Eigenvalues and eigenspaces of a matrix with a given property

#linear_algebra

Suppose $A$ is a real $n \times n$ matrix that satisfies $A^{2} v = 2 A v$ for every $v \in R^{n}$ .
(a) Show that the only possible eigenvalues of $A$ are 0 and 2.
(b) For each $λ \in R$ , let $E_{λ}$ denote the $λ$ -eigenspace of $A$ , i.e., $E_{λ} = {v \in R^{n} ∣ A v = λ v}$ . Prove that $R^{n} = E_{0} \oplus E_{2}$ . (Hint: For every vector $v$ one can write $v = (v - \frac{1}{2} A v) + \frac{1}{2} A v$ .)

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Suppose $A$ is a real $n\times n$ matrix that satisfies $A^2 {\bf v} = 2A{\bf v}$ for every ${\bf v}\in {\bf R}^n$.
\begin{enumerate}[label=\alph*)]
	\item Show that the only possible eigenvalues of $A$ are 0 and 2.
	\item For each $\lambda\in {\bf R}$, let $E_{\lambda}$ denote the $\lambda$-eigenspace of $A$, i.e., $E_{\lambda} = \{{\bf v}\in {\bf R}^n\mid A{\bf v}=\lambda {\bf v}\}$. Prove that ${\bf R}^n = E_0\oplus E_2$. ({\itshape Hint:} For every vector ${\bf v}$ one can write ${\bf v}=({\bf v}-\frac{1}{2}A{\bf v})+\frac{1}{2}A{\bf v}$.)
\end{enumerate}