# Applied Mathematics 1411A/B Lecture Notes - Lecture 20: Pythagorean Theorem, Unit Circle, Elementary Matrix

## Document Summary

Recall: last day, we introduced the concept of a weighted inner product, and how to find distance and norm using this inner product. Definition: if v is an inner product space, then the set of points in v that satisfy. 1|| is called the unit sphere or sometimes the unit circle in v. Example: for the standard euclidean inner product in r2, the unit circle is a geometric circle. For the weighted inner product (geometrically) an ellipse. Verify the above theorem holds for this product on r2 generated by. Example: given the matrices below, find the inner product and the norm of each matrix. Theorem (properties of inner products): if u, v, and w are vectors in a real inner product space, and k is any scalar, then: v0 wvu wvu wvu vu k. 0 vu vu wu vu k wu wu wv. Angle and orthogonality in inner product spaces (6. 2; pg.