4450.001,  Spring 2001
The Theory of Matrices
Conley

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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.

It is the responsibility of students with certified disabilities to provide the instructor with appropriate documentation from the Dean of Students Office.

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