Question 1. Let V and W be finite dimensional vector spaces over a field F with...

Question 1. Let V and W be finite dimensional vector spaces over a field F with dimF(V ) = dimF(W) and let T : V → W be a linear map. Prove there exists an ordered basis A for V and an ordered basis B for W such that [T] A B is a diagonal matrix where every entry 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 can you 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 and all other entries are zero, what can you deduce?

Hint 3. Now construct bases with the properties found in Hint 1 and Hint 2.

Hint 4. Theorem 18 part 5 is your friend.

Hint 5. The proof of the Rank-Nullity Theorem is your best friend.

Expert Solution

here V and W be finite dimensional vector space and dim V= dim W

let the dimension=n imply in the basis we should have n elements

hence A={v₁,v₂,....,v_n} and B={w₁,w₂,.....,w_n} be the basis of V and W respectively

then T(v₁)=w₁=1.w₁+0.w₂+.....+0.w_n

T(v₂)=w₂=0.w₁+1.w₂+.......+0.w_n

_:

similarly T(v_n)=w_n=0.w₁+0.w₂+......+1.w_n

so by these basis properties we get the matrix

C= which is a diagonal matrix with entries 0 and 1

milcah answered 2 years ago

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