Identity Matrix and Projection

Question

Hey is there a way to proof if I is the identity Matrix and P is the projection of some subspace U with $R^{nxn}$ that $I - P$ is the projection Matrix of the orthogonal subspace $U^⊥$.

Ok I know this:

$A^T*B = 0$

so I can write this:

$B(B^TB)^{-1}B^T = I - A(A^TA)^{-1}A^T$

is that the right approach: $B*B^{-1}(B^T)^{-1}B^T = I - A*A^{-1}(A^T)^{-1}A^T$

Answer 1

This is only true if $P$ is symmetric. In this case we have $Im(P)=U$ and $Im(I-P)=U^{\perp}.$

Reason: $Im(I-P)= ker (P).$