Math 2406: Abstract Vector
Spaces
Lecture 20 Plan
(Thursday, November 1, 2007).
Apostol 6.5-6.8.
Today we learn why any linear transformation of a space with complex
scalars has an
eigenvalue . Next we
sketch the triangularization theorem,
which has a long proof, but one which lets us practice our
concepts of bases, and matrix representations. Finally we learn how to
compute eigenvalues and eigenvectors, along the way defining the
characteristic polynomial (for eigenvalues), and using the null space
(for eigenvectors).
Review:
1. Why doesn't the proof of
Theorem 6.4 work if we have real scalars?
2. If T:R^2 to R^2 is given by [0 1; -1 0], it can't be triangularized
as mentioned in Thm. 6.5. Why not?
3. Does the zero matrix have any eigenvalues or eigenvectors?
No homework due November 8.
Practice problems for
Quiz Thursday, November 8, 2007.
(Not to be handed in; instead,
to be discussed in
office hours: 11-12 Tuesday 11/6,
10-11 Wednesday 11/7)
Apostol 4.4: 10, 14, 22, 30.
Apostol 4.8: 10, 19, 29, 31.
Apostol 4.12: 2, 3, 8, 13, 19.
Homework 9
due at 5 p.m.
Thursday, November 15, 2007.
1. Apostol 6.4 Exercise 6.
2. Apostol 6.4 Exercise 7.
3. Apostol 6.10 Exercise 8.
4. Apostol 6.10 Exercise 13.
5. Apostol 6.12 Exercise 1.
6. Apostol 6.12 Exercise 8.