Advanced Linear Algebra - Advanced Linear Algebra, Lecture 3.1: Determinant prerequisites

Advanced Linear Algebra, Lecture 3.1: Determinant prerequisites In this series of lectures, we will study the determinant of a linear map T from X to X in a basis-free way, i.e., without reference to a matrix. It will end up being a certain type of n-linear function. But to do this, we need some basic facts about permutations, which we cover in this lecture. We define the sign of a permutation using the discriminant of n variables. We also give an intuitive "unofficial" definition of the determinant -- the signed volume of the image of the unit cube, which describes the scaling factor the linear map. We explain why some some fundamental properties of the determinant follow immediately from this geometric concept. Our goal with this is to get some context and motivation as we work toward the abstract definition of the determinant. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html