We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$. (a) Let $s _ { 1 } : E \longrightarrow E$ be the orthogonal reflection with respect to the line $\mathbb { R } \omega _ { 1 }$. Show that the matrix of $s _ { 1 }$ in the basis $\mathcal { C }$ is $\left( \begin{array} { c c } 1 & 1 \\ 0 & - 1 \end{array} \right)$. (b) Let $s _ { 2 } : E \longrightarrow E$ be the orthogonal reflection with respect to the line $\mathbb { R } \omega _ { 2 }$. Show that the matrix of $s _ { 2 }$ in the basis $\mathcal { C }$ is $\left( \begin{array} { c c } - 1 & 0 \\ 1 & 1 \end{array} \right)$.
We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$.\\
(a) Let $s _ { 1 } : E \longrightarrow E$ be the orthogonal reflection with respect to the line $\mathbb { R } \omega _ { 1 }$. Show that the matrix of $s _ { 1 }$ in the basis $\mathcal { C }$ is $\left( \begin{array} { c c } 1 & 1 \\ 0 & - 1 \end{array} \right)$.\\
(b) Let $s _ { 2 } : E \longrightarrow E$ be the orthogonal reflection with respect to the line $\mathbb { R } \omega _ { 2 }$. Show that the matrix of $s _ { 2 }$ in the basis $\mathcal { C }$ is $\left( \begin{array} { c c } - 1 & 0 \\ 1 & 1 \end{array} \right)$.