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:

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 X square. Zeros would also be prepended to Y

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

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

Thank you!!