Here's the problem:P
Find the Jacobian matrix of the following vector-valued multi-variable functions.
$f \colon \mathbb{R}^n \to \mathbb{R}^m$ is defined by $f(x) = \mathbf{A}\boldsymbol{x} − \boldsymbol{b}\,$, where $\boldsymbol{x} \in \mathbb{R}^n\,$, $\mathbf{A} \in \mathbb{R}^{m \times n}\,$, $\boldsymbol{b} \in \mathbb{R}^m$.
What I Know
I am lost here. I know how to calculate a Jacobian Matrix for a given function. But I don't know how to get start for a vector-valued multi-variable functions like this one. Should I start with converting the function to matrix form first or? I just need a tip to start with. Thanks.
Writing out our matrix $\mathbf{A}$, our vector $\boldsymbol{v}$, and our vector $\boldsymbol{b}$:
$$\mathbf{A} = \begin{bmatrix}\boldsymbol{a}_1\\ \boldsymbol{a}_2\\ \vdots\\ \boldsymbol{a}_m \end{bmatrix} = \begin{bmatrix} a_{11} &a_{12} &\cdots &a_{1n}\\ a_{21} &a_{22} &\cdots &a_{2n}\\ \vdots &\vdots &\ddots &\vdots\\ a_{m1} &a_{m2} &\cdots &a_{mn}\end{bmatrix}, \qquad\boldsymbol{x}=\begin{bmatrix}x_1\\ x_2\\ \vdots\\ x_n\end{bmatrix}, \qquad\boldsymbol{b} = \begin{bmatrix}b_1\\ b_2\\ \vdots\\ b_m\end{bmatrix}$$
Where each $\boldsymbol{a}_i = \begin{bmatrix} a_{i1} &a_{i2} &\cdots &a_{in}\end{bmatrix}$ is a $1 \times n$ matrix (it can be seen as an $n$-dimensional covector if you are familiar with those).
Thus,
$$\boldsymbol{f}(\boldsymbol{x}) = \mathbf{A}\boldsymbol{x}-\boldsymbol{b} = \begin{bmatrix}\boldsymbol{a}_1\boldsymbol{x} - b_1\\ \boldsymbol{a}_2\boldsymbol{x} - b_2\\ \vdots\\ \boldsymbol{a}_m\boldsymbol{x} - b_m\end{bmatrix} = \begin{bmatrix}f_1(\boldsymbol{x})\\ f_2(\boldsymbol{x})\\ \vdots\\ f_m(\boldsymbol{x}) \end{bmatrix}$$
Where $\boldsymbol{a}_i\boldsymbol{x} = {\boldsymbol{a}_i}^\intercal\cdot\boldsymbol{x}\,$.
For any $f_i(\boldsymbol{x})$, we have
$$f_i(\boldsymbol{x}) = \left(\sum_{j=1}^n{a_{ij}x_j}\right) - b_i$$
Using this, can you work out the Jacobian matrix $\mathbf{J}_{\boldsymbol{f}}\,$?