Let $S=\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$ be a set of linearly dependent vectors in $V$, and let $T$ be a linear transformation from $V$ into $V$. Prove that the set $\left\{T\left(\mathbf{v}_{1}\right), T\left(\mathbf{v}_{2}\right), \ldots, T\left(\mathbf{v}_{n}\right)\right\}$ is linearly dependent.