Q2. Let A, B € Mnxn (F) and assume that A is similar to B. In Proposition 3.2.5 of the course

notes, you learned that rank(A) = rank(B) and nullity(A) = nullity (B). In this problem you

will give another proof of these equalities.

€

(a) Suppose that A = PBP-¹, where P Mnxn (F). Show that if Col(A) then

P-¹7 € Col(B).

(b) Construct an isomorphism L: Col(A) → Col(B). (You must prove that your map L is

an isomorphism.) Explain why this implies that rank(A) = rank(B).

(c) Prove that nullity(A) = nullity(B). [Hint: There is a quick proof.]