**4450.001, Spring 2001**

**The Theory of Matrices**

**Conley**

the University of North Texas. |

**INSTRUCTOR:** Charles Conley, GAB
475,
(940) 565 - 3326

**OFFICE HOURS:** Monday 2:00-3:00, Wednesday
10:00-11:00 and 2:00-3:00, Friday 12:00-1:00

**CLASS MEETS:** MWF 11:00-11:50, GAB 438

**EXAMS, HOMEWORK, AND GRADING:** There will be two 100 point
midterms, on Wednesday, Feb. 21 and Wednesday, Apr. 4, and a
comprehensive 180 point final, on Monday, May 7, 10:30-12:30.
There will also be twelve 10 point
homeworks, due Fridays at the beginning of class. There will be
no
make-up exams except for emergencies, and late homework will be worth
half-credit.

**TEXT, PREREQUISITES, AND OTHER REFERENCES: **The text is *Linear
Algebra,* by S. Lang. The prerequisite is Math 2700, which
covers vector spaces, linear transformations, and inverses,
eigenvalues, and eigenvectors of matrices. Let me know if you
would like a recommendation for a supplementary
book. You can find many other references in the science library,
for
example in call numbers QA 251 and QA 184. The books by G.
Strang,
D. Lay, M. O'Nan and H. Enderton, and K. Hoffman and R. Kunze may be
particularly
useful.

**TOPIC:** Our central goal will be the spectral
theorem, which states that a real symmetric matrix has an orthonormal
basis of eigenvectors. We will work through Chapters 1-8 of the
text essentially in order, omitting a few sections and perhaps adding a
few from Chapters 9-11 if there is time
at the end of the course. The main points of each chapter are
given
below. As you can see, there is some overlap with the material
from
Math 2700, which we will review and cover in greater depth.

*Chapter 1.* Vector spaces: bases, dimension, sums, and
direct sums.

*Chapter 2.* Matrices and linear equations.

*Chapter 3.* Linear maps: their kernels, images, and
inverses.

*Chapter 4.* The relationship between matrices and linear
maps.

*Chapter 5.* Scalar products, orthogonal bases, rank,
congruence, bilinear forms, and Sylvester's theorem.

*Chapter 6.* Determinants, Cramer's rule, and
triangulation of matrices.

*Chapter 7.* Symmetric matrices, Hermitian matrices, and
unitary
matrices.

*Chapter 8.* The characteristic polynomial, eigenvectors,
eigenvalues,
diagonalization of matrices, and the spectral theorem.

*Extra Topics.* If there is time, we will discuss the
Cayley-Hamilton theorem and minimal polynomials.

FINAL EXAM: Monday, May 7.

The final will cover chapters 5, 7, and 8; see
the
review sheet for more detail.

HOMEWORK 12, Due Friday, May 4:

Section 8.3: In problems 1 and 2, find the largest
eigenvalue and the associated eigenvector. The diagonalize the
matrices in problem 2ab.

Section 8.4: Do 7, 10, 11, and 14, always assuming that V
is
R^n and A is a real n by n matrix. Then diagonalize the matrix A
of
problem 9: find B so that BAB^-1 is diagonal. You need not find
B^-1,
but do find BAB^-1.

HOMEWORK 11, Due Friday, April 27:

Section 8.1: 1, 2, 3 (for n=3 only), 4, 5, 6, 7 (assume
V
= R^n and A and B are matrices);

Section 8.2: 1 (n=3 only), 2 (n=3 only), 3a, 4d, 5a, 6b,
7a,
8b, 9 (n=2 only), 11, 10*.

Extra credit problem: 8.2.15.

HOMEWORK 10, Due Friday, April 20:

Section 7.1: 1bc, 2, 3, 4, 9, 13 (for n=1 and 2 only),
15;

Section 7.3: 1a (for the usual dot product), 2, 3, 6, 8,
10-12
(do 10-12 only for V=R^2).

In all of these problems, whenever Lang refers to an operator *A*
on
a vector space *V*, assume that *V = *R^n* *and *A
*is an *n* by *n* matrix.

HOMEWORK 9, Due Friday, April 13:

Section 5.2: 0, 1a, 2, 9, 10;

Section 5.3: 1c-h, 2, 3, 4abd, 5c;

Extra problems handed out in class.

EXAM 2: Wednesday, April 4.

This exam will cover Sections 3.3-4, 4.1-2,
5.1, 6.1-6.5, and the notes passed out in class.

HOMEWORK 8, Due Wednesday, April 4:

Section 3.3: 4, 12;

Section 3.4: 4, 8b;

Section 4.2: 1b, 3, 5;

Section 6.3: 5b for n=4;

Repeat Homework 7, Problem A for 1de and 2d.

HOMEWORK 7, Due Friday, March 16:

Section 6.8: 2, 3, 4 (it is not necessary to read the
section for these);

Section 6.5: 1, 2 (both only for 3 by 3 matrices);

Problems A and B from class.

HOMEWORK 6, Due Friday, March 8:

Section 4.2: 1adef, 2, 4, 6-10;

Section 4.3: 11 (it is not necessary to read this section
yet);

Section 6.2: 1;

Section 6.3: 1cde, 2ab, 3, 4, 5a, 6ehi.

HOMEWORK 5, Due Friday, March 1:

Section 3.3: 1, 2, 3, 10, 11abce, 14-16;

Section 3.4: 1, 2, 5, 6, 8a, 10, 16, 19;
Section 4.1:
1d.

EXAM 1: Wednesday, February 21.

This exam will cover Sections 1.1 to 3.2.

HOMEWORK 4, Due Friday, February 16:

Section 3.1: 2, 3;

Section 3.2: 1fg, 2, 4, 6, 7, 14, 15, 16, 18a;

Section 2.3: 25a, 27, 28, 29, 32, 34, 35, 36, 39, 18de.

HOMEWORK 3, Due Friday, February 9:

Section 1.4: 1, 2;

Section 2.2: 1, 2;

Section 2.3: 1, 2, 3b, 4, 5, 8, 9, 11, 13b, 15, 18abc, 19,
21-24.

HOMEWORK 2, Due Friday, February 2:

Section 2.1 I: 2, 4, 5, 9, 10, 12;

Section 2.1 II: 1, 3-6

HOMEWORK 1, Due Friday, January 26:

Section 1.1: 1, 2, 4, 5, 8c, 9a, 10, 13;

Section 1.2: 1bch, 2ad, 3b, 4, 9, 10