Q:

When does the projection matrix become the identity? How to show it mathematically?

I want to show mathematically that if I project a vector b onto a space that is spanned by a matrix A, which is invertible, the projection matrix P is the identity matrix.
I can explain it geometrically that in that case b is element of the space, hence, the projection of b is b, but how do I show this in a concise mathematical fashion?
Thanks for any help!

A:

When A is invertible, the following holds:
$P=A*(A^{T}*A)^{-1}*A^{T}=A*(A^{-1}*(A^{T})^{-1})*A^{T}=I$
because:
$(A^{T}*A)*(A^{T}*A)^{-1}=A^{T}*A*A^{-1}*(A^{T})^{-1}=I$

