Total Derivative as a Linear Map

Question

I am currently studying Calculus on Manifolds .I am studying Spivacks book along with Munkres and J.Shurmans notes since i might not understand something from one book to another.What i noticed is so far no one mentions a way to calculate the total derivative of a function f except using the definition ."finding a linear(multilinear) map such that a limit exists." What i wanted to ask is that if we know The Matrix of this map (jacobian matrix) and the Basis of the Vector Spaces .Then i can find that map=the total derivative with the standard way of finding a linear function "reversing" its matrix right? I tested it and was right for a couple of functions but i wanted to be sure since there is not on books or wikipedia.

Answer 1

Theorem. A map $T$ from a subset of $\Bbb{R}^{n}$ to $\Bbb{R}^{m}$ is linear iff there is some $m \times n$ matrix $M$ such that $T(x) = Mx$ for all $x$ in the domain of $T$.

Definition. The Jacobian matrix of a differentiable map $f$ from a set open in $\Bbb{R}^{n}$ to $\Bbb{R}^{m}$ at any $x$ in the domain of $f$ is the matrix of the derivative of $f$ at $x$ with respect to the usual bases.

Can you now deduce from the above an answer to your question?