$G = \langle \begin{bmatrix}1&1\\1&1\\ \end{bmatrix}, \begin{bmatrix}1&1\\1&0\\ \end{bmatrix}, \begin{bmatrix}2&-3\\1&1\\ \end{bmatrix},\begin{bmatrix}4&-1\\3&2\\ \end{bmatrix}$

First I write all the matrices as vectors:

$M\_1 = \begin{bmatrix}1\\1\\1\\1 \end{bmatrix}$

$M\_2 = \begin{bmatrix}1\\1\\1\\0 \end{bmatrix}$

$M\_3 = \begin{bmatrix}2\\-3\\1\\1 \end{bmatrix}$

$M\_4 = \begin{bmatrix}4\\-1\\3\\2 \end{bmatrix}$

Then I found the rref:

$$\begin{bmatrix}1 & 1 & 1 & 1\\1 & 1 & 1 & 0\\2 & -3 & 1 & 1\\4 & -1 & 3 & 2 \end{bmatrix} \rightarrow (...) \rightarrow \begin{bmatrix}1 & 0 & \frac{4}{5} & 0\\0 & 1 & \frac{1}{5} & 0\\0 & 0 & 0 & 1\\0 & 0 & 0 & 0 \end{bmatrix}$$

The basis I get is

$$(\begin{bmatrix}1 &0\\4/5&0 \end{bmatrix},\begin{bmatrix}0 &1\\\frac{1}{5}&0 \end{bmatrix}),\begin{bmatrix}0 &0\\0&1 \end{bmatrix}$$

Is this correct? It doesn't even come close to the solution in my book which is

$$(\begin{bmatrix}1 &1\\1&1 \end{bmatrix},\begin{bmatrix}1 &1\\1&0 \end{bmatrix}),\begin{bmatrix}2 &-3\\1&1 \end{bmatrix}$$