TLDR: Given that $A=X\cdot W + b$ where $A \in \mathbb{R}^{N\times M}, X\in \mathbb{R}^{N\times D}, W \in \mathbb{R}^{D\times M}$ and $b \in \mathbb{R}^{1\times M}$, where does $\frac{\partial A}{X_{i,j}}$, $\frac{\partial A}{W_{i,j}}$, $\frac{\partial A}{b_{i}}$ lie, in terms of their "index", in their respective Jacobian matrices? For example, would $\frac{\partial A}{X_{3,4}}$ be at (row 3, column 4) of its Jacobian matrix?
On a smaller note, what is the difference between $\frac{\partial F_{a,b}}{b_{c}}$ and $\frac{\partial F}{b_{c}}$? Specifically, is it correct to write
$$\frac{\partial F_{a,b}}{\partial b_{c}} = \sum_{i,j}\frac{\partial F_{a,b}}{\partial A_{i,j}} \cdot \frac{\partial A_{i,j}}{\partial b_{c}} \hspace{3mm}\text{ or }\hspace{3mm}\frac{\partial F}{\partial b_{c}} = \sum_{i,j}\frac{\partial F}{\partial A_{i,j}} \cdot \frac{\partial A_{i,j}}{\partial b_{c}}$$
Context: I'm currently doing CS231N as a hobby. In assignment 2, I'm asked to produce the Jacobian for several operations. Note I have not taken any formal linear algebra course, which means that my logic is not rigorous.
Consider case 1:
$W = \begin{bmatrix} W_{1,1} & W_{1,2} & W_{1,3} \\ W_{2,1} & W_{2,2} & W_{2,3} \\ W_{3,1} & W_{3,2} & W_{3,3} \\ \end{bmatrix}$, $X = \begin{bmatrix} X_{1,1} & X_{1,2} & X_{1,3} \\ X_{2,1} & X_{2,2} & X_{2,3} \\ X_{3,1} & X_{3,2} & X_{3,3} \\ \end{bmatrix}$, $b = \begin{bmatrix} b_{1}& b_{2}& b_{3} \end{bmatrix}$
Let $A = X\cdot W + b$ $\hspace{1mm}$(where b is "broadcasted" to be of dimension $3\times 3$).
More explicitly, $A = \begin{bmatrix} \sum_{j=1}^{3}X_{1,j}\cdot W_{j,1} + b_{1} & \sum_{j=1}^{3}X_{1,j}\cdot W_{j,2} + b_{2} & \sum_{j=1}^{3}X_{1,j}\cdot W_{j,3} + b_{3}\\ \sum_{j=1}^{3}X_{2,j}\cdot W_{j,1} + b_{1} &\dots & \dots \\ \sum_{j=1}^{3}X_{3,j}\cdot W_{j,1} + b_{1} &\dots & \dots \\ \end{bmatrix}$
Suppose that $A$ is put through an unknown function to yield $F \in \mathbb{R}^{3\times 3}$ and that we are given $\frac{\partial{F}}{\partial{A}} \in \mathbb{R}^{3\times 3}$. If my notation of $\frac{\partial F_{a,b}}{\partial b_{c}}$ means what I think it means then (but someone please correct me if I'm wrong lol),
$$\frac{\partial F_{a,b}}{\partial b_{c}} = \sum_{i,j}\frac{\partial F_{a,b}}{\partial A_{i,j}} \cdot \frac{\partial A_{i,j}}{\partial b_{c}}$$
Since $A_{a,b} = A'_{a,b}+b_{b}$ which means the partial derivative $\frac{\partial A_{i,j}}{\partial b_c}$ is $1$ when $j=c$ and $0$ otherwise,
$$\frac{\partial F_{a,b}}{\partial b_{c}} = \sum_{i}\frac{\partial F_{a,b}}{\partial A_{i,c}}$$
In this case, my intuition tells me that the partial derivatives go into the Jacobian following the index of $b$ in $\frac{\partial A}{b_{c}}$ ($c$):
$$\frac{\partial F}{\partial b} = \begin{bmatrix} \frac{\partial F}{\partial b_{1}}\\ \frac{\partial F}{\partial b_{2}}\\ \frac{\partial F}{\partial b_{3}}\\ \end{bmatrix} $$
Consider case 2:
Here is where my intuition of where partial derivatives should go fails. In particular, $\frac{\partial f}{W_{1,2}} = 2q_1 x_2 = 0.44 \times 0.4 = 0.176$ is located in (row 2, column 1) and not (row 1, column 2)! What confuses me more is that somehow the equation is also wrong because
$$2q\cdot x^T = \begin{bmatrix}0.44\\0.52\end{bmatrix}\cdot\begin{bmatrix}0.2&0.4\end{bmatrix}=\begin{bmatrix}0.088 &0.176\\0.104 & 0.208 \end{bmatrix} \not = \begin{bmatrix}0.088 &0.104\\0.176 & 0.208 \end{bmatrix}$$
This brings me to my source of confusion: how do I know which index to put any partial derivative in the Jacobian?
I can only partially answer my question. Looks like they corrected the slide in 2018 but lecture videos for 2018 were never released and I didn't think to look at it. At the moment I will say that it seems to be that $\frac{\partial A}{\partial b_{i,j}}$ should be in (row $i$, column $j$).
However, I still do not know how to interpret the difference between $\frac{\partial F}{b_{c}}$ and $\frac{\partial F_{a,b}}{b_{c}}$ (assumming $F$ is a matrix)...