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Linear Algebra

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Prerequisites: Qualifying math score and completion of Calculus BC or equivalent

Course Format: Individually Paced

Course Length: Typically 6 months

Recommended School Credit: One full year of high school credit equal to or greater than an AP class or one semester of college credit

Course Code: LIN

Course Description


Linear Algebra is an online and individually-paced course equivalent to a first-year college linear algebra course. This course covers the entire syllabus from the Johns Hopkins semester-based, in-person Linear Algebra course, plus several additional topics. Computer based interactives, homeworks and quizzes help to reinforce concepts taught in the class. Projects covering advanced applications will introduce students to mathematical typesetting with LaTeX. Online course materials supplement the required textbook.

Each student is assigned to a CTY instructor to help them during their course. Students can contact their instructor via email with any questions or concerns at any time. Live one-on-one online review sessions can be scheduled as well to prepare for the graded assessments, which include quizzes, homework, midterm exams, and a cumulative final. Instructors use virtual classroom software allowing video, voice, text, screen sharing and whiteboard interaction.

Topics include:

  • Linear Equations
  • Matrix Algebra
  • Determinants
  • Vector Spaces
  • Eigenvalues
  • Orthogonality
  • Least Squares
  • Symmetric Matrices
  • Quadratic Forms

For a detailed list of topics, click the "List of Topics" tab.

Parallelograms image from Math Insight
Image from Math Insight

Materials Needed

A textbook purchase is required for this course.

Linear Algebra and its Applications, 4th edition, by David C. Lay.

ISBN 13: 978-0-321-38517-8

Linear Algebra textbook

List of Topics

Upon successful completion of the course, students will be able to demonstrate mastery over the following 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


  • 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

CTY Online 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 requires high-speed Internet access (such as Cable or DSL) for online lesson videos. Your browser will need to allow javascript, login cookies, and popup windows from CTY course web sites.

You may also need the Java Runtime Environment.