← All courses A v ⃗ A\vec v A v Algebra
Matrices, vector spaces and linear maps, determinants, eigenvalues and diagonalization, orthogonality and least squares, quadratic forms, the SVD, and Markov chains.
Before this: Algebra 2 What you'll be able to do
Solve linear systems by row reduction and read off the solution set Find eigenvalues and diagonalize Project onto subspaces and fit least-squares models Interpret the SVD Start the course
Go through each topic step by step — prose, worked examples, quick checks, and a timed practice round after every section.
Every test ends with a score by topic and a list of what to revisit. Test out of a topic with 80% or better to mark it mastered.
Topics 1. Matrices & Systems
5 lessons Arithmetic with grids of numbers
2. Row Reduction & Rank
5 lessons Gaussian elimination and the shape of solution sets
3. Determinants & Inverses
5 lessons One number that decides invertibility
4. Vector Spaces & Bases
5 lessons Span, independence, basis, dimension
5. Linear Transformations
5 lessons Kernel, image, rotations and reflections
6. Orthogonality & Least Squares
5 lessons Projections, distance, and best fit
The directions a matrix merely stretches
8. Diagonalization & Matrix Powers
5 lessons A = PDP⁻¹ and everything it unlocks
9. Symmetric Matrices & Quadratic Forms
5 lessons The spectral theorem and definiteness
10. Singular Values & the SVD
5 lessons Every matrix is rank-one layers, sorted by importance
11. Markov Chains & Long-Run Behavior
5 lessons Steady states, mixing, and absorbing states