We have been going through how to solve the system of equations known as $Ax=b$, where $A$ is a matrix, $x$ is a vector and $b$ is a vector. I understand that if we have $A$ and $b$ we must find out what $x$ is, this happens via Gauss-Jordan elimination, back substitution, etc.
What does solving the linear system of equations actually mean though? Is it the point in space where all of the vectors of $A$ intersect, and what will this be useful for? Real-life examples are appreciated!
Best Answer
A system $$Ax=b\tag{1}$$ of equations in unknowns $x_1$, $x_2$, $\ldots$, $x_n$ implicitly defines the subset $$S:=\{x\in{\mathbb R}^n\>|\>Ax=b\}\quad\subset{\mathbb R}^n\ .$$ "Implicitly" means that for any given $x\in{\mathbb R}^n$ it is easy to test whether it is an element of $S$ or not (just compute $Ax$ and check whether this is $=b$); but you don't have a-priori a complete overview over this set $S$.
Solving the system $(1)$ (or a similar system containing equations of a more complicated nature) means obtaining such an overview. If $S$ is in fact the empty set you want a proof of this fact; if $S$ contains just finitely many points you want an explicit list of these points, etc.
In linear algebra the favorite case is when $S$ is a one-element set $\{a\}$; we then call $a$ "the" solution of $(1)$. But it very often happens that $S$ is an infinite set; say, a two-dimensional plane embedded in ${\mathbb R}^n$. In such a case you want an explicit "production scheme" with a certain number of free variables, in other words: a parametric representation of $S$, which generates every point of $S$ exactly once. In the case where $S$ is a two-dimensional plane such a parametric representation looks like $$S:\quad(u,v)\mapsto x:=a+up+vq \qquad\bigl((u,v)\in{\mathbb R}^2\bigr)\ ,\tag{2}$$ whereby the vectors $a$, $p$, $q$ have to be computed from the data $A$ and $b$. (Note that the same $S$ has many different representations $(2)$.)