Math 2406: Abstract Vector Spaces
Lecture 9 Plan
(Tuesday, September 18
Apostol 3.14-3.16.
Today we prove the Gram-Schmidt Theorem, and use the Gram-Schmidt Process to turn (1, t, t^2, ...) into an orthogonal set on [-1, 1], the Legendre polynomials. We have already seen projection along an element; now we introduce projection on a subspace. We use this concept to prove the Approximation Theorem, which tells us that projections are the best approximants.Review questions:
1. Let V = R^4. What are three
possible subspaces S and the corresponding orthogonal complements
S^(perp) ?
2. Let V = R^3 and S be the subspace of vectors normal to (1, 1, 1).
Given any x in V, what are the steps needed to produce p(x)?
3. In the case of trigonometric polynomials (p. 116), also called
"finite Fourier Series", why is the dimension said to be 2n+1? Does
this make sense according to the definition of dimension?
Test 1
Test 1 is in one week: Tuesday,
September 25. The test is closed book, with no notes permitted.
The test will cover material through today's lecture, including: material presented in class, in the book sections noted in each
lecture plan, in Chris Heil's notes, and in the homework.
Recommended procedures for
studying:
1. Read through the lecture
plans and write an outline of the course material. This outline should
include
lists of each of the theorems and
definitions, and
examples for each theorem or definition involving vectors (for Chap. 1
and 3), and
functions (Chap. 3). Save this
outline for the final exam.
2. Review all homework problems
and posted solutions (I plan for HW 3 and 4 solutions to be posted this
week and will notify you by email when they are posted).
3. Work through some of the following additional practice problems, and read and think about the rest:
0.13 #2; 1.4 #5, 12; 1.8 #24;
1.11 #14; 1.15 #9, 13, 17; 1.17 #9, 10;
3.5 #4, 13, 23; 3.10 #7, 23d, 24h; 3.13 #1e, 12, 13d; 3.17 #1a,
3, 7.