is the collection of all $n-$tuples of real numbers for a positive inetger $n$. $P_V$ is the projection operator onto $V$.