Let $P:\mathbb R^m \to \mathbb R^m$ is orthogonal linear transformation on some subspace $U$ of $\mathbb R^m$. Execute the term for matrix of linear transformation P using standard base. Explain the procedure.
I know matrix for orthogonal linear transformation is P=$A(A^TA)^{-1}A^T$, I saw proof that this matrix is matrix of projection and it is orthogonal but,how I get that matrix, I could not find anything how to get that matrix, can you help me?