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- Jun 20, 2014

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I was sick for some time, so I had not posted any new problems for either the uni POTW or the grad POTW for a couple weeks. Just this time, there will be a special of two problems posted today for both the university and graduate levels! Here is this week's two POTW:

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1. Suppose $f$ is a continuous, complex-valued function on the complex plane $\Bbb C$ such that $\lim\limits_{\lvert z\rvert \to \infty} \lvert f(z)\rvert = 0$. Prove that $f$ has maximum modulus in $\Bbb C$.

2. If $X$ and $Y$ are $n\times n$ matrices over a field $F$, show that the trace of $X\otimes Y$ is the product of the traces of $X$ and $Y$.

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You may submit a solution one of the two problems or submit solutions to both of the problems. Remember to read the POTW submission guidelines to find out how to submit your answers!