I took a Linear Algebra course last year: the relationship between linear transformations, matrices and basis was carefully explained and also given a clear geo

I took a Linear Algebra course last year: the relationship between linear transformations, matrices and basis was carefully explained and also given a clear geometrical interpretation but the determinant was introduced as some magical matrix function with all the property we needed (multilinear, alternating, ...) and the just proceeded to prove its existance and uniqueness. I have never even thought it was related to area before. Also the justification (in chapter 8 part 2) for the formal determinant calculation to obtain the cross product is amazing. I'm still not sure what the essence of the cross product is: how it is related to divisions algebra (quaternions and octonions) and how bivectors fit in the context of general vector spaces. This was not in the scope of the course however.