let T:R^n -> R^m be a linear transformation, and let {v1, v2, v3} be a linearly dependent set in R^n. Explain why the set {T(v1),(v2),(v2)} is linearly dependent.
first, the latter set contains two elements which are equal..
assuming my corrections below...
if $\displaystyle \{v_1, v_2, v_3\}$ is linearly dependent, then WLOG, $\displaystyle v_1 = r_2v_2+ r_3v_3$ where $\displaystyle r_2$ and $\displaystyle r_3$ are scalars..
$\displaystyle T(v_1) = T(r_2v_2+ r_3v_3) = r_2T(v_2) + r_3T(v_3)$ since $\displaystyle T$ is a linear transformation.
Thus you were able to express, WLOG, $\displaystyle T(v_1)$ as a linear combination of $\displaystyle T(v_2)$ and $\displaystyle T(v_3)$. Hence $\displaystyle \{T(v1),T(v_2),T(v_3)\}$