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 \(A \mathbf{z} \in W^{\perp} .\)