Prove that $tr: M_n(k)\to k$ is continuous.

I did continuity of determinant map using induction, but how to prove trace map is continuous. please give a thorough answer. My analysis is not too good.

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    $\begingroup$ An easy method is as follows: this is a linear map, and all linear maps on finite dimensional spaces are continuous. $\endgroup$ – Ben Grossmann Sep 9 '14 at 20:26

Is it clear that the map $$k^{n^2}\to k \ \ \ \ \ \ \ \ \ \ (a_{11},a_{12},\dots,a_{1n},a_{22},a_{21},\dots,a_{nn})\mapsto a_{ii}$$ is continuous for $i=1,\dots,n$?

Also, the sum of continuous functions is continuous.


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