grandes-ecoles 2023 Q8

grandes-ecoles · France · x-ens-maths-a__mp Groups Group Homomorphisms and Isomorphisms

Let $\theta \in \mathbb{R}$ and $v \in \mathbb{H}^{\mathrm{im}} \cap S$, and let $u = (\cos\theta)E + (\sin\theta)v$. a) Show that $u \in S$ and that $u^{-1} = (\cos\theta)E - (\sin\theta)v$. b) Let $w \in \mathbb{H}^{\mathrm{im}} \cap S$ be a vector orthogonal to $v$. Describe the matrix of $C_u$ in the direct orthonormal basis $(v, w, vw)$ of $\mathbb{H}^{\mathrm{im}}$.

Let $\theta \in \mathbb{R}$ and $v \in \mathbb{H}^{\mathrm{im}} \cap S$, and let $u = (\cos\theta)E + (\sin\theta)v$.\\
a) Show that $u \in S$ and that $u^{-1} = (\cos\theta)E - (\sin\theta)v$.\\
b) Let $w \in \mathbb{H}^{\mathrm{im}} \cap S$ be a vector orthogonal to $v$. Describe the matrix of $C_u$ in the direct orthonormal basis $(v, w, vw)$ of $\mathbb{H}^{\mathrm{im}}$.

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