Exercise 1, § 1

Let $(e_1, e_2)$ be the canonical basis of $\mathbf{R}^2$ and $u$ the element of $A =\mathscr{L}(\mathbf{R}^2)$ defined by $u(e_1) =e_2,u(e_2) =-e_1$. Prove that $Sp_A(u) =\emptyset$ and Sp$_A(u^2) =$ $\{-1\}$.