I'm trying to get a better intuition regarding Jacobians. I think I have a decent understanding of gradients and what their directions mean, but I'm trying to connect all the dots for the generalisation of The Jacobian.
I understand that taking the Jacobian of a vector field in $\mathbb{R}^2$ would give us a 2 by 2 matrix that can be viewed as the best linear transformation at a certain point $v$, but what if the dimension of the function output is different from the input, as f.e with the gradient and scalar functions, would that give us the best approximated linear transformation in that other dimension?
What I'm asking is, if we view the gradient as a 1 by 2 matrix $A = \begin{bmatrix}\partial f_x, \partial f_y \end{bmatrix}$ that we can use to transform other vectors, would that give us the best 1 dimensional approximated linear transformation at a certain point?