I'm writing a proof for a take-home exam... I don't want the answer outright, but I was wondering if it was possible to zero pad orthogonal basis matrices in order to sum them?

e.g. can I do this:

$\displaystyle X_{N\times N}=[\vec{x_1}, \vec{x_2}, \hdots , \vec{x_p}, \vec{0}, \hdots, \vec{0}$ where zeros vectors are added to make $\displaystyle X$ square. Zeros would also be prepended to $\displaystyle Y$

Equivalently, if $\displaystyle X\perp Y$ and $\displaystyle X+Y$ is a basis of a larger subspace $\displaystyle S$, does $\displaystyle X_{N\times N}\equiv X_{N\times p}$?

Is the projection matrix $\displaystyle X(X^TX)^{-1}X^T$ the same after zero vectors have been added?

Thank you!!