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Given $P : R^n → R^n$ is a linear transformation. Show that there is an integer $k$ such that $R(P^k)=R(P^{k+1})=R(P^{k+2})=...$($R(P)$ denotes the range of $P$.)

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Well, we have inclusions $R(P) \supseteq R(P^2) \supseteq \ldots$And by finite dimensionality we have to stabilize (if not these subspaces decrease in dimension infinitely often).

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    @drawar: To stabilize for a sequence means to become constant from a certain point on, here exactly the condition $R(P^k)=R(P^{k+1})=R(P^{k+2})=\ldots$ of the question.2012-11-18