Skip Navigation

Linear Algebra

Arrow image that is used as button to enroll in courses. Enroll in this Course

Eligibility: CTY-level or Advanced CTY-level math score required

Prerequisites: Successful completion of AP Calculus BC or equivalent; completion of Multivariable Calculus preferred

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, 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 with several additional topics. Computer based interactives, homework and quizzes help reinforce concepts taught in 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 may be scheduled to prepare for 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.

This course does not have any synchronous class meetings, but students may schedule one-on-one virtual meetings directly with the instructor to answer questions or concerns.

Videos from YouTube or other web providers may be present in the course. Video recommendations or links provided at end of videos are generated by the video host provider and are not CTY recommendations.

Image of Linear Algebra banner.

Proctor Requirements

For enrollment starting on or after January 1, 2019, a proctor is required for this course if the student’s goal is to obtain a grade. Please review Proctor Requirements for more details.

Materials Needed

A textbook purchase is required for this course:

Linear Algebra and its Applications, 5th edition, by David C. Lay, Steven R. Lay, and Judi J. McDonald

ISBN 13: 978-0-321-98238-4

Image of Linear Algebra Text Book Cover.

List of Topics

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

Linear Equations in Linear Algebra

  • Systems of Linear Equations
  • Row Reduction and Echelon Form
  • Vector Equations
  • The Matrix Equation Ax->=b->
  • Solution Sets of Linear Systems
  • Applications of Linear Systems
  • Linear Independence
  • Introduction to Linear Transformations
  • The Matrix of a Linear Transformation

Matrix Algebra

  • Matrix Operations
  • The Inverse of a Matrix
  • Characterizations of Invertible Matrices
  • Subspaces of Rn
  • Dimension and Rank


  • Introduction to Determinants
  • Properties of Determinants
  • Cramer's Rule, Volume, and Linear Transformations

Eigenvalues and Eigenvectors

  • Eigenvalues and Eigenvectors
  • The Characteristic Equation    
  • Diagonalization    
  • Eigenvalues and Linear Transformations    
  • Complex Eigenvalues    
  • Discrete Dynamical Systems

Vector Spaces

  • Vector Spaces and Subspaces
  • Null Spaces, Column Spaces, and Linear Transformations
  • Linearly Independent Sets; Bases
  • Coordinate Systems
  • The Dimension of a Vector Space
  • Rank
  • Change of Basis
  • Applications to Difference Equations
  • Applications to Markov Chains

Orthogonality and Least Squares

  • Inner Products Length, and Orthogonality
  • Orthogonal Sets
  • Orthogonal Projections
  • The Gram-Schmidt Process
  • Least-Squares Problem
  • Inner Product Spaces

Symmetric Matrices and Quadratic Forms

  • Diagonalization of Symmetric Matrices
  • Quadratic Forms
  • The Singular Value Decomposition

Back to top


Sample Video Lecture

Sample Video

Image of Video Lecture


Technical Requirements

This course requires a properly maintained computer with high-speed internet access and an up-to-date web browser (such as Chrome or Firefox). The student must be able to communicate with the instructor via email. Visit the Technical Requirements and Support page for more details.

Zoom online virtual classroom
This course uses an online virtual classroom which can be used for instructor-student communication if the student has any questions about the course or curriculum. The classroom works on standard computers with the Zoom desktop client and also tablets or handhelds that support the Zoom Mobile app. Students will need a computer with the Zoom desktop client installed to watch any recorded meetings. The Zoom desktop client and Zoom Mobile App are both available for free download.