In the book of Analysis on Manifolds by Munkres at page 235, it is given that
However, shouldn't the rows of the matrix have 4 row since the vectors are from $\mathbb{R}^4$ ?
As my main question (where I think the problem is), in the same book and in the same page, it is given also that
However, if I'm not mistaken, the author defined a determinant function for $n\times k$ matrices, which is something that I thought is not defined until now, so just to be clear, is there anything wrong with this definition and with my understanding, or am I not getting what I'm reading (again) ?