TITLE: Understanding the derivative as a linear transformation
QUESTION [17 upvotes]: It's been a while now I am studying multivariable calculus and the concept of differentiation in space (or higher dimension). I saw relative posts but one question remains. I can't understand the concept of linear transformation that we use to define the Frechet derivative. In single variable the derivative is the best linear approximation of the function, so I guess this extends to multivariable but we can't use a number for this (why?) and instead we use a matrix. Can someone clears this for me in plain english?

REPLY [4 votes]: I have little to add to Michael's excellent answer. However, Dieudonne said it best: this is the introduction to his chapter on differentiation in Modern Analysis Chapter VIII.

The subject matter of this Chapter is nothing else but the elementary theorems of Calculus, which however are presented in a way which will probably be new to most students. That presentation, which throughout adheres strictly to our general "geometric" outlook on Analysis, aims at keeping as close as possible to the fundamental idea of Calculus, namely the "local" approximation of functions by linear functions. In the classical teaching of Calculus, the idea is immediately obscured by the accidental fact that, on a one-dimensional vector space, there is a one-to- one correspondence between linear forms and numbers, and therefore the derivative at a point is defined as a number instead of a linear form. This slavish subservience to the shibboleth of numerical interpretation at any cost becomes much worse when dealing with functions of several variables...

In other words, the confusion you face follows from thinking of the derivative wrongly in calculus I. "Wrong" in the sense that the idea does not generalize to higher dimensions directly. The derivative of a function from $\mathbb{R}^n \rightarrow \mathbb{R}^m$ is not another function from $\mathbb{R}^n \rightarrow \mathbb{R}^m$. Instead, it's a linear transformation, or if you prefer the Jacobian viewpoint, a matrix of functions.