Answer
Any point x on the line through p in the direction of v satisfies the parametric equation $ x = p + tv $ for some value of t. By linearity, the image T(x) satisfies the parametric equation $T(x) = T (p+tv) = T (p) + tT(v) $
IF $ t(V) = 0$, then $T(x) = T(p)$ for all values of $t$, and the image of the original line is just a single point. Otherwise,$ (*) $is the parametric equation of a line through $T(p)$ in the direction of $T(v)$.
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