Linear Algebra: Residual $A x - b$ I'm doing a course on Applied Linear Algebra. Which, for the most part, is about applications in Octave, so there isn't much about the happenings behind a lot of the mechanics of everything.
I'm faced with the statement: 
"You can use the condition number to estimate the accuracy at which Octave solves for $x$ in $A x = b$. "First, we define the residual which is $A x - b$."
And I'm battling to understand this residual "$A x - b$". 
I'm trying to find some answers as to what this residual is and what to do with it.
Any help or reading material I can explore will be greatly appreciated.
 A: By numerical solution of $Ax=b$ we obtain $\bar x$.
If $\bar x$ was an exact solution it should be $$A\bar x=b \implies A\bar x-b=0$$
since we always have  some kind of numerical error or approximation we have a residual (vector) that is
$$r=A \bar x-b\neq 0$$
We can extimate the error by the norm |r|.
A: You can consider a matrix $A \in \mathbb{R}^{m{\times}n}$, as a procedure to move point (or vector) x in $\mathbb{R}^n$ to a point b in  $\mathbb{R}^m$, denoted $Ax=b$. Usually, $m=n$, meaning the matrix is square. 
An example would be to have a matrix $A \in \mathbb{R}^{3{\times}3}$ and x in $\mathbb{R}^3$, which can be considered our space. The center of the sun could be defined as the origin i.e. $(0, 0, 0)$. If we compute $Ax=b$ , we compute how the matrix $A$ moves point $x$ in space to $b$. The trajectory of Heavy Falcon can then be described as $x_{i+1}=Ax_{i}$ and $x_{0}$ would then be Cape Canaveral.
A linear system of equation $Ax=b$ can be reformulated as: given the "moving procedure" $A$ and my final position $b$, where did I start? 
The residual is the difference between $Ax_{guess}$ and $b$, meaning we guess that we started from $x_{guess}$ and since we know the "moving procedure" $A$, we end up at $Ax_{guess}$. If your $x_{guess}$ is correct, we end up at $b$. Usually the distance (euclidean norm) is used: $||Ax_{guess}-b||_2$. 
