How the set of solutions of system of linear equations over finite field GF(2) is expressed?

Question

I already know how the set of solutions of system of linear equations over real numbers infinite field $\Bbb{R}$ is expressed.

When there is only single solution then it is just a vector of scalars, where each scalar is a real number.

When there are more than 1 solution, and actually infinite solutions, then it is just a parametric linear vector space in the form: $\forall t \in \Bbb{R}: \vec p+\vec v \cdot t$ where $\vec p \in \Bbb{R}^n \land \vec v \in \Bbb{R}^n$ where $n$ denotes the number of real variables in each linear equation thus $n \in \Bbb{N}$.

But my question is how the set of solutions of system of linear equations over the finite field $\Bbb{Z}_2$ or galois field GF(2) is expressed?

I already know that when there is only single solution then it is also just a vector of scalars, but where each scalar is a binary number either zero or one in $\Bbb{Z}_2$ where $\Bbb{Z}_2=$ {0,1}, but when there are more than 1 solution, but always finite number of solutions, then how are they expressed?

Is this similar to how real solutions are expressed by parametric linear vector space by modulo 2? Or something else? I don't know. I am trying to google the answer for this question for days but I don't find the answer anywhere. It seems like nobody talks about this topic.

Do you know how?

Answer 1

Theory of linear equations works exactly the same way no matter the field. Let $k$ a field and $A\in M_n(k)$.

The most relevant result is Kronecker-Capelli theorem:

Linear system $Ax = b$ has solution if and only if $\operatorname{rank} A = \operatorname{rank}[A|b]$ where $[A|b]$ denotes augmented system matrix.

In that case, let $x_0$ be a particular solution of the system, i.e. $Ax_0 = b$. Then the solution set $S$ of homogeneous linear system $Ax = 0$ is vector space of dimension $n-\operatorname{rank} A$ and all solutions of the system $Ax = b$ are given by $$\{x_0+x\mid Ax = 0\} = x_0+S.$$

These results are elementary and will be found in any Linear Algebra textbook. However, it is important to note that ground field $k$ is completely irrelevant in this context, be it $\mathbb R$, $\mathbb C$, $\mathbb Z/p\mathbb Z$ or whatever. It becomes relevant in spectral theory, though.