I'm trying to determine ifSis a basis forVifS={(1,2,1,0),(2,1,3,1),(2,2,1,2)} inV=spanS

so I made the matrix $\displaystyle \left[ \begin{array}{cccc} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 1 & 3 & 1 \\ 0 & 1 & 2 \end{array} \right]

$ and put it in (not reduced) row echelon form which gives $\displaystyle \left[ \begin{array}{cccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right]

$ and this means that (1,2,1,0),(2,1,3,1) and (2,2,1,2) form a basis for $\displaystyle R^4$ and the dimension is 3.

I have a feeling I'm doing something wrong. Can someone tell me if this is right?