MTH 525, FALL 2006

Brief Notes for Week 15

(very tentative)

Monday, December 4

We study positive forms (sections 9.3, 9.4) headed toward the spectral theorem.

Wednesday, December 6

The spectral theorem and its corollaries (section 9.5.) If time permits, we discuss further properties of normal operators, finishing chapter 9.

Monday, December 11

Final Exam (200 pts.) over chapters 1-8.

Last modified October 1, 2006

Linear Algebra can be summarized (in this class) as having three major components:

I. Abstract vector spaces

II. Linear Transformations (= vector space homomorphisms)

III. “Good” bases (various canonical forms, or orthogonal bases.)

In part I we introduce

0. Some abstract algebra background: groups/fields/rings/vector spaces

1. Systems of linear equations

2. EROs

3. Matrices, RREF

4. Abstract vector spaces

5. Subspaces

6. Bases, dimension

7. Coordinates with respect to a basis

8. Matrix computation in this light.

Part I will take us about two weeks and covers chapters 1 & 2 of our textbook (Hoffman & Kunze.)

In part II we represent linear transformations by matrices (usually) and look at invariants of linear transformations. This material is in chapters 3-5 and is the first half of our course.

In part III we look represent a linear transformation with respect to a “good” basis. This leads to concepts of diagonalization, triangulation, cyclic vectors, Jordan form and rational form of matrices. We will also look at inner product spaces. This material will be covered in the last half of our course, chapters 6-9.