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

Theorem
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.

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