## Proof That the Image of a Linearly Independent Set By a One to One Linear Transformation is Linearly Independent

Theorem
If
$T$
is a one to one linear transformation with domain
$D$
and
$S \subset D$
is linearly independent then
$T(S)$
is also linearly independent.
Proof
Suppose
$\left\{ \mathbf{v_1} , ..., \mathbf{v_n} \right\}$
be a linearly independent set of vectors so that
$\alpha_1 \mathbf{v_1} + ...+ \alpha_n \mathbf{v_n}\neq 0$
for any scalars
$\alpha_1 , ..., \alpha_n$
.
Transforming by
$T$
gives
\begin{aligned} 0=T(0) & \neq 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}

$T$
is one to one hence
$T(x)=T(0)=0 \rightarrow x=0$
so that
$\alpha_1 \mathbf{v_1} + ...+ \alpha_n \mathbf{v_n}\neq 0$
$\mathbf{v_1}, ..., \mathbf{v_n}$