MTH 525, FALL 2006

Brief Notes for Week 3

Week 3:

Monday, September 11

We finish section 2.4 on Coordinates.

I review Theorem 4 (giving a more coherent proof.)

We then examine Theorems 7 & 8 on coordinates of a vector with respect to a basis. We introduce a “change of basis” matrix P, which translates a coordinate vector (with respect to one basis) into another coordinate vector (with respect to a second basis.) This matrix is invertible; theorem 8 says that every invertible matrix P can be viewed as a change of basis matrix.

This “change of basis” matrix P will have columns which are the coordinates of basis #2 written with respect to basis #1 (see p. 91, in chapter 3.)

In preparation for Wednesday’s class, you should review the important definitions -- see notes from Wednesday, September 6.

Begin Assignment 2, problem 1. For each set V, list (if possible) at least three nonzero elements of the set. Then answer the questions: Is this set closed under addition? Is the set closed under scalar multiplication? Is there a zero element in the set?

Wednesday, September 13

There is a short quiz over definitions at the beginning of class.

We discuss section 2.5, summary of row-equivalence, proving (Theorem 11) that rref(A) is a (unique) representative of the space of matrices row equivalent to A. I discuss theorems 9, 10, 11 and their corollaries.

We will briefly discuss section 2.6, computations using vector spaces.

Do 66: 2, 6.

Friday, September 15,

I will hold office hours for this class, MTH 525, from 3 to 4 PM. This is a good time to make sure you have finished Assignment 2.

Last modified October 1, 2006