MTH 525, FALL 2006

Brief Notes for Weeks 12 & 13

(tentative)

Monday, November 13

Wednesday, November 15

Assignment 9 is collected.

We study cyclic subspaces, cyclic decomposition and the rational form (sections 7.1 and 7.2)

Monday, November 20

Assignment 9 is collected before 3 PM on Wednesday. (There is no class Wednesday.)

We finish chapter 7, looking at the Jordan form (7.3.) We summarize the chapter by computing invariant forms (7.4) and discussing semi-simple operators (7.5).

We introduce inner products in preparation for the study of inner product spaces, Chapter 8.

We define inner product and give some examples. We define quadratic form.

Do p. 275.

We define innter product spaces and basic properties (Theorem 1) of an inner product on an inner product space. We define orthogonal and orthonormal. Theorems 2 (a set of orthogonal vectors is independent) and 3 motivate the study, since they show that inner products give us “good” sets of bases. We study orthogonal projections (Theorem 4.)

We study linear functionals and adjoints (section 8.3) and unitary operators (section 8.4)

We study unitary and normal operators (sections 8.4 and 8.5.)

Last modified October 1, 2006