Proving Null Spaces and Transformations

[redyelloworange]

Homework Statement

Let T:V  W be a linear transformation. Prove the following results.

(a) N(T) = N(-T)
(b) N(T^k) = N((-T)^k)
(c) If V = W and t is an eigenvalue of T, then for any positive integer k
N((T-tI)^k) = N((tI-T)^k) where I is the identity transformation

The Attempt at a Solution

(a) for every x in V:
If T(x) = y, then –T(x) = -y
So then, T(0) = 0 = -T(0)
Is this right? On the right track?

I’m not sure how to approach the rest of them?

Thanks for your help!

 

[AKG]

a) is right. b) is done essentially the same way. If T^kx = y, then (-T)^kx = ___? It should be pretty easy. c) follows immediately from b).

 

[Hurkyl]

a) is right.

Well, everything he said is right, but he hasn't proven what he set out to prove: that N(T) and N(-T) are equal sets.

 

[AKG]

Sorry, my mistake. Hurkyl is correct, redyelloworange has not yet answered part a) fully.