what I want to prove is that if $P \in \mathbb{R}^{n \times n}$ is a projection matrix onto $\mathbb{S}_1$ along $\mathbb{S}_{2}^{\perp}$ , $v_1, v_2, \ldots, v_m$ and $w_1, w_2, \ldots, w_m$ constitute a basis of $\mathbb{S}_1$ and $\mathbb{S}_2$ , then $$ P=V\left(W^{\top} V\right)^{-1} W^{\top}, $$
in which $V=\left[v_1, v_2, \ldots, v_m\right], W=\left[w_1, w_2, \ldots, w_m\right]$. I want to use geometric meaning to prove this, and I konw $Ax$ is a orthogonal projection onto the subspace spanned by the columns of $A$, then I don't know how to proceed