Does multiplying a set of linearly dependent vectors in $\mathbb R^n$ by a $n \times n$ matrix $A$ result in a still linearly dependent set?

Question

Does multiplying a set of $k$ linearly dependent vectors in $\mathbb R^n$ by a $n \times n$ matrix $A$ result in a still linearly dependent set of vectors?

I believe this is true but I am not able to prove it. Any help would be greatly appreciated.

Answer 1

Supoose for a particular vector $c \in \mathbb{R}^k$, $$\sum_{i=1}^k c_iv_i=0$$

where $\exists$ index $j$ such that $c_j \ne 0$,

Then we still have , $$\sum_{i=1}^k c_iAv_i=0$$