grandes-ecoles 2023 Q22

grandes-ecoles · France · x-ens-maths-a__mp Groups Algebra and Subalgebra Proofs

We fix an element $i_A$ of $A$ such that $i_A^2 = -1$. We denote $U = \mathbb{R} + \mathbb{R}i_A$ and we define the map $$T : A \rightarrow A,\quad T(x) = i_A x i_A.$$ We denote $\mathrm{id} : A \rightarrow A$ the identity map of $A$. a) Show that $T(xy) = -T(x)T(y)$ for all $x, y \in A$. b) Calculate $T^2 = T \circ T$ and deduce that $A = \ker(T - \mathrm{id}) \oplus \ker(T + \mathrm{id})$.

We fix an element $i_A$ of $A$ such that $i_A^2 = -1$. We denote $U = \mathbb{R} + \mathbb{R}i_A$ and we define the map
$$T : A \rightarrow A,\quad T(x) = i_A x i_A.$$
We denote $\mathrm{id} : A \rightarrow A$ the identity map of $A$.\\
a) Show that $T(xy) = -T(x)T(y)$ for all $x, y \in A$.\\
b) Calculate $T^2 = T \circ T$ and deduce that $A = \ker(T - \mathrm{id}) \oplus \ker(T + \mathrm{id})$.

Paper Questions

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