The linear code $C \cong \mathbb{F}^5_2$ is given by $C = \{(x_1, x_2, x_3, x_4, x_5) | x_1 + x_2 + x_3 = 0, x_4 + x_5 = 0$ in $\mathbb{F}_2\}$.
Write down a parity check matrix and a generator matrix for $C$.
For the parity check matrix I've let $\underline{x} = (x_1, x_2, x_3, x_4, x_5)$.
So for the condition $ x_1 + x_2 + x_3 = 0$ we have $\underline{x}.(1 1 1 0 0)= 0$.
And for the condition $x_4 + x_5 = 0$ we have $\underline{x}.(0 0 0 1 1)= 0$
So a parity check matrix is: \begin{bmatrix} 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ \end{bmatrix}
How do I then go on to construct a generator matrix? I'm struggling to understand my notes and don't know how to begin. I just know that the dimension of $C$ is $5 - 2 = 3$ so the generator matrix will have $5$ columns and $3$ rows.