Why can the Jacobian represent transformations?

[Coffee_]

Why is it so that I can write:

$x'_i=A_{ij}x_j$ where $A_{ij}=\frac{\partial x'_i}{\partial x_j}$?

Yes if the first expression is assumed it is clear to me why the coefficients have to be the partial derivatives, but why can we assume that we can always write it in a linear fashion in the first place? I assume this is something similar to any function being linearly apporximated close enough to a point but I'd like to hear it to be sure.

 

[Orodruin]

I assume this is something similar to any function being linearly apporximated close enough to a point but I'd like to hear it to be sure.

When you ask questions like this it would be good to have some context. In many cases, you will be dealing with linear transformations only.

 

[Coffee_]

When you ask questions like this it would be good to have some context. In many cases, you will be dealing with linear transformations only.

This is supposed to be a general 'math for physics' chapter in my mechanics course. We seem to have not made any type of restrictions on what kind of transformations we make or at least, I don't remember the prof mentioning it. This is an introduction to concepts like the metric tensor,covariance and such. I was just a bit confused why formally we could assume that a general transformation could be expressed as $\vec{x'}=A\vec{x}$ where $A$ is a matrix with the partial derivatives. Once we assume that, obviously I see why A is supposed to be filled with partial derivatives, but the assumption in the first place isn't clear.