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2. Consider the following two bases for \( \mathbb{R}^{2} \) : \[ \begin{aligned} \mathcal{B} &=\left\{\alpha_{1}, \alpha_{2}\right\}=\{(-7,3),(2,-1)\} \\ \mathcal{B}^{\prime} &=\left\{\alpha_{1}^{\prime}, \alpha_{2}^{\prime}\right\}=\{(5,-1),(1,6)\} . \end{aligned} \]
(a) Determine the coordinate matrix \( [\alpha]_{\mathcal{B}} \) for the vector \( \alpha=(-4,9) \). (b) Compute the unique transition matrix \( P \) between \( \mathcal{B} \) and \( \mathcal{B}^{\prime} \). (c) Use the result of Part (a) to determine the coordinate matrix \( [\alpha]_{\mathcal{B}^{\prime}} \) for the vector \( \alpha=(-4,9) \).