Answer
See explanation
Work Step by Step
since $\left\{\mathrm{v}_{1}, \mathrm{v}_{2}, \mathrm{v}_{3}\right\}$ is a linearly dependent set, there exist weights $c_{1}, c_{2}, c_{3}$ such that
\[
c_{1} \mathrm{v}_{1}+c_{2} \mathrm{v}_{2}+c_{3} \mathrm{v}_{3}=0
\]
Thus,
\[
T\left(c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}+c_{3} \mathbf{v}_{3}\right)=T(\mathbf{0})
\]
By the properties of linear equations,
\[
c_{1} T\left(\mathbf{v}_{1}\right)+c_{2} T\left(\mathbf{v}_{2}\right)+c_{3} T\left(\mathbf{v}_{3}\right)=\mathbf{0}
\]
Thus, $\left\{T\left(\mathbf{v}_{1}\right), T\left(\mathbf{v}_{2}\right), T\left(\mathbf{v}_{3}\right)\right\}$ is a linearly dependent set
