I've just started to learn about generator matrices and I'm having some trouble understanding how to obtain codewords from them. I missed a lecture in class and am now studying for my midterm. I can't really find good answers online regarding the technique used to obtain codewords from a generator matrix - just a few examples in which the process is mostly omitted for brevity. For example, say we have a $7$-ary linear code $C$ with generator matrix
$$ G = \left[ \begin{matrix} 0 & 2 & 4 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 6 & 3 & 2 & 1 \\ \end{matrix} \right] $$
I know how to find equivalent generator matrices, but I have no clue as to how we actually extract codewords from a matrix like this. What are the actual steps one needs to go through to obtain the codewords of $C$?
The code $C$ is generated by the row space of the matrix $G$. Or said in another way each codeword is a linear combination of the rows of G. If you want to encode say $\mathbf{u}=(u_1,\ldots,u_k)$ with the matrix $G$ then the codeword $\mathbf{x}=(x_1,\ldots,x_k,x_{k+1},\ldots,x_n)$ will be
$$ \mathbf{x}=\mathbf{u}G = u_1G_1 + u_2G_2+\ldots + u_kG_k $$
where $G_i$ is the $i$-th row of $G$.