Question: An \(n \times n\) matrix \(M\) defines a linear map from \(\mathbb{R}^{n}\) to \(\mathbb{R}^{n}\). Note that \(M^{2}\) also defines a linear map from \(\mathbb{R}^{n}\) to \(\mathbb{R}^{n}\).

a: Show that if \(v \in \operatorname{im}\left(M^{2}\right)\), then \(v \in \operatorname{im}(M)\), meaning we have the inclusion of subspaces \(\operatorname{im}\left(M^{2}\right) \subset \operatorname{im}(M)\)

b: What are the corresponding statements about \(\operatorname{ker}(M)\) and \(\operatorname{ker}\left(M^{2}\right) ?\)

c: Find a \(3 \times 3\) matrix \(M\) with \(\operatorname{im}(M) \neq \operatorname{im}\left(M^{2}\right)\) and also (necessarily - why?!) \(\operatorname{ker}(M) \neq \operatorname{ker}\left(M^{2}\right)\)

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