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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?

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