Math 2406: Abstract Vector Spaces
Lecture 6 Plan
(Thursday, September 6
Apostol 3.7-3.9.
Today we will generalize some concepts from the vector spaces of Chapter 1 to linear spaces: linear independence (and dependence), basis, dimension, and components. We will then apply these generalized terms to real functions of a real variable. Functions introduce for the first time infinite-dimensional spaces.Review questions:
1. Is the induction hypothesis
used in Example 7, p. 99? Why or why not?
2. What are the differences between the proofs of Theorem 3.5 and 1.8,
or between the proofs of Thms 3.6 and 3.7, and 1.10?
3. Can an element in a finite-dimensional space have an infinite number
of components (see section 3.9)?
Homework 3
due
Thursday, September 13, 2007 at 5 p.m. in Skiles 238A door box.
1. Apostol 3.5, Exercise 21.
2. Apostol 3.5, Exercise 23d.
3. Apostol 3.10, Exercise 10--your argument for
whether S is a subspace should be in proof form, but
your computation of dim(S) need not be in proof form.
4. Apostol 3.10, Exercise 12--same comment as
above.
5. Apostol 3.10, Exercise 25b.
6. Apostol 3.10, Exercise 26d.
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