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Let \(A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}\) and \(B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}\).

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Let \(A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}\) and \(B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}\). Prove that \(\operatorname{rank} A+\operatorname{rank} B-\ell \leq \operatorname{rank} A B \leq \min \{\operatorname{rank} A, \operatorname{rank} B\} .\)
\(\left[\right.\) HINT: Observe that \(\left.\operatorname{rank}(A B)=\left.\operatorname{rank} A\right|_{\operatorname{Image}(B)} \cdot\right]\)
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