525, FALL 2006

Brief Notes for Week 5

Week 5:

Monday, September 25

Assignment 3 is collected.

We look at representations of linear transformations (section 3.4.) We use the “change of basis” matrix P to change the representation of a linear transformation with respect to a new basis.

Do p. 95: 1, 2, 6, 7. (Problem 10 is also good.)

We look at linear functionals, chapter 3.5. The concept of linear functional is a simple one, but leads to some nice ideas, including the dual vector space and eventually, the double dual.

Wednesday, September 27

Quiz 4 (on definitions) is given.

We look at sections 3.5 (linear functionals) 3.6, the dual and double dual

Theorem 15 proves that a basis B of a vector space has a “dual basis”; there is a unique basis in the dual corresponding to the original basis.

We define the annihilator S₀ of a set S.

We introduce the double dual (section 3.6).

(Other good problems: p. 105: 1, 2, 3 (on trace), 4, p. 111: 1.)

I am out of town Thursday and Friday and so will not have office hours those days.

Last modified October 1, 2006