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Linear Algebra (NCAA Approved)

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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: Session Based. See calendar for session dates and application deadlines.

Course Length: 30 weeks

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

Student Expectations: Students are strongly encouraged to work at least 1 hour a day, 5 days a week.

Course Code: LINY

Course Description


Linear Algebra is an online and course equivalent to a first-year college linear algebra course. This course covers the entire syllabus from the Johns Hopkins one-semester, in-person Linear Algebra course, plus several additional topics. Online interactives and assignments 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.

Online course materials, such as videos, notes, interactive webpages, and practice problems with solutions, are provided for the student. Students are expected to watch videos and review notes regularly. Each student is assigned to a CTY instructor to help them during their course.

In this course, participation in forums and synchronous online virtual sessions are required as part of the final grade. Discussion forums are located within the course where students will be required to respond to a given prompt and then comment on responses by other students. Online sessions, led by a CTY instructor, are held in Adobe Connect, which allows for video, voice, text, screen sharing, and whiteboard interaction. 

Students can contact their instructors via email with 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, projects, unit exams, and a cumulative final.

The textbook must be purchased separately.

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.

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Proctor Requirements

Proctor Requirements

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

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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

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Sample Video Lecture

Sample Video

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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.