Linear Algebra: Lecture 26: motivation, calculation and interpretation of determinants

I introduce the determinant. Motivated first by our search for a formula to capture invertibility then later, and even more significantly, as a means to calculate generalized volumes and orientation. An orientation is some device that allows us to answer the question: given n-1 vectors in order, then which n-th direction is up ? (thinking of the n-1 directions as filling out some hyperplane ). Some properties of determinants mentioned. Follows start of Part III of my notes. (we will transition to the text after this topic is concluded)
***Correction: there should just be an e_j in the top row of the determinant formula at the 35:55 minute mark, similar comments apply to the i,j,k formula to the left of it, each entry in the matrix I intend is just a number, unlike the trick-formula one often sees in calculus III***