How can I determine the Jacobi matrix for:
Let $A \in \mathbb R^{m \times n}$ and $b\in \mathbb R^m$
$h: \mathbb R^n \rightarrow \mathbb R^m, x \mapsto Ax-b$
For a mapping like the following it is clear to me:
$f(x,y,z) = \binom{x^2+y^2+z \cdot sin x}{z^2+z \cdot sin y}$
Question: How can I get the Jacobi matrix for a general mapping like $h$?
I have tried to understand it with the help of an example:
$$ \begin{pmatrix} 3 & 4 & 5 & 6 \\ 7 & 8 & 9 & 10 \\ 11 & 12 & 13 & 14 \end{pmatrix} \cdot \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \begin{pmatrix} 3x_1+4x_2+5x_3+6x_4 \\ 7x_1+8x_2+9x_3+10x_4\\ 11x_!+12x_2+13x_3+14x_4 \end{pmatrix} = \begin{pmatrix} \sum_{j=1, i=1}^m A_{ij}x_i \\ \vdots \\ \sum_{j=1, i=n}^m A_{ij}x_i \end{pmatrix} $$
The expression you wrote at the end of your question is really close to the solution. You have $$Ax+b=\begin{pmatrix} \sum_j A_{1j}x_j+b_1\\ \vdots\\ \sum_j A_{nj}x_j+b_n \end{pmatrix}.$$ The Jacobian matrix $J$ is the matrix whose coordinates are $J_{ij}=\frac{\partial h_i}{\partial x_j}$. Here $h_i=\sum_j A_{ij}x_j+b_i$ and $\frac{\partial h_i}{\partial x_j}=A_{ij}$. Thus $J_{ij}=A_{ij}$ and $J=A$.