trace map is continuous

Question

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.

Answer 1

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.