Math 2406: Abstract Vector Spaces

Lecture 8 Plan  (Thursday, September 13, 2007).

Apostol 3.14-3.15.

Having defined bases and inner products, we will use the inner product to construct bases--in particular, orthogonal/orthonormal bases.  We take a set of elements one by one, and project each element onto the space perpendicular to those elements that came before it.  Interesting function examples are the sines and cosines of Fourier series and the Legendre polynomials.
fg

Partial sums of the Fourier series for f(x) = x/2      The first six Legendre polynomials (p. 114 Apostol)

(p. 117 Apostol)


Review questions:

1. What's the Gram-Schmidt formula (3.17) when the y's are orthonormal?
2. Does the Gram-Schmidt process work if the x's are a dependent set?
3. What's the projection of (the projection of x along y) along y?

Homework 4 due Thursday, September 20, 2007 at 5 p.m. in Skiles 238A door box.

1. Apostol 3.13, Exercise 2.
2. Apostol 3.13, Exercise 14.
3. Apostol 3.13, Exercise 17--Here you need only show that (f,g) satisfies the complex inner product axioms--including 1' and 3' on p. 104.
4. Apostol 3.17, Exercise 2b.
5. Apostol 3.17, Exercise 5.