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tinynerdi
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Homework Statement
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)
Homework Equations
T(-v) = -T(v)
N(T) = {v in V: T(v)=0} in V hence T(v) = 0 for all v in V.
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)