Let $L$ be a subspace of $\mathbb{R}^N$ with dimension $k$ and such that $L=\left \{ A\beta \mid \beta\in \mathbb{R}^k\right \}$, where $A$ is a $N\times k$ matrix. Let $C$ be a $1\times k$ non-zero matrix and define $L^*=\left \{ A\beta \mid \beta\in \mathbb{R}^k \wedge C\beta=0 \right \}$. It is clear that $L^*$ has dimension $k-1$ and is a subspace of $L$.
I need to show that $p^*(\mathbb{R}^N)=L^*$ where
$p^*(x)=P^*x = A(A^TA)^{-1}A^Tx-A(A^TA)^{-1}C^T(C(A^TA)^{-1}C^T)^{-1}C(A^TA)^{-1}A^Tx$
I have already shown $p^*(\mathbb{R}^N)\supseteq L^*$, but I do have problem with the other inclusion.