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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