Chapter 6.1, Problem 24E

Let $x_1,...,x_r$ be eigenvectors of an $n\times n$ matrix A and let S be the subspace of ${ℝ}^n$ spanned by $x_1,x_2,...,x_r.$ Show that S is invariant under A (i.e., show that $Ax\in S$ whenever $x\in S).$