Question

Let \(\vec{e}_{1}, \vec{e}_{2}\), and \(\vec{e}_{3}\) be the standard basis for \(\mathbb{R}^{3}\) and let \(L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) be a linear transformation with the properties
\[
L\left(\vec{e}_{1}\right)=\vec{e}_{2}, \quad L\left(\vec{e}_{2}\right)=2 \vec{e}_{1}+\vec{e}_{2}, \quad L\left(\vec{e}_{1}+\vec{e}_{2}+\vec{e}_{3}\right)=\vec{e}_{3} .
\]
Find a vector \(\vec{v}\) such that \(L(\vec{v})=k \vec{v}\) for some real number \(k\).