Let $A\in \mathbb{R^{n\times n}}$ be an invertible matrix, $b\in \mathbb{R^n}$ and let $\|.\|$ be a given norm on $\mathbb{R^n}$ with an induced matrix norm. For the relative error $\delta_x$ holds the following inequality: $$\kappa(A)^{-1}\frac{\| r\|}{\|b\|}\leq \delta_x=\frac{\|x'-x\|}{\|x\|}\leq \kappa(A)\frac{\|r\|}{\|b\|} $$ Where $\kappa(A)=\|A^{-1}\|\|A\|$ is the condition number and $r=b-Ax'$ is the residual approximate of $x'$.
I came across this in my lecture notes of numerical analysis and the professor treated it as a very obvious thing but I don't really see how its supposed to so, could someone provide a proof.
You know $$\|Av\|\le \|A\|\,\|v\|$$ by the definition of the induced operator norm. With the inverse matrix you can go the other way, $$ \|v\|=\|A^{-1}Av\|\le\|A^{-1}\|\,\|Av\|. $$ Now combine $$ \|A\|^{-1}\|Av\|\le \|v\|\le\|A^{-1}\|\,\|Av\| $$ for numerator and denomiator in $\frac{\|x'-x\|}{\|x\|}$ and you get the claim.