SYLLABUS
I. MATH 361-362 Linear Algebra I, II 3, 3 credit hours
II. 2001-2002 academic year INSTRUCTOR: Dr. Keith Stafford
Office: 103 Farr-Chinnock, ext. 6413
e-mail: kstafford@inetlmu.lmunet.edu
Conference Hours (reserved times to assist students):
Monday – 1:00 p.m. - 2:15 p.m.
Tuesday – 2:15 p.m. - 3:30 p.m.
Wednesday – 1:00 p.m. - 2:15 p.m.
Thursday – 2:15 p.m. - 3:30 p.m.
Friday – by appointment only
III. PREREQUISITE: Math 231-232 or equivalent.
IV. Course description/GOALS: The major emphasis is on the theory of matrices and their determinants, vector spaces and linear transformations. Other topics include linear programming and other various applications, and inner product spaces. Goals: 1) to foster respect for systematic approaches to problem solving; 2) to nourish the ability to reason with abstract concepts in order to proceed logically from hypothesis to conclusion; 3) to learn the basic concepts of linear algebra so that they may be used to model important problems in the sciences; 4) to strengthen the ability to visualize spatial relations and to relate them to the algebra; 5) to improve the oral and written communication of mathematics.
V. RELATIONSHIP OF COURSE TO CONTENT AREA KNOWLEDGE AND SKILLS: This course develops a knowledge of algebraic properties of a vector space, especially the real number system, and matrix, mapping, and polynomial spaces. The course includes solutions of systems of linear equations, determinants, and eigenvalues and eigenvectors. It applies matrix theory and linear transformations to a variety of fields. The theory is related to applications in linear programming, graph theory, Markov chains, least squares, and differential equations among others. Some concepts of the calculus of vectors may be covered. Finally, it develops the ability to reason inductively and deductively, provides the opportunity to make and test conjectures, and to prove theorems.
VI. TEXT: Kolman Bernard and Hill, Elementary Linear Algebra with Applications, Seventh Edition, Prentice Hall, Upper Saddle River, New Jersey, 2000.
VII. COURSE OBJECTIVES: 1) Understand the concept of a mathematical
system and how relations apply to its operations; 2) Solve systems of linear equations by matrix methods and use
the ideas to become acquainted with linear spaces; 3) Develop the idea of linear spaces in relationship to two- and
three-space, and to extend this idea to 
; 4) Be able to test
and prove conjectures and theorems of linear algebra; 5) Understand and visualize geometrically the notions of bases,
linear independence, dimension, matrix representations of linear mappings,
rank, and nullity; 6) Perform matrix
algebra, determine ranks of matrices, and singularity of square matrices; 7) Use eigenvalues, eigenvectors, and the
characteristic polynomial in the diagonalization of matrices; 8) Utilize similar matrices and change of
basis to obtain a new matrix representation of a linear operator; 9) See how linear algebra is applied to
real-world problems and other areas of mathematics; 10) Understand the notions of congruent (equivalent) matrices in
relation to quadratic forms; 11) Recognize
how complex numbers can be incorporated into linear algebra; 12) Learn about inner product spaces and the
metric relationship to them.
VIII. METHODS OF INSTRUCTION AND LEARNING: Lectures and problem solving will constitute the principal method of instruction, with attention given to the nature of proofs and analysis of theorems. Homework, class discussion, and some board work will be expected of all students. All are encouraged to ask questions pertaining to assignments and course material in and out of class.
IX. UNITS OF INSTRUCTION / OUTLINE OF COURSE CONTENT:
1. Linear Equations and Matrices: linear systems, matrix operations, algebraic properties of matrix operations, special types and partitioning of matrices, Echelon form, elementary and invertible matrices, equivalent matrices.
2. Real Vector Spaces: vectors
in the plane (
) and 
, vector spaces, subspaces, span and linear independence,
basis and dimension, homogeneous systems, coordinates and isomorphisms, rank.
3. Inner Product Spaces:
standard inner product on 
and 
, cross product in 
, inner product
spaces, Gram-Schmidt process, orthogonal complements, least squares (if
time permits).
4. Linear Transformations and Matrices: definition and examples, the kernel and range, matrix of a linear transformation, vector space of matrices and of linear transformations, similarity.
5. Determinants: definition, properties, cofactor expansion, inverse of a matrix, other applications, a computational point of view.
6. Eigenvalues and Eigenvectors: definitions and examples, diagonalization of similar matrices, Markov processes, diagonalization of symmetric matrices, real quadratic forms, conic sections, quadric surfaces.
7. Complex Numbers: definition, algebraic operations, geometric representation, complex numbers in linear algebra, including Hermitian matrices.
X. REQUIRED READINGS: Sections of the textbook as they are covered in class.
XI. SUGGESTED READINGS/BIBLIOGRAPHY: Appendix of the textbook.
XII. COURSE REQUIREMENTS / METHOD OF ASSESSMENT AND EVALUATION: Class attendance is required and students are expected to participate in discussion. Attendance may affect grades as follows: One-half (1/2) point is deducted from your final grade percentage for each fifteen (15) minutes of class missed. Collected homework, a midterm, and a final exam are used to evaluate subject knowledge and comprehension, and arrive at a performance percentage. Each of these three graded works, comprising the main measure of grades, is equally weighted. Weighting of tests and final exam will be at the discretion of the professor, but conform to the following scheme:
88 & up A 67 to 71 B- 46 to 50 D+
83 to 87 A- 62 to 66 C+ 40 to 45 D
78 to 82 B+ 56 to 61 C 35 to 39 D-
72 to 77 B 51 to 55 C- 00 to 34 F
Only one final exam is given on the date fixed by the university, designated in the LMU current schedule of classes, excepting the requirement of an early grade for a graduating senior.
XIII. DATE OF REVISION: August 20, 2001.