(Solved): Verify that this \( \operatorname{map} f: \mathbb{R}^{3} \rightarrow \mathcal{M}_{2 \times 2} \) i ...

Verify that this \( \operatorname{map} f: \mathbb{R}^{3} \rightarrow \mathcal{M}_{2 \times 2} \) is one-to-one. \[ f\left(\left(\begin{array}{l} a \\ b \\ c \end{array}\right)\right)=\left(\begin{array}{cc} a+2 b & -c \\ a-b & 0 \end{array}\right) \]