SUNY Fredonia
Department of Mathematical Sciences
MATH 231 – Linear Algebra
Fall Semester, 2004
Instructor:
Dr. Iyad Abu-Jeib
Class Meeting Days, Times, and Locations:
MWF 2:00 - 3:10 Fenton 180.
Catalog Description:
|
MATH 231 |
Linear Algebra |
|
|
Thorough treatment of linear algebraic systems. Matrix
algebra; determinants; vector spaces; linear independence; basis and
dimension; inner product spaces; least squares approximation; eigenvalues
and eigenvectors; diagonalization. Selected applications to physics,
economics, geometry, statistics, and differential equations. |
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Prerequisite: MATH 121 or 123; MATH 210
recommended. |
|
Rationale:
The concepts and results of linear algebra are
fundamental to the study of mathematics, showing the power and beauty of
mathematical abstraction. For this reason, the Mathematical Association
of America, through its Committee on the Undergraduate Program in
Mathematics (CUPM), has consistently recommended that a course in linear
algebra be required of all math majors.
The most recent curriculum recommendations from the
CUPM (2003) stipulate that courses designed for mathematical sciences
majors should ensure that students:
- Develop mathematical thinking and communication skills.
- Progress from a procedural/computational understanding of mathematics
to a broad understanding encompassing logical reasoning, generalization,
abstraction, and formal proof;
- Become able to convey their mathematical knowledge in a variety of
settings, both orally and in writing.
- Develop skill with a variety of technological tools.
- Departments should ensure that majors learn to use a variety of tools,
including computer algebra systems, visualization software, statistical
packages, and a programming language.
- Provide a broad view of the mathematical sciences.
- Departments should ensure that all majors have significant experience
working with ideas representing the breadth of the mathematical sciences,
including topics that are: continuous and discrete; algebraic and
geometric; deterministic and stochastic; theoretical and applied.
- Majors should understand that mathematics is an engaging field, rich
in beauty, with powerful applications to other subjects, and contemporary
open questions.
- Study a single area in depth, drawing on ideas and tools from previous
coursework and making connections, by completing two related courses or a
year-long sequence at the upper level;
- Work on a senior-level project that requires the analysis and creation
of mathematical arguments and leads to an oral and written
report
Along with MATH 210, MATH 231 helps to provide a bridge
from the more computational and procedural courses at the 100-level to the
more abstract and proof-oriented courses at the upper-level. In
particular, MATH 231 provides an introduction to some of the key examples
of algebraic structures that are studied in more depth in MATH 331 –
Abstract Algebra I.
The TI-89 calculator is highly recommended, and will be
used frequently to aid with various matrix computations. In addition,
Maple will be used to help visualize systems of linear equations and their
solutions.
One of our main objects of study will be the vector
space of all n-tuples over the real numbers. This space is
continuous, but has a finite basis, thus providing a nice link between the
discrete and the continuous. Moreover, when n = 2 or 3, we may
visualize vectors geometrically in the plane or in space, allowing us to
connect the algebraic with the geometric. Finally, although our focus
will be mainly on the theory of vector spaces, we will endeavor to mention
some of the more important applications of the subject.
The CUPM makes several additional recommendations
concerning programs of study to prepare future teachers. Courses for
pre-service elementary and middle school teachers should help candidates
develop:
- a deep understanding and mastery of the following mathematical topics
through several grade levels above what they will be certified to teach:
number and operations, algebra and functions, geometry and measurement,
and data analysis, statistics, and probability;
- mathematical common sense, including a broad range of examples and
explanations, well-developed reasoning and communication skills, and
facility in separating and reconnecting the component parts of concepts
and methods;
- an understanding of and extensive experience with the uses of
mathematics in a variety of areas;
- the knowledge, confidence, and motivation to pursue career-long
professional mathematical growth.
In addition to the skills to be developed in K-8
teachers, the CUPM recommends that mathematical sciences majors preparing
to teach secondary mathematics:
- learn to make appropriate connections between the advanced mathematics
they are learning and the secondary mathematics they will be
teaching;
- fulfill their requirements for a mathematics major by including topics
from abstract algebra, analysis, geometry, probability and statistics
(with an emphasis on data analysis), discrete mathematics, and number
theory;
- experience many forms of mathematical modeling and a variety of
technological tools, including graphing calculators and geometry
software;
- learn about the history of mathematics and its applications, including
recent work.
MATH 231 is directly relevant to the attainment of the
above goals. More particularly, in its 2002 report, The Mathematical
Preparation of Teachers, the Conference Board of the Mathematical
Sciences states that, to be well-prepared to teach the new
Standards-based curricula, middle and high school mathematics
teachers need, in the area of Algebra and Number Theory:
- understanding of the properties of the natural, integer, rational,
real, and complex number systems;
- understanding of the ways that basic ideas of number theory and
algebraic structures underlie rules for operations on expressions,
equations, and inequalities;
- understanding and skill in using algebra to model and reason about
real-world situations;
- ability to use algebraic reasoning effectively for problem solving and
proof in number theory, geometry, discrete mathematics, and
statistics.
In is noted that, "Calculus and linear algebra
courses provide an opportunity to give undergraduates extensive practice
with algebraic manipulation. Making this an explicit goal for these
courses helps to assure that future teachers have technical ‘know how’ in
high school algebra." In addition, the report states that for middle
grades teachers who are likely to teach algebra, "coursework in
linear algebra and modern algebra would be appropriate."
The National Council of Teachers of Mathematics, in its
Principles and Standards for School Mathematics, has identified
five content standards and five process standards that "specify the
understanding, knowledge, and skills that students should acquire from
prekindergarten through grade 12." MATH 231 is directly relevant to
the following NCTM standards:
- Number and Operations Standard
- Understand meanings of operations and how they relate to one
another.
- Understand patterns, relations, and functions.
- Represent and analyze mathematical situations and structures using
algebraic symbols.
- Analyze characteristics and properties of two- and three-dimensional
geometric shapes and develop mathematical arguments about geometric
shapes.
- Specify locations and describe spatial relationships using coordinate
geometry and other representational systems.
- Apply transformations and use symmetry to analyze mathematical
situations.
- Use visualization, spatial reasoning, and geometric modeling to solve
problems.
- Build new mathematical knowledge through problem solving.
- Solve problems that arise in mathematics and in other contexts.
- Apply and adapt a variety of appropriate strategies to solve
problems.
- Monitor and reflect on the process of mathematical problem
solving.
- Reasoning and Proof Standard
- Recognize reasoning and proof as fundamental aspects of
mathematics.
- Make and investigate mathematical conjectures.
- Develop and evaluate mathematical arguments and proofs.
- Select and use various types of reasoning and methods of
proof.
- Organize and consolidate mathematical thinking through
communication.
- Communicate mathematical thinking coherently and clearly to peers,
teachers, and others.
- Analyze and evaluate the mathematical thinking and strategies of
others.
- Use the language of mathematics to express mathematical ideas
precisely.
- Recognize and use connections among mathematical ideas.
- Understand how mathematical ideas interconnect and build on one
another to produce a coherent whole.
- Recognize and apply mathematics in contexts outside of
mathematics.
- Create and use representations to organize, record, and communicate
mathematical ideas.
- Select, apply, and translate among mathematical representations to
solve problems.
Textbook: Introductory Linear Algebra: An Application-Oriented First
Course, 8/E, by Bernard Kolman and David R. Hill, Prentice Hall,
2005.
Calculator: The TI-89 is strongly recommended.
Objectives: Students who complete the course should be able
to:
- correctly apply the operations of matrix addition, scalar
multiplication, transpose, and matrix multiplication;
- solve a system of linear equations using matrices and interpret the
solution set geometrically;
- express a square matrix as a product of elementary matrices and find
its inverse;
- conjecture and prove theorems about matrices, linear systems, vector
spaces, linear transformations, etc.;
- compute the determinant of a matrix by various methods;
- know and be able to apply the various conditions that are equivalent
to a square matrix being invertible;
- find the subspace of n-dimensional space spanned by a given set of
vectors and interpret the result geometrically;
- determine whether a given subset of a vector space is a subspace;
- construct a basis for vector space;
- find the row space, column space, kernel, and consistency space of a
given matrix and know the fundamental results relating these
subspaces;
- find the eigenvalues of a given square matrix and know their
significance.
Instructional Methods and Activities:
Lecture/Discussion
Refer to instructor syllabus for additional details.
Evaluation and Grade Assignment: Typically will involve
homework, short quizzes, exams, and a comprehensive final exam.
Refer to instructor syllabus for additional details.
Grading Scale:
Refer to instructor syllabus.
Bibliography:
- H. Joseph Straight, Linear Algebra: An Invitation.
- Howard Anton, Elementary Linear Algebra, 7th ed., Wiley,
1994.
- David Poole, Linear Algebra: A Modern Introduction,
Brooks/Cole, 2003.
- Gilbert Strang, Introduction to Linear Algebra, 2nd ed.,
Cambridge-Wellesley, 1998.
- Fred Szabo, Linear Algebra: An Introduction Using MAPLE,
Harcourt/Academic Press, 2002.
In addition, the following journals frequently contain
articles related to linear algebra and accessible to
undergraduates:
- The American Mathematical Monthly
, published by the
Mathematical Association of America.
- The College Mathematics Journal
, Mathematical Association of
America.
- Mathematics Magazine
, Mathematical Association of America.
- Mathematics Teacher
, National Council of Teachers of
Mathematics.
- Mathematics Teaching in the Middle School
, National Council of
Teachers of Mathematics.
- Linear Algebra and Its Applications
, Elsevier.
- Numerical Linear Algebra with Applications, Wiley
Interscience.
- Linear and Multilinear Algebra
, Taylor & Francis.
- Electronic Journal of Linear Algebra
, Internat. Linear Algebra
Soc.
Topics:
Matrix algebra; systems of linear equations;
determinants; vector spaces; linear transformations; eigenvalues and
eigenvectors; other topics as time permits.
Policies:
Refer to instructor syllabus.