when writing vectorized form for dw, can we rewrite this as dw= (1/m) * np.sum(np.dot(X,dz.T))???
Sorry, but that is not equivalent to the formula as written. Note that your formula will produce a scalar value. We need dw to be a vector with the same dimensions as w.
Also note that matrix multiplication involves multiplication followed by addition, so there is an implied addition in the np.dot.
You can also gain some insight by doing the “dimensional analysis”. What are the shapes of the objects involved? We have:
X is n_x x m where n_x is the number of features and m is the number of samples.
Z is 1 x m, so that means that dZ is also 1 x m, but you can also see that because both A and Y are 1 x m.
So what will happen when we compute X \cdot dZ^T? It will be n_x x m dotted with m x 1 (because of the transpose), so the result will be n_x x 1. That is correct, since w is n_x x 1.
If we write out the formula in python/numpy, it is:
dw = 1./m * np.dot(X, dZ.T)
If you then take the sum of that, you’ll end up with a scalar value, which is not what we want.