Review key linear algebra concepts through focused video lessons, worked examples, and guided notes. These resources are drawn from a university-level linear algebra course and are designed to help you build understanding, reinforce skills, and prepare for exams.

Strengthen Your Linear Algebra Skills

Completed Notes

Linear Algebra Video Lessons with Guided Notes

  • Watch the video lessons for each section.

  • Download the guided notes PDF to follow along as you watch.

  • Engage actively by completing the notes during the lesson.

  • Review your completed notes as a study resource for exams.

1. Linear Equations

▶️ Video‍ ‍📝 Notes

  1. Solutions to a Linear System with Two Equations and Two Unknowns

  2. Coefficient and Augmented Matrices

  3. Elementary Row Operations, Echelon, and Reduced Echelon Form

  4. Gaussian Elimination and Gauss Jordan Elimination

  5. Consistent and Inconsistent Linear Systems

  6. Linear Combinations and Span

2. Matrix Algebra

▶️ Video‍ ‍ ‍📝 Notes

  1. Addition, Subtraction, and Multiplication of Matrices

  2. Transpose and Trace of a Matrix

  3. Special Matrices - Triangular, Symmetric, and Diagonal

  4. Inverse of a 2x2 and 3x3 Matrix

  5. Solutions to a Linear System Using Matrix Inversion

3. Transformations

▶️ Video‍ ‍📝 Notes

  1. Linear Transformations - Domain and Codomain

  2. Linear Transformations - Standard Matrix

  3. Matrix Transformations - Reflection, Projection, Rotation, Expansion/Contraction

  4. One-To-One and Onto Transformations

4. Determinants

▶️ Video‍ ‍📝 Notes

  1. Determinant of a 2x2 Matrix

  2. Determinant of a 3x3 Matrix - Cofactor Expansion, Shortcut, Row Reduction

  3. Properties of Determinants

  4. Cramer’s Rule

5. Vector Spaces

▶️ Video‍ ‍ ‍📝 Notes

  1. Linear Independence

  2. Basis

  3. Coordinate Vectors

  4. Dimension

  5. Row Space, Column Space, and Null Space

  6. Rank of a Matrix and Rank Theorem

6. Eigenvalues and Eigenvectors

▶️ Video‍ ‍ ‍📝 Notes

  1. Introduction to Eigenvalues and Eigenvectors

  2. How to Find Eigenvalues

  3. How to Find Eigenvectors and a Basis for an Eigenspace

  4. Similar Matrices

  5. Diagonalization

  6. Complex Eigenvalues and Eigenvectors

7. Orthogonality

▶️ Video‍ ‍ ‍📝 Notes

  1. Norm of a Vector, Unit Vector, Standard Unit Vector

  2. Distance in R^n

  3. Dot Product

  4. Orthogonality of Vectors

  5. Orthogonal Projection

  6. Linear Combination of Orthogonal Vectors

  7. Gram Schmidt Process

  8. Orthogonal Matrices

  9. Orthogonal Diagonalization

Copyright & Usage

© Understand The Math LLC. All Rights Reserved. These materials are freely provided for non-commercial educational use. Redistribution is permitted with proper attribution to UnderstandTheMath.com.

About Dr. Cheryl Hile

Understand The Math was created by Dr. Cheryl Hile, who earned a Ph.D. in Engineering Science and Applied Mathematics from Northwestern University. She has more than 25 years of experience teaching mathematics at the university level, including 20 years at Penn State. The lessons and guided notes on this site are based on materials she developed for her university courses.