Suppose that the matrix \(H\) in \((1.5)\) is given by $$ H=\frac{1}{\sqrt{6}}\left(\begin{array}{ccc} \sqrt{2} & \sqrt{2} & \sqrt{2} \\ -2 & 1 & 1 \\ 0 & -\sqrt{3} & \sqrt{3} \end{array}\right) $$ Check that \(H^{\mathrm{t}} H=H H^{\mathrm{t}}=I\). Write down the components of \(\boldsymbol{e}_{1}\), \(e_{2}\), and \(e_{3}\) in \(\tilde{T}\), and the components of \(\tilde{e}_{1}, \tilde{e}_{2}\), and \(\tilde{e}_{3}\) in \(\mathcal{T}\).

The transpose of the matrix \(H\) is: $$ H^{\mathrm{t}}= \frac{1}{\sqrt{6}}\left(\begin{array}{ccc} \sqrt{2} & -2 & 0 \\\ \sqrt{2} & 1 & -\sqrt{3} \\\ \sqrt{2} & 1 & \sqrt{3} \end{array}\right) $$ 2. What are the components of \(e_1, e_2, e_3\) in \(\tilde{T}\)? The components of \(e_1, e_2, e_3\) in \(\tilde{T}\) are: $$ \boldsymbol{e_1}_{\tilde{T}}= \frac{1}{\sqrt{6}}\left(\begin{array}{c} \sqrt{2} \\ -2 \\ 0 \end{array}\right), \quad \boldsymbol{e_2}_{\tilde{T}} = \frac{1}{\sqrt{6}}\left(\begin{array}{c} \sqrt{2} \\ 1 \\ -\sqrt{3} \end{array}\right), \quad \boldsymbol{e_3}_{\tilde{T}} = \frac{1}{\sqrt{6}}\left(\begin{array}{c} \sqrt{2} \\ 1 \\ \sqrt{3} \end{array}\right) $$ 3. What are the components of \(\tilde{e}_1, \tilde{e}_2, \tilde{e}_3\) in \(\mathcal{T}\)? The components of \(\tilde{e}_1, \tilde{e}_2, \tilde{e}_3\) in \(\mathcal{T}\) are: $$ \boldsymbol{\tilde{e}_1}_{\mathcal{T}}= \frac{1}{\sqrt{6}}\left(\begin{array}{c} \sqrt{2} \\ -2 \\ 0 \end{array}\right), \quad \boldsymbol{\tilde{e}_2}_{\mathcal{T}} = \frac{1}{\sqrt{6}}\left(\begin{array}{c} \sqrt{2} \\ 1 \\ -\sqrt{3} \end{array}\right), \quad \boldsymbol{\tilde{e}_3}_{\mathcal{T}} = \frac{1}{\sqrt{6}}\left(\begin{array}{c} \sqrt{2} \\ 1 \\ \sqrt{3} \end{array}\right) $$