volume in linear algebra

Question

I'm reading linear Algebra Done Right, and I don't understand why we come up with the formula inside the blue circuit. Someone please help me.

Answer 1

By definition of our set $\Gamma$ we have that if $\omega \in \Gamma$ then $\omega = x + y$ for some $y$ such that $\Vert y\Vert$ small.

In particular it follows by using the derivative as an approximation and the linearity of the derivative that \begin{align*} \sigma(\omega) &\approx \sigma(x) + \sigma'(x)(y) \\ &= \sigma(x) + \sigma'(x)(\omega-x) \\ &= \sigma(x) + \sigma'(x)\omega - \sigma'(x)x \\ &= \sigma(x)- \sigma'(x)x + \sigma'(x)\omega \end{align*}

This means that the following sets are approximately the same $$ \sigma(\Gamma) \approx \sigma( x) + \sigma'(x)x + \sigma'(x)(\Gamma)$$ Since adding constant vectors preserves volume we have

$$ \text{ Volume } \sigma(\Gamma) \approx \text{Volume }\sigma'(x)\Gamma$$