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- Jan 26, 2012

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This week's problem was again proposed by yours truly (it would be nice if more people proposed some problems! ).

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**Problem**: Let $\phi:V\rightarrow V$ be a linear operator on a finite-dimensional vector space $V$. Let $k=\text{dim}(\ker \phi)$, and let $d=\text{dim}(V)$. Let $\phi^2:V\otimes V\rightarrow V\otimes V$ be the unique linear operator which satisfies $\phi^2(v_1\otimes v_2) = \phi(v_1)\otimes \phi(v_2)$ for all $v_1, v_2\in V$. Prove that $\text{dim}(\ker\phi^2) = 2dk-k^2$.

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I will provide some hints for this week's problem:

Remember to read the POTW submission guidlines to find out how to submit your answers!