Proposition:
Suppose that $T : R^n \to R^m$ is the linear transformation defined by
$T(x) = Mx$
for some m × n matrix $M$. Then
$DT(x) = M$
for all points $x \in R^n$
where $D$ is the partial derivative matrix. (Jacobian?)
Question:
I don't understand what is being said. $T(x)$ is a linear transformation on $x$. How does the partial derivative of $T(x)$ lead to the transformation matrix. Neither do I have an algebraic intuition nor a geometric one.
Further, How is the total derivative of $g(x,y,z)$ equal to $ Dg(x,y,z) \begin{pmatrix} x \\ y\\ z \end{pmatrix}$?
This is stated without proof. There is a chance, I made a wrong interpretation so I am pasting the portion of the text where it appears.
Is it that the change in $x$ in all dimensions of the output of $g(x,y,z)$ multiplied by $x$ and similar for y and z gives a total derivative. I don't seem to understand.I know the total derivative is a derivative taking into account that other variables are not constant during differentiation by one variable.