Learn Linear Algebra:
Free Course
with Videos
and Guided Notes
Welcome to our Linear Algebra Videos and Guided Notes page! This free resource is designed to help you learn and review fundamental linear algebra concepts. Whether you’re studying linear algebra for the first time or preparing for an exam, our step-by-step video lessons and downloadable guided notes PDFs will help you build confidence and mastery.
What’s Included in This Free Linear Algebra Course?
Video Lessons
Our videos cover all the essential topics in linear algebra, following the structure of a standard college course. Each lesson includes clear explanations, worked-out examples, and applications so you can strengthen your understanding of concepts such as:
Systems of linear equations and row reduction
Matrix algebra and determinants
Vector spaces and subspaces
Eigenvalues and eigenvectors
Orthogonality and least-squares problems
Applications in engineering, data science, and more
Free Guided Notes for Linear Algebra
Each video comes with linear algebra guided notes PDFs that you can download and complete as you watch. These notes help you:
Reinforce understanding by filling in key formulas and steps
Improve retention through active participation
Stay organized with a structured resource for quizzes, exams, and future reference
How to Learn Linear Algebra Effectively
Watch the videos for each section to follow along with step-by-step lessons.
Download the guided notes PDFs to complete while watching.
Work through examples in the notes to practice problem-solving.
Review your completed notes to prepare for exams or to strengthen your foundation for future courses.
Start Learning Linear Algebra Now!
With our free linear algebra course, video lessons, and guided notes, you’ll develop a strong understanding of key concepts like matrices, determinants, vector spaces, eigenvalues, and orthogonality. These resources are designed to help you succeed in linear algebra and beyond.
Get started today with clear lessons and guided notes PDFs that make even the most challenging linear algebra topics approachable.
© 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.
Review Materials are available at the end of each section
0. Introduction to Linear Algebra
1. Linear Equations in Linear Algebra
1.1 Systems of Linear Equations - Introduction
1.1 Systems of Linear Equations - Part 1
1.1 Systems of Linear Equations - Part 2
1.2 Row Reduction and Echelon Forms
1.3 Vector Equations- Part 1
1.7 Linear Independence
1.3 Vector Equations - Part 2 (Span)
1.4 The Matrix Equation Ax=b
1.5 Solution Sets of Linear Systems
1.6 Applications of Linear Systems
1.8 Introduction to Linear Transformations
1.9 The Matrix of a Linear Transformation - Standard Matrix
1.9 The Matrix of a Linear Transformation - Geometric Transformations in R^2
1.9 The Matrix of a Linear Transformation - One-to-One and Onto Transformations
1.10 Difference Equations (Population Models)
Review Materials
Exam Concept Review - Linear Systems of Equations
Exam Concept Review - Linear Transformations
Exam Review Problems - Linear Systems of Equations
YouTube Video Exam Review Problems
Exam Review Problems - Linear Transformations
YouTube Video Exam Review Problems
2. Matrix Algebra
2.1 Matrix Operations
2.2 The Inverse of a Matrix Part 1 - Inverse of a 2x2 Matrix and Solving Linear Systems
2.2 The Inverse of a Matrix Part 2 - Algorithm for Finding the Inverse of a Matrix
2.2 The Inverse of a Matrix Part 3 - Inverse Properties and Matrix Expression Simplification
2.3 Characterizations of Invertible Matrices Part 1 - The Invertible Matrix Theorem
2.3 Characterizations of Invertible Matrices Part 2 - Inverse Linear Transformations
Review Materials
Exam Review Problems - Matrix Algebra
YouTube Video Exam Review Problems
3. Determinants
3.1 Introduction to Determinants
3.2 Properties of Determinants
Review Materials
Exam Review Problems - Determinants
YouTube Video Exam Review Problems
4. Vector Spaces
4.1. Vector Spaces and Subspaces
4.2 Null Space and Column Space
4.3 Linearly Independent Sets - Basis
4.3 Linearly Independent Sets - Basis of Null Space and Column Space
4.5 Dimension of a Vector Space
Review Materials
Exam Review Problems - Vector Spaces
YouTube Video Exam Review Problems
5. Eigenvalues and Eigenvectors
5.1. Eigenvalues and Eigenvectors
5.2. The Characteristic Equation
5.3. Diagonalization
5.3 Application Problem - Long Term Behavior of a Population
Review Materials
Exam Review Problems - Eigenvalues and Eigenvectors
YouTube Video Exam Review Problems
6. Orthogonality and Least Squares
6.1 Inner Product, Length, and Orthogonality
6.2. Orthogonal Sets
6.3. Orthogonal Projections
6.4. The Gram-Schmidt Process
6.5. Least-Squares Problems
7.4. Singular Value Decomposition
Review Materials
Exam Review Problems - Orthogonality and Least Squares
YouTube Video Exam Review Problems
Final Exam Review
YouTube Video - How to Study for a Math Exam
Part 1 - Linear Systems of Equations YouTube Video
Part 2 - Linear Transformations YouTube Video
Part 3 - Matrix Algebra YouTube Video
Part 4 - Determinants YouTube Video
Part 5 - Vector Spaces YouTube Video
Part 6 - Eigenvalues and Eigenvectors YouTube Video
Part 7 - Orthogonality and Least Squares YouTube Video