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**bmp05** Let $\displaystyle f: M_{22} (\mathbb{R}) \rightarrow \mathbb{R}[T]$ defined by $\displaystyle f \left[\begin{matrix} a & b\\ c & d \end{matrix}\right] = (a+b) + (a + b)T + (a + b + c + d)T^2$ for all $\displaystyle \left[\begin{matrix} a & b\\ c & d \end{matrix}\right] \in M_{22}$.

1. Prove that $\displaystyle f$ is linear.

2. Calculate the basis of the null-space of $\displaystyle f$ and of the image of $\displaystyle f$.

*How do I even prove that $\displaystyle f$ is linear?*