$X_{N\times N}=[\vec{x_1}, \vec{x_2}, \hdots , \vec{x_p}, \vec{0}, \hdots, \vec{0}$ 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?