Derivative of a Linear Transformation

Question

I am having trouble grasping the concept of derivatives in relation to linear transformations. If I have a function f(x)=Tx which is the linear transformations given by the matrix T. Then what is the derivative Df(x)?

Answer 1

The derivative (or, better said, the differential) of a function $f:D\subseteq\mathbb R^n\to\mathbb R^m$ at the point $x_0\in D$ is, by definition, a linear transformation $L$ such that $$f(x_0 + x) = f(x_0) + Lx + o(x)$$ where $$\lim_{x\to 0}\frac{o(x)}{\|x\|} = 0$$

In particular, if $f$ is a linear transformation, then at any point $x_0$, the derivative of $f$ is simply $f$ itself, which can be easily proven, since $o(x)$ then becomes a very very simple function.