If
\[T\]
is a one to one linear transformation with domain \[D\]
and \[S \subset D\]
is linearly dependent then \[T(S)\]
is also linearly dependent.Proof
Suppose
\[\left\{ \mathbf{v_1} , ..., \mathbf{v_n} \right\}\]
be a linearly dependent set of vectors so that \[\alpha_1 \mathbf{v_1} + ...+ \alpha_n \mathbf{v_n}=0\]
for some scalars \[\alpha_1 , ..., \alpha_n\]
.Transforming by
\[T\]
gives\[\begin{equation} \begin{aligned} 0=T(0) &= T(\alpha_1 \mathbf{v_1} + ...+ \alpha_n \mathbf{v_n}) \\ &= T(\alpha_1 \mathbf{v_1}) + ...+ T(\alpha_n \mathbf{v_n})\\ &=\alpha_1 T(\mathbf{v_1}) + ...+ \alpha_n T(\mathbf{v_n}) \end{aligned} \end{equation}\]
Hence
\[ T(\mathbf{v_1}) , ...+, T(\mathbf{v_n})\]
are linearly dependent.