Given the system of equations:
$ x - y + 2z = 0 \\\\ x + 2y - z = 0 $
Using RREF yields the basis to the solution:
z*$\begin{bmatrix}-1\\\\1\\\\1\end{bmatrix}$
However if you set the equations to the following:
x = y - 2z
y = z
(follows from equation 1 replacing x in equation 2) -> the basis for this solution set is 2-dimensional with the following vectors:
$ y*\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \end{bmatrix} \And \ z*\begin{bmatrix} -2 \\\\ 1 \\\\ 1\end{bmatrix}$
This basis is independent and 2-dimensional, but is there anyway to get to the 1 dimensional basis from this set of 2 vectors, namely (1,0,0) and (-2,1,1)?