First, your projection matrix makes sense if . Otherwise, . If so, maps to the -dimensional linear space spanned by the vectors orthogonal to the columns of . Now, when you consider , this operator maps to the linear subspace orthogonal to the column space of . Here, you can have different options depending on the rank of . Let, for instance, . In this case, the column space of is the same as the column space of and . But this will not be the case if .