Question: Linearity. Given \(A, B, C, X \in \mathbb{C}^{n \times n}\), determine if the following maps from \(\mathbb{C}^{n \times n} \rightarrow \mathbb{C}^{n \times n}\) are linear. Justify your answers.

1. \(X \mapsto A X+X B\)
2. \(X \mapsto A X+B X C\)
3. \(X \mapsto A X A-X\)
4. \(X \mapsto A X+X B X\)

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