Gradient and transpose I have simple question, when taking the Jacobian of a linear vector function, do you have to transpose the matrix $A$?
$b\in\mathbb{R}^{n},x\in\mathbb{R}^{m}\\J(Ax+b) = A^T, A\in\mathbb{R}^{n\times m}$

$J_x(f)$ is the, in case of existence, uniquely determined matrix with the property 
$$f(x+h)=f(x)+J_x(f) h+\mathrm o(\Vert h\Vert).$$
When plugging in $f(x)=Ax+b$, you are looking for a $J$ with
$$ A(x+h)+b=Ax + b + J_x(f)h+\mathrm  o(\Vert h\Vert).$$
Since the left side is equal to $Ax + Ah + b$ (because of linearity), you see, that the unique solution is $J_x(f)=A$, and you can even drop the $\mathrm o$-term.
