Linear Algebra
Free Online Course -- Self-Paced Learning -- College Level
Welcome to Linear Algebra!
This comprehensive course covers the core topics of a college-level linear algebra course: systems of equations, matrix algebra, determinants, vector spaces, linear transformations, eigenvalues, inner product spaces, and real-world applications. Each module includes detailed lessons, worked examples, practice problems, quizzes, and printable study materials.
Self-Paced
Learn on your schedule. All materials available 24/7.
Interactive
Practice problems with instant feedback and detailed solutions.
Comprehensive
Study guides, quick reference cards, and worked examples.
Research-Based
Designed using evidence-based learning principles.
Course Modules
Systems of Linear Equations
Master augmented matrices, Gaussian elimination, and solution sets for systems of equations.
- Systems and augmented matrices
- Gaussian elimination and row echelon form
- Gauss-Jordan and reduced row echelon form
- Solution sets: unique, infinite, and none
Matrix Algebra
Learn matrix operations, properties, inverses, and LU factorization.
- Matrix addition, scalar multiplication, matrix multiplication
- Properties of matrix arithmetic and transpose
- The inverse of a matrix
- Elementary matrices and LU factorization
Determinants
Explore determinant computation, properties, Cramer's Rule, and geometric meaning.
- Determinant: definition and cofactor expansion
- Properties of determinants
- Cramer's Rule
- Geometric interpretation: area, volume, orientation
Vector Spaces
Understand vectors in R^n, subspaces, span, linear independence, basis, and dimension.
- Vectors in R^n
- Subspaces, span, and linear combinations
- Linear independence
- Basis and dimension
Linear Transformations
Study linear maps, their matrix representations, kernel, range, and the Rank-Nullity Theorem.
- Linear transformations: definition and examples
- The matrix of a linear transformation
- Kernel and range
- The Rank-Nullity Theorem
Eigenvalues & Eigenvectors
Discover eigenvalues, eigenvectors, diagonalization, and applications to Markov chains.
- Eigenvalues and eigenvectors: definitions
- The characteristic equation
- Eigenspaces and diagonalization
- Applications: Markov chains and matrix powers
Inner Product Spaces
Explore dot products, orthogonality, projections, Gram-Schmidt, and least squares.
- Dot product, length, orthogonality
- Orthogonal projections
- Gram-Schmidt process
- Least squares approximation
Applications of Linear Algebra
Apply linear algebra to change of basis, SVD, data science (PCA), and computer graphics.
- Change of basis and similar matrices
- Introduction to SVD
- Linear algebra in data science: PCA
- Linear algebra in computer graphics
Ready to Get Started?
Begin with Module 1 to build a strong foundation in solving systems of linear equations. All materials are completely free!
Learning Tips
- Work through modules in order -- later topics build on earlier ones
- Do every worked example by hand before reading the solution
- Practice, practice, practice! Complete all practice problems before the quiz
- Use the study guide to review before quizzes and exams
- Visualize! Think of matrices as transformations, not just grids of numbers
- Connect ideas -- determinants, invertibility, linear independence, and rank are all related
About This Course
This linear algebra course covers the standard topics of a first college-level course. It is designed for students in mathematics, engineering, computer science, data science, and the physical sciences. The material progresses from concrete computation (solving systems, matrix arithmetic) to abstract reasoning (vector spaces, linear transformations) and back to applications (PCA, computer graphics).