Advanced Linear Algebra - Advanced Linear Algebra, Lecture 5.1: Inner products and Euclidean structure

Advanced Linear Algebra, Lecture 5.1: Inner products and Euclidean structure Up until now, most of the previous theory have have developed has been algebraic in flavor, because we haven't had a metric. In this lecture, we will review the dot product in Euclidean space, and how it leads to concepts such as length and angle. We will derive the law of cosines, the Cauchy-Schwarz, and triangle inequalities. It turns out that all of this works for an arbitrary symmetric bilinear form, as long as it is additionally positive-definite. This is called an inner product, and a vector space endowed with an inner product is called an inner product space. We conclude with some non-standard examples (and non-examples) of inner products on R^2, the space of linear maps, and the space of continuous functions. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html