Linear Algebra

Course Description

Description

This course presents the main concepts and terminology of linear algebra. It is a full introductory linear algebra course equivalent to a first-year college linear algebra course.

Topics include:

• linear equations
• matrix algebra
• determinants
• vector spaces
• eigenvalues
• orthogonality
• least squares
• symmetric matrices
• quadratic forms

As illustrated throughout the course, the topics presented play an essential role in areas such as computer science, engineering, environmental science, economics, statistics, business management, and the social sciences. This course provides an excellent foundation for Multivariable Calculus.

Assignments are based on a textbook that is purchased separately by the student.

Materials Needed

A textbook purchase is required for this course:

Linear Algebra and its Applications by David Lay, 3rd ed; Addison-Wesley.

ISBN: 0321287134

A graphing calculator is required, such as:

• TI-83 PLUS
• TI-84 PLUS
  
• TI-85
  
• TI-86
  
• TI-89*
  

*Recommended

List of Topics

Linear Equations

• Systems of Linear Equations
• Row Operations
• Echelon Form
• Existence and Uniqueness of Solutions
• Vector Equations
• The Matrix Equation Ax=b
• Balancing Chemical Equations
• Network Flow
• Linear Independence
• Linear Transformations
• Superposition Principle
• Matrix of a Transformation
• Geometric Transformations
• One-to-one Transformations

Matrix Algebra

• Matrix Arithmetic
• Row-Column Rule for AB
• Matrix Inversion
• Inversion and Ax=b
• Properties of Inverse
• Elementary Matrices
• Row Reduction and Inverses
• Invertible Matrix Theorem
• Partitioned Matrices
• Column-Row Expansion for AB
• LU Factorization
• LU and Electrical Circuits
• Leontief Input-Output Model
• Basic Computer Graphics

Determinants

• Definition of Determinants
• Cofactor Expansion
• Properties of Determinants
• Row Reducing for Determinants

Vector Spaces

• Definition of a Vector Space
• Examples: Polynomials, Arrows, Sequences, Functions, Matrices
• Definition of Subspaces
• Spanning Sets
• Null Space of a Matrix
• Column Space of a Matrix
• Kernel and Range of Transformations
• Basis of a Subspace
• Pivot Columns as Basis
• Unique Representation Theorem Coordinates from a Basis
• Euclidean Coordinates
• Dimension of a Vector Space
• Basis Theorem
• Row Spaces
• Rank Theorem
• Change of Coordinates
• Difference Equations
• Markov Chains

Eigenvalues and Eigenvectors

• Definition of Eigenvectors
• Triangular Matrices
• Distinct Eigenvalues
• Characteristic Equation
• Similar Matrices
• Diagonalization of Matrices
• Eigenvectors and Transformations
• Complex Eigenvalues
• Predator-Prey Systems

Orthogonality and Least Squares

• Inner Products
• Length of a Vector
• Distance between Vectors
• Orthogonal Vectors
• The Pythagorean Theorem
• Orthogonal Complement
• Angles and Vectors
• Orthogonal Basis
• Orthogonal Projection
• Orthonormal Sets
• Orthogonal Matrices
• Orthogonal Decomposition
• Geometric View of Projections
• Best Approximation Theorem
• Gram-Schmidt Process
• QR Factorization
• Least Squares Problem
• Solution to Least Squares Problem
• Normal Equations
• Inverse of ATA
• QR Factorization and Least Squares
• Least-Squares Line of Best Fit
• Least-Squares Fitting of Curves

Symmetric Matrices and Quadratic Forms

• Diagonalization of Symmetric Matrices
• Spectral Theorem
• Quadratic Form
• Change of Variable in a Quadratic Form
• Principal Axes
• Quadratic Forms and Eigenvalues

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System Requirements

CTYOnline courses require a properly-maintained computer with Internet access and a recent-version web browser (such as Firefox, Safari, or Internet Explorer) with the Adobe Flash plugin. Students are expected to be familiar with standard computer operations (e.g. login, cut & paste, email attachments, etc).

This course uses an online mathematical whiteboard for individual or group discussions with the instructor. The whiteboard web site requires cookies, popup windows, and the Java Runtime Environment. (Note: iOS & Android devices cannot run Java applets.)