Proving Null Spaces and Transformations

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Homework Help Overview

The discussion revolves around proving properties of null spaces related to linear transformations, specifically focusing on the transformation T:V → W. The participants are examining three statements regarding the null spaces of T and its negative, as well as powers of these transformations.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to establish the equality of null spaces for T and -T, questioning their approach and whether they are on the right track. Some participants suggest that the reasoning for part (a) is correct but note that the original poster has not fully proven the equality of the sets. Others indicate that parts (b) and (c) can be approached similarly, though details are not fully explored.

Discussion Status

The discussion is ongoing, with some participants affirming the correctness of certain statements while others point out that further proof is needed for the original poster's claims. There is a mix of agreement and clarification regarding the steps taken, indicating a productive exploration of the topic.

Contextual Notes

Participants are navigating the complexities of proving set equality in the context of linear transformations, with some uncertainty about the completeness of the original poster's proofs. The nature of the homework problem imposes a requirement for rigorous justification of each statement.

redyelloworange
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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!
 
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a) is right. b) is done essentially the same way. If Tkx = y, then (-T)kx = ___? It should be pretty easy. c) follows immediately from b).
 
AKG said:
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.
 
Sorry, my mistake. Hurkyl is correct, redyelloworange has not yet answered part a) fully.
 

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