MTH300 Linear Algebra

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COURSE DESCRIPTION

This course covers the following topics: matrix algebra, systems of linear equations, vector spaces, linear transformations, eigenvalues, eigenvectors and applications.

GENERAL COURSE GOALS

To have students:
  • expand their understanding of the nature of mathematics
  • improve their mathematical problem-solving skills, including their skill in presenting rigorous mathematical arguments in proofs and derivations
  • learn to use mathematics to gain insight into areas of application, for example in the physical and life sciences, and economics
  • appreciate the importance of mathematics in human civilization
  • learn the appropriate use of technology including how to use it and when to be skeptical of its results
  • develop critical thinking and a basic understanding of logic essential for lifelong learning.

CORE STUDENT LEARNING OUTCOMES

This course addresses the following two core student learning outcomes:
  • 2.1. Demonstrate knowledge of the truth as it is pursued in the disciplines of mathematics and the sciences; the social sciences; the humanities; and the fine arts.
  • 3.1. Utilize the liberal arts skills to analyze and evaluate significant texts and investigate mathematical and scientific processes.

SPECIFIC COURSE LEARNING OUTCOMES

By successfully completing this course, students should be able to:
  • row-reduce a matrix using elementary row operations
  • relate an augmented matrix to a system of linear equations
  • understand the meaning of existence and uniqueness of solutions
  • solve a system of equations by reducing a matrix to echelon form
  • solve a system with free variables
  • use technology (e.g., MATLAB) to row-reduce a matrix
  • understand and use basic properties of vectors in ℝ2, ℝ3, ℝn,
  • write a linear system as a vector equation
  • understand and use the definitions of linear combination and Span{v1, v2, ..., vn}
  • understand equivalent statements related to solutions of Ax = b
  • compute Ax, A( u + v ), A (c u )
  • understand matrix factorization
  • understand and compute the LU factorization and its role in solving linear systems
  • understand and use the definition of linear independence
  • understand the definition and properties determinants
  • understand the relation of determinants and existence of solutions of linear systems
  • find all solutions of homogeneous and nonhomogenous systems, if they exist
  • determine whether solutions exist and, if so, are unique
  • understand the concept of a linear transformation
  • perform matrix operations of addition, subtraction, and multiplication
  • compute the transpose of a matrix and the inverse of a matrix
  • know and use the Invertible Matrix Theorem
  • know the definition of a vector space and a subspace
  • know and be able to find the null space and the column space of a matrix
  • identify when a set of vectors forms a basis for a subspace
  • find bases for Nul A and Col A
  • understand the concept of a coordinate system and find coordinates of a vector with respect to a given basis
  • understand the concepts of dimension and rank of a vector space
  • know and use the rank theorem
  • understand the concept of eigenvalues and eigenvectors, and find them for a given matrix
  • find the characteristic equation for a matrix
  • diagonalize a matrix
  • determine inner product, length, and orthogonality of vectors and sets of vectors
  • find the orthogonal projections of vectors
  • apply the Gram-Schmidt algorithm to find an orthogonal basis for an inner product space, if time permits.
  • as time permits, use theory in applications such as the following: economics ("input-output" models), balancing chemical equations, network flow (traffic), temperature distribution in a metal plate, Markov chains, difference equations, dynamical systems
  • understand additional topics, as time permits.

SUPPORT OF THE UNIVERSITY MISSION

This course supports the mission of the university as follows: "...through teaching ... prepares men and women for responsible lives by imparting and expanding knowledge, developing skills, and cultivating enduring values. ... students develop their abilities for thinking clearly and creatively, enhance their capacity for sound judgment, and prepare for the challenge of learning throughout their lives."

PREREQUISITES

MTH202 Calculus II or equivalent.

WHERE AND WHEN

  • Secion 01 : Course meeting time/place: T, Th 12:30 P.M. - 1:45 P.M. Location: O'Hare 109

TEXTBOOK AND MATERIALS

  • Required: Linear Algebra and its applications, 5th Edition by David and Steven Lay, and Judi McDonald, Pub. Addison-Wesley.
  • Strongly recommended: TI-83Plus or TI-84Plus graphing calculator, MATLAB (Note: If calculators will be allowed on examinations, only the following models will be permitted: TI-83, TI-83Plus, TI-84Plus.)

TEACHING METHODOLOGY

Lecture, question and answer, readings, homework exercises.

GENERAL REMARKS AND ADVICE

This web site is designed to, among other things, make it clear what is expected from you and what you can expect from the course and from me. This course will be challenging; it is fast-paced, it requires a great deal of meticulous attention to detail as well as imagination and creativity, and just about everything in it depends on your understanding of everything else in the course that preceded it. Nevertheless, if you work hard, do not allow yourself to fall behind, and seek help when you need it, you should be successful in this course.

Reading assignments and homework problems for nearly each class meeting are posted on the course web site. It is your responsibility to check the web site frequently (i.e., at least once a day) for the homework. It is extremely important that you complete the reading assignments and try the homework problems before the following class meeting.

Last modified: 9/5/2019