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:

where zeros vectors are added to make square. Zeros would also be prepended to

Equivalently, if and is a basis of a larger subspace , does ?

Is the projection matrix the same after zero vectors have been added?

Thank you!!