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Let $A$ be an $n \times n$ real self-adjoint matrix and $\mathbf{v}$ an eigenvector with eigenvalue $\lambda$. Let $W=\operatorname{span}\{\mathbf{v}\}$.

a) If $\mathbf{w} \in W$, show that $A \mathbf{w} \in W$
b) If $\mathbf{z} \in W^{\perp}$, show that $A \mathbf{z} \in W^{\perp} .$
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