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Let $L: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a linear map. Show that
$\operatorname{dim} \operatorname{ker}(L)-\operatorname{dim}\left(\operatorname{ker} L^{*}\right)=n-k .$
Consequently, for a square matrix, $\operatorname{dim} \operatorname{ker} A=\operatorname{dim} \operatorname{ker} A^{*}$. [In a more general setting, ind $(L):=\operatorname{dim} \operatorname{ker}(L)-\operatorname{dim}\left(\operatorname{ker} L^{*}\right)$ is called the index of a linear map $L$. It was studied by Atiyah and Singer for elliptic differential operators.]
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