Exercise 16

Show that any non-$\vec{0}$ vector in any nontrivial vector space can be an eigenvector. That is, given a $\vec{v}\neq\vec{0}$ from a nontrivial $V$, show that there is a transformation $t\colon V\to V$ having a scalar eigenvalue $\lambda\in\mathbb{R}$ such that $\vec{v}\in V_{\lambda}$.

What if we are given a scalar $\lambda$? Can any non-$\vec{0}$ member of any nontrivial vector space be an eigenvector associated with $\lambda$?