I have the following system of equations:
x + y + z = 2
x - y + z + t = 1
I have converted this into a following matrix:
\begin{bmatrix}1&1&1&0|2\\1&-1&1&1|1\\\end{bmatrix}
After applying Gauss elimination method I get this:
\begin{bmatrix}1&1&1&0|2\\0&-2&0&1|-1\\\end{bmatrix}
From this matrix I see that it has 3 non pivotal columns on the left side (all but the first). Therefore according to Rouche theorem I conclude that the dimension of its basis needs to be 3. I need not go further because the textbook solution says 4.
Where have I made a mistake?