Question 1. Let V and W be finite dimensional vector spaces overa field F with dimF(V ) = dimF(W) and let T : V ? W be a linearmap. Prove there exists an ordered basis A for V and an orderedbasis B for W such that [T] A B is a diagonal matrix where everyentry along the diagonal is either a 0 or a 1.
Hint 1. Suppose A = {~v1, . . . , ~vn} and B = { ~w1, . . . ,~wn}. If the k th column of [T] A B consists of all zeros, what canyou deduce?
Hint 2. Suppose A = {~v1, . . . , ~vn} and B = { ~w1, . . . ,~wn}. If the k th column of [T] A B has a one in the k th entry andall other entries are zero, what can you deduce?
Hint 3. Now construct bases with the properties found in Hint 1and Hint 2.
Hint 4. Theorem 18 part 5 is your friend.
Hint 5. The proof of the Rank-Nullity Theorem is your bestfriend.
