let the dimension of "Q" be "s" ≤ "r". Each vector in "Q" is an eigenvector of "A' "corresponding to eigenvalue "λ". So the Jordan form of "A' " must contain "s" Jordan chains corresponding to "s" linearly independent eigenvectors. So the basis {"p"1, ..., "p""r"} must contain "s" vectors, say {"p""r"−"s"+1, ..., "p""r"}, that are lead vectors in these Jordan chains from the Jordan normal form of "A. We can "extend the chains" by taking the preimages of these lead vectors. (This is the key step of argument; in general, generalized eigenvectors need not lie in Ran("A" − λ I'").) Let "q""i" be such that