I. Number and title of course

Mathematics 603 (1968) Theory of Matrices

A. There has been increasing growth of applications of matrix theory recently, in diversified fields such as education, psychology, chemistry, physics, business, and engineering. Within mathematics itself there has been widespread application of matrices to other mathematical disciplines.

B. This course is an excellent beginning graduate course to be used in connection with masters degree programs in the teaching of mathematics since matrix algebra is currently being taught in many high schools in compliance with recommendations of various study groups on high school mathematics curricula.

A. To further study the theory of matrices and vectors from a point of view which will help the student gain an insight into the theory as well as the techniques of matrix algebra.

B. To give the student a firm foundation in matrix algebra upon which he may base further study in his applied field, such as physics, education, psychology, or mathematics.

A. Introductory concepts 1. Simultaneous linear equations 2. Substitutions and matrix multiplication 3. Transpose and associativity 4. Diagonal matrices

B. Vector spaces 1. Subspaces 2. Linear independence and bases 3. Dimension 4. Sum of subspaces

C. Equivalence, rank and inverses

1. Equivalent systems of equations

2. Elementary row operations

3. Solution of linear systems

4. Canonical sets

5. Left and right inverses

D. Determinants

1. Transpositions

2. Expansion by cofactors

3. The adjoins

4. Non singularity

E. Congruence and Hermitian congruence

1. Quadratic forms

2. Congruent symmetric matrices

3. Congruent semi definite matrices

4. Skew matrices

5. Hermitian matrices

Ailken. A.C. Determinants and Matrices.Edinburgh: Oliver and Boyd Company 1948.

Albert, A.A. Introduction to Algebraic Theories. Chicago: The University of Chicago Press 1937.

Birkhoff, G., and S. MacLane. A Survey of Modern Algebra. New York: MacMillan Company 1941.

Campbell, Hugh G. An Introduction to Matrices, Vectors, and Linear

Programming. New York: Appleton Century Crofts 1965.

Dickson, L. E. Modern Algebraic Theories. Chicago: Sanborn Company 1926.

Halmos, P.R. Finite Dimensional Vector Spaces. Princeton D. Van Nostrand Company 1942.

Hohn, Franz A. Elementary Matrix Algebra. New York: The Macmillan Company 1958.

MacDuffee, C.C. The Theory of Matrices. New York: Chelsea Publishing Company 1946.

MacDuffee, C. C. Vectors and Matrices.

Carus Mathematical  Photography No.7. Buffalo: The Mathematical Association of America 1943.

Turnbull, H. W., and A.C. Ailken. An Introduction to the Theory of

Canonical Matrices. London: Blackie and Son 1948.

Lectures, class discussions, problem assignments, and written examinations.

The successful completion of the course work usually required for an undergraduate major in mathematics. (At least 24 semester hours of undergraduate mathematics)

Three semester hours.

This course was examined in accord with recommended procedures and was approved by the Graduate Faculty of the Department of Mathematics on September 24, 1965. ______________

MATH. 603 THEORY OF MATRICES

Concepts; vector spaces; diagonal matrices, equivalence; rank; bases; determinants; congruence; Hermitian congruence.

CLASSIFICATION: Elective for graduate mathematics majors

PREREQUISITE: Completion of undergraduate major in mathematics of at least 24 semester hours in mathematics.

CREDIT: Three semester hours - three class hours.

NAME PREPARATION EXPERIENCE

Montgomery, M. A. Ph.D., SUNY at Buffalo 23 years

Smith, Sigmund Ed. D., Penn State University 10 years

Stern, Samuel Ph.D., SUNY at Buffalo 8 years