Math 2406: Abstract Vector
Spaces
Lecture 16 Plan
(Thursday, October 18, 2007).
Apostol 4.18-4.19.
We will discuss the Gauss-Jordan method for (1) solving linear systems
of equations and (2) finding the inverse of a matrix. We will
look at the step-by-step procedure (which you can practice on the HW,
below), and we will prove why it works. The proof is simple and
elegant--applying the method is equivalent to left multiplying the
original matrix by a series of elementary
matrices. The product of these matrices is the inverse of the
original matrix. That's it. Using the Gauss-Jordan method, we can also
see whether we have zero, one, or an infinite number of solutions to a
system.
Review:
1. How is the method for
solving a linear system of equations similar to the method for finding
an inverse of a matrix?
2.
Say we apply the Gauss-Jordan method to solve a system of m equations
in n variables, with m > n. Can you rule out having zero, one, or an
infinite number of solutions? And what if m
< n?
3. How could you use the
Gauss-Jordan method to find a basis for the null space of a linear
transformation?
Practice problems for
HW 7 (Not to be handed in; instead, to be discussed in
office hours, 11-12 Tuesday 10/23, 11-12 Wednesday 10/24)
Apostol 4.16: 8, 11, 14.
Apostol 4.20: 2, 4, 11, 15
Apostol 4.21: 4, 5
Homework 7
due at 5 p.m.
Thursday, October 18, 2007.
1. Apostol 4.16 Exercise 2.
2. Apostol 4.16 Exercise 9.
3. Apostol 4.16 Exercise 15.
4. Apostol 4.20 Exercise 10.
5. Apostol 4.20 Exercise 14.
6. Apostol 4.21 Exercise 1.
7. BONUS: Apostol 4.21 Exercise 2.