This is an alternative proof of Lemma 2.9. Given an n n matrix H, fix a

Question:

This is an alternative proof of Lemma 2.9. Given an n × n matrix H, fix a domain V and codomain W of appropriate dimension n, and bases B, D for those spaces, and consider the map h represented by the matrix.
(a) Show that h is onto if and only if there is at least one RepB() associated by H with each RepD().
(b) Show that h is one-to-one if and only if there is at most one RepB() associated by H with each RepD().
(c) Consider the linear system HRepB() = RepD(). Show that H is nonsingular if and only if there is exactly one solution RepB() for each RepD().