It is confusing when the matrix W is first referred to as already transposed:

but later on it seems that the definition of the matrix W has changed in the sense that W is not transposed, and needs to be transposed, and thus the superscript βTβ is used:

In the first slide, W has dimensions (4,3), which means, it is *already* transposed, since a single example X has the dimensions (3, 1) and thus, every row vector in W must have 3 elements/parameters, each for each of X.

However, in the next video, the superscript βTβ is used, which is confusing. Has the matrix W become (3,4) again, so that we need to transpose it?

It becomes even more confusing in the quiz, when it is not clear which version of W (transposed or not) is assumed. So a note in the lecture page would help.