Show that the non zero rows of an echelon form matrix form a linearly independent set.
That’s what it is?
(1) (0) (0) = (0)
c1(0) + c2 (1) +c3 (0) = (0) so c1=0, c2=0, c3=0
(0) (0) (1) = (0)
I think you are trying to write
$\displaystyle c_1\begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}+ c_2\begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix}+ c_3\begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}= 0$
and derive from that that those vectors are independent. But that is the identity matrix, not a general "row echelon matrix". A three by three row echelon matrix would be of the form:
$\displaystyle \begin{bmatrix}a & b & c \\ 0 & d & e \\ 0 & 0 & f\end{bmatrix}$
so you should be looking at
$\displaystyle c_1\begin{bmatrix} a \\ 0 \\ 0\end{bmatrix}+ c_2\begin{bmatrix}b \\ d \\ 0\end{bmatrix}+ c_3\begin{bmatrix}c \\ e \\ f\end{bmatrix}$.
Also, I see nothing in your question that restricts this to 3 by 3 matrices. You should be able to do a general proof.