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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 $V$.

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    In statistical terms perpedicular means independent. For a counterexample you just need to make the sets dependent that is find V and V0 such that P(V|V0) ≠ P(V).2012-09-18

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