$T^2 (v)=-v \, \, \, \forall v \in\mathbb R ^n $ a linear transformation , find $n$ is even.

Question

Let $T:\mathbb R^n\to\mathbb R^n $ be a linear transformation satisfying $ T ^2 (v)=-v $ for all $v\in\mathbb R ^n $.

Then show that:

(A) $n$ is always even.

(B) $T$ is an isomorphism

(C) $T$ becomes a $\mathbb C$-linear isomorphism.

I proceed as

If we define $T $ by $T:\mathbb R^2\to \mathbb R^2$ by $T(x_1, x_2)=(-x_2,x_1)$ Then $T^2=-Id $.

Answer 1

Taking the $\det$: $$\det(T^2)=\det(T)^2=\det(-I_n)=(-1)^n$$ so $(-1)^n \geq 0$ i.e $n$ is even and $\det(T) \neq 0$ as $\det(T)^2 \neq 0$.