Answer
Suppose $\{ \mathbf{v}_1, ..., \mathbf{v}_p \}$ is a linearly dependent subset of $V$. Then by the definition of linear dependence, there exist scalars $c_1, ..., c_p$, not all zero, such that $c_1 \mathbf{v}_1 + \cdots + c_p \mathbf{v}_p = \mathbf{0}$. Applying $T$ to both sides yields
$$T(c_1 \mathbf{v}_1 + \cdots + c_p \mathbf{v}_p) = T(\mathbf{0}).$$ But $T$ is a linear transformation, so the superposition principle and the identity $T(\mathbf{0})=\mathbf{0}$ both hold. Hence, the above equation implies
$$c_1 T(\mathbf{v}_1) + \cdots + c_p T(\mathbf{v}_p)=\mathbf{0}.$$ Since $c_1, ..., c_p$ are not all zero, this is a linear dependence relationship among the elements of $\{ T(\mathbf{v}_1), ..., T(\mathbf{v}_p)\}$. Therefore, this set must be linearly dependent in $W$. $\blacksquare$
Work Step by Step
Equations $(3)$ and $(4)$ on page 67 state the superposition principle and the identity $T(\mathbf{0})=\mathbf{0}$ for linear transformations.
