Suppose Vis finite-dimensional, T = L(V), and U is a subspace of V. Prove that U and U are both invariant under T if and only if Pu T = TPU.

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Suppose \( V \) is finite-dimensional, \( T \in \mathcal{L}(V) \), and \( U \) is a subspace of \( V \). Prove that \( U \) and \( U^\perp \) are both invariant under \( T \) if and only if \( P_U T = T P_U \).
Transcribed Image Text:Suppose \( V \) is finite-dimensional, \( T \in \mathcal{L}(V) \), and \( U \) is a subspace of \( V \). Prove that \( U \) and \( U^\perp \) are both invariant under \( T \) if and only if \( P_U T = T P_U \).
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