I already know how the set of solutions of system of linear equations over real numbers infinite field PP 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: ∀t∈R:p→+v→⋅t where p→∈Rnlandv→∈Rn where n denotes the number of real variables in each linear equation thus n∈NBut my question is how the set of solutions of system of linear equations over the finite field Z2 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 Z2 where Z2={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.How?

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I already know how the set of solutions of system of linear equations over real numbers infinite field PP 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: ∀t∈R:p→+v→⋅t where p→∈Rnlandv→∈Rn where n denotes the number of real variables in each linear equation thus n∈NBut my question is how the set of solutions of system of linear equations over the finite field Z2 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 Z2 where Z2={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&#039;t know. I am trying to google the answer for this question for days but I don&#039;t find the answer anywhere. It seems like nobody talks about this topic.How?

Halle Hansen · Accepted Answer

Theory of linear equations works exactly the same way no matter the field. Let k a field and A∈Mn(k)  The most relevant result is Kronecker-Capelli theorem:  Linear system Ax=b has solution if and only if rankA=rank[A∣b] where [A|b] denotes augmented system matrix.  In that case, let x0 be a particular solution of the system, i.e. Ax0=b. Then the solution set S of homogeneous linear system Ax=0 is vector space of dimension n−rankA and all solutions of the system Ax=b are given by  {x0+x∣Ax=0}=x0+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 R,C,ZpZ or whatever. It becomes relevant in spectral theory, though.

I already know how the set of solutions of system of linear equations over real numbers infinite fie

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