Let a be a nonzero vector in \( \mathbb{R}^{2} \). Define \( \tau_{\mathbf{a}}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \) by \[ \tau_{\mathbf{a}}(\mathbf{x})=-\mathbf{x}+2 \frac{\mathbf{a} \cdot \mathbf{x}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a} \] 2. Let a be a nonzero vector of \( \mathbb{R}^{2} \). Answer the following questions. (iii) Find the matrix \( \left[\tau_{\mathbf{a}}\right]_{\mathcal{B}}^{\mathcal{B}} \) of \( \tau_{\mathbf{a}} \) with respect to \( \mathcal{B} \) and \( \mathcal{B} \).

To find the matrix \(\left[\tau_{\mathbf{a}}\right]_{\mathcal{B}}^{\mathcal{B}}\) of the transformation \(\tau_{\mathbf{a}}\) with respect to the basis \(\mathcal{B}\) and \(\mathcal{B}\), we can start by expressing \(\tau_{\mathbf{a}}(\mathbf{x})\) in terms of a chosen basis. Let’s first identify the basis \(\mathcal{B} = \{\mathbf{b}_1, \mathbf{b}_2\}\) with \(\mathbf{b}_1\) and \(\mathbf{b}_2\) as standard basis vectors in \(\mathbb{R}^2\). For convenience, let’s set \(\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}\) and define \(\tau_{\mathbf{a}}\) as follows. 1. Calculate \(\tau_{\mathbf{a}}(\mathbf{b}_1)\) and \(\tau_{\mathbf{a}}(\mathbf{b}_2)\): - \(\tau_{\mathbf{a}}(\mathbf{b}_1) = -\mathbf{b}_1 + 2 \frac{\mathbf{a} \cdot \mathbf{b}_1}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a}\) - \(\tau_{\mathbf{a}}(\mathbf{b}_2) = -\mathbf{b}_2 + 2 \frac{\mathbf{a} \cdot \mathbf{b}_2}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a}\) 2. After evaluating these expressions, we obtain the transformed vectors \(\tau_{\mathbf{a}}(\mathbf{b}_1)\) and \(\tau_{\mathbf{a}}(\mathbf{b}_2)\) in terms of the basis vectors. 3. The matrix \(\left[\tau_{\mathbf{a}}\right]_{\mathcal{B}}^{\mathcal{B}}\) is formed by writing these transformed vectors as linear combinations of the basis vectors. To illustrate, if we denote \(\tau_{\mathbf{a}}(\mathbf{b}_1) = c_1 \mathbf{b}_1 + d_1 \mathbf{b}_2\) and \(\tau_{\mathbf{a}}(\mathbf{b}_2) = c_2 \mathbf{b}_1 + d_2 \mathbf{b}_2\), the matrix becomes: \[ \left[\tau_{\mathbf{a}}\right]_{\mathcal{B}}^{\mathcal{B}} = \begin{pmatrix} c_1 & c_2 \\ d_1 & d_2 \end{pmatrix} \] By computing \(c_1, d_1, c_2, d_2\) accordingly, we can get the matrix representation of the transformation corresponding to the specified basis vectors in \(\mathbb{R}^2\). Without loss of generality, let's assume \(\mathbf{b}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\mathbf{b}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\), and follow through with the steps to find exact values for \(c_1, d_1, c_2, d_2\) based on the specific components of \(\mathbf{a}\). This matrix will encode how the vector transformation acts in the aforementioned basis.