In Exercises 1-3, let T: V W be a linear transformation between finite-dimensional vector spaces V

Question:

In Exercises 1-3, let T: V → W be a linear transformation between finite-dimensional vector spaces V and W Let B and C be bases for V and W, respectively, and let A = [T] C←B.
1. Show that nullity(T) = nullity(A).
2. Show that rank(T) = rank(A).
3. If V = W and B = C, show that T is diagonalizable if and only if A is diagonalizable.