(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let T:V->W be a linear transformation. Prove that if V=W (So that T is linear operator on V) and λ is an eigenvalue on T, then for any positive integer K

N((T-λI)^k) = N((λI-T)^k)

2. Relevant equations

T(-v) = -T(v)

N(T) = {v in V: T(v)=0} in V hence T(v) = 0 for all v in V.

3. The attempt at a solution

we know that (T-λI)^k(-v) = -(T-λI)^k(v) = (λI-T)^k(v). So when (T-λI)^k(v) = 0 so does (λI-T)^k(v). Hence N((T-λI)^k) = N((λI-T)^k)

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