Orthogonal projection of a vector on a linear subspace

Question

I know this question has already been asked before but I'm not quite sure I've understood them well enough.

I want to find the projection $\vec{v}_{proj}$ of $\vec{v}\in V$ on the span of the vectors $\vec{a_i}\in V$, which is a subspace of $V$.

What is the matrix that represent this projection in an arbitrary basis $\mathscr{B}$, and if it isn't entirely intuitive, why?

Answer 1

Choose an orthonormal basis of that subspace and extend to an orthonormal basis of the whole space. Then the projection matrix with respect to this basis will of the form $P=\left({I\mid Z\over Z\mid Z}\right)$ where $I$ is the identity matrix of size = dimension of the subspace, and $Z$ represent zero matrices of appr0priate sizes to make the whole matrix $n\times n$.

Now for an arbitrary basis assuimg $Q$ is the change-of-basis matrix the projection matrix would be $Q^{-1}PQ$.