# For back propagation in neural networks , how do we calculate vector by matrix derivative?

I am following the course deep learning ai by Andrew NG. In course1 week4, 04-06-Forward and Backward Propagation, he calculates backward propagation for layer $l$ in neural networks as follows (a single training sample case): $$dz^{[l]}=da^{[l]}*g^{[l]'}(z^{[l]}),(1)$$ $$dw^{[l]}=dz^{[l]}.a^{[l-1]}.(2)$$ The corresponding forward propagation is: $$z^{[l]}=w^{[l]}a^{[l-1]}+b^{[l]},(3)$$ $$a^{[l]}=g^{[l]}(z^{[l]}).(4)$$ I have some problems with Equation $(2)$: I think $$dw^{[l]}=dz^{[l]}.\frac{dw^{[l]}a^{[l-1]}}{dw^{[l]}},(5)$$ where $w^{[l]}$ is of shape $(n^{[l]},n^{[l-1]})$($n^{[l]}$ is the number of neurons in layer $l$) and $a^{[l-1]}$ is of shape $(n^{[l-1]},1)$, so $w^{[l]}a^{[l-1]}$ is $(n^{[l]},1)$, a column vector.

According to vector differentiation by a matrix, vector by matrix derivative is not defined. Then how can Equation $(5)$ be calculated and equal to $a^{[l-1]}$ in Equation (2)?

By the way, I think it should be $(a^{[l-1]})^T$ in Equation $(2)$, otherwise the shapes of $dz^{[l]}$ $(n^{[l]},1)$ $a^{[l-1]}$ $(n^{[l]}-1,1)$do not match at all.

It just so happens that the VJP of a function like $$f(W,a) = Wa$$ with respect to the weights $$W$$ is $$ga^T$$, where $$g$$ is the backpropagated gradient.