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Show that \(\mathcal{L}\) and \(\mathcal{R}\) are linear spaces and compute their dimensions.

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Say \(A \in M(n, \mathbb{F})\) has rank \(k\). Define
\[
\mathcal{L}:=\{B \in M(n, \mathbb{F}) \mid B A=0\} \quad \text { and } \quad \mathcal{R}:=\{C \in M(n, \mathbb{F}) \mid A C=0\} .
\]
Show that \(\mathcal{L}\) and \(\mathcal{R}\) are linear spaces and compute their dimensions.
in Mathematics by Platinum (102k points) | 241 views

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