Let be $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ the operator given by $T=3ref_{x_2=2x1}$, where $ref_{x_2=2x1}$ is the reflection on the line $x_2=2x_1$. Define the operator $U:{(\mathbb{R}^2)}^n \rightarrow {(\mathbb{R}^2)}^n$ by $U(v_1,..,v_n):=(T(v_1),...,T(v_n))$ for all $(v_1,..,v_n) \in {(\mathbb{R}^2)}^n$. I need to determine if $U$ is ortogonal, normal, or self-adjoint, and find $det U$.
I try to use the canonical basis of $R^{2n}$, $e_1, e_2,...,e_{2n}$, and find the matrix representation of $U$ in this base, to determine if $U$ is ortogonal, normal, or self-adjoint.
