*rot90* and *rot180* are geometric operations. What does that have to do with computing and applying gradients? Do you mean “transpose”? That’s a completely different thing.

Prof Ng has specifically designed these courses not to require any knowledge even of univariate calculus, let alone higher dimensional and matrix calculus. So any questions about the derivations of the formulas given to us are beyond the scope of this course. Here’s a thread that has links to some material on the web that goes into the math behind all this. (Please note that thread is linked from one of the topics on the DLS FAQ Thread, which is also worth a look if you haven’t seen it yet).

The one high level thing worth pointing out is that Prof Ng has done one sort of “simplification” in the way he presents the results of the math: he uses the convention that the gradient of an object and the base object are the same shape. If you really take the math literally and carefully, it turns out that the gradient of an object has the shape of the *transpose* of the base object. But for the way we are applying the gradients, it’s simpler to formulate things the way Prof Ng does. If you’re not actually doing the underlying math, his way is just simpler and cleaner. I’m guessing that is the real point that answers your question here.