Let $M$ be an $m \times n$ matrix, where the rows are linearly independent.
Let $M^{+}$ be the pseudo-inverse of $M$ (which I think is $M^{+} =M^T\left(MM^T\right)^{-1}$, at least when $m\leq n$).
Does $\ker(M) = Span(I - M^{+}M)$? Sometimes? Always? Under what circumstances?
Context: I am a musician working on an algorithmic composition system. I am not a student or a mathematician. This is not homework.