Suppose we have linear operators $S, T$ over a finite-dimensional $V.$ We are also given that $ST = TS.$ How can we prove that null($T - \lambda I$) is invariant under $S$ for any lambda in the field?
Proving invariance
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linear-algebra
1 Answers
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Let $\xi$ be inside $null(T-\lambda I)$. Then $(T-\lambda I) S \xi = TS \xi - \lambda S \xi = ST \xi - \lambda S \xi = S [(T - \lambda I) \xi] = S 0 = 0$ and so $S \xi$ is inside $null(T-\lambda I)$.