Math 2406: Abstract Vector Spaces
Lecture 19 Plan
(Tuesday, October 30
Apostol 6.1-6.5.
We'll forgo the remaining parts of Chap. 5 on cofactors and Cramer's rule (which are not used very often). Instead, we begin the very important material in Chap. 6 on eigenvalues and eigenvectors. The eigenvectors of a linear transformation are the vectors which the transformation acts very simply: it multiplies them by a scalar. We'll study the relationship between eigenvectors and linear independence.Review questions:
1. Why doesn't rotation in the
plane by 90 degrees have any real eigenvalues?
2. Say we allowed the zero vector to be an eigenvector. For which linear
transformations would it be an eigenvector, and what would its
eigenvalue be?
3. In the proof of Thm. 6.4, why do we let x_k be
the last nonzero element in
the sequence?
Quiz
Nov. 8, covers: Apostol 4.1 to 4.12.
Recommended
study procedures:
1. Add these sections to your outline of important definitions
and theorems (useful for the Final).
2. Work the exercises in these sections with answers in the back of the
book.
3. Bring questions on the examples and exercises in these sections to
office hours.
Test
2, Nov. 20, covers:
Apostol 4.13 to 4.21, 5.1 to 5.13, some of Chap. 6 (to be made precise).