Table for the general solution of the system of linear equations

Question

On this wikipedia page: https://en.wikibooks.org/wiki/Linear_Algebra/General_%3D_Particular_%2B_Homogeneous#th:GenEqPartPlusHomo, after the proof of corollary 3.11 there is a table that says this: no particular solution + infinite solutions to the homogenous system = no solutions. How can this be the case when if we are given a homogenous system at the start, an derive infinite number of solutions, there won't be a particular solution, and we can conclude that the solution set is infinite. Why does this say that there are no solutions? Can someone explain the table in general? Thanks.

Answer 1

I'm not quite sure I have understood your post, but you seem to be considering the case where the homogeneous system has no particular solution.

This case is impossible since a homogeneous system means a system $$A{\bf x}={\bf0}\ ,$$ and it always has a solution, for example, ${\bf x}={\bf0}$.

Answer 2

For any solution $\mathbf v$ of $A\mathbf x=\mathbf b$, $\mathbf v+\mathbf w$, where $\mathbf w$ is any solution of the homogeneous equation $A\mathbf x=0$, is also a solution. The vector $\mathbf v$ is called a particular solution of the equation. In fact, all solutions of the original equation are of this form. That’s the gist of the Wikipedia article that you reference in your question. The set of solution to the homogeneous equation is called the null space of $A$.

If there’s a nonzero solution to the homogeneous equation, the dimension of the null space is greater than zero and the original equation potentially has a multitude of solutions. However, unless $\mathbf b=0$, elements of the null space do not solve the original equation, so you need still some other vector that’s outside of the null space as a starting point for its solution set. (Indeed, for the equation to have any solution at all, $\mathbf b$ must be an element of $A$’s column space.) So it’s entirely possible for the homogenous equation to have an infinite number of solutions, but for the inhomogeneous equation to have none.