Matrix algebra
Undergraduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Undergraduate Matrix Algebra — scope drawn from Boyd & Vandenberghe Introduction to Applied Linear Algebra, Hefferon Linear Algebra, and related OER (see docs/reference/matrix-linear-algebra-books.md).
Linear systems and elimination
- Represent linear systems with augmented matrices — Represent linear systems with augmented matrices
- Perform row operations to reach row echelon — Perform row operations to reach row echelon
- Reduced row echelon form — Reduced row echelon form
- Classify systems as unique, none, or infinitely many solutions — Classify systems as unique, none, or infinitely many solutions
- Interpret solutions in applied STEM contexts — Interpret solutions in applied STEM contexts
Matrix operations
- Matrix addition and scalar multiplication — Matrix addition and scalar multiplication
- Transpose — Transpose
- Matrix multiplication — Matrix multiplication
- When products are defined — When products are defined
- Matrix-vector products as linear combinations of columns — Matrix-vector products as linear combinations of columns
- Special matrices: identity, diagonal, triangular, sparse patterns — Special matrices: identity, diagonal, triangular, sparse patterns
Inverses and determinants
- Invertible matrices — Invertible matrices
- The Invertible Matrix Theorem (computational view) — The Invertible Matrix Theorem (computational view)
- Finding inverses via row reduction — Finding inverses via row reduction
- Formulas for 2x2 — Formulas for 2x2
- Determinants and cofactor expansion — Determinants and cofactor expansion
- Geometric scaling — Geometric scaling
- Cramer's rule — Cramer's rule
- Volume/area interpretations — Volume/area interpretations
Least squares and applications
- Orthogonal projections onto column spaces (computational) — Orthogonal projections onto column spaces (computational)
- Normal equations — Normal equations
- Least-squares solutions — Least-squares solutions
- Fitting lines and simple models to data — Fitting lines and simple models to data
- Markov chains or discrete dynamical systems with matrices (intro) — Markov chains or discrete dynamical systems with matrices (intro)
Eigenvalues for applications
- Characteristic polynomial and eigenpairs for 2x2 and 3x3 — Characteristic polynomial and eigenpairs for 2x2 and 3x3
- Diagonalization when possible — Diagonalization when possible; powers of matrices
- Discrete dynamical systems and long-run behavior — Discrete dynamical systems and long-run behavior
- Symmetric matrices — Symmetric matrices
- Orthogonal diagonalization (computational) — Orthogonal diagonalization (computational)
SVD and PCA (intro)
- Singular values and low-rank intuition — Singular values and low-rank intuition
- Thin SVD for data matrices (statement and use) — Thin SVD for data matrices (statement and use)
- Principal components as leading singular directions — Principal components as leading singular directions
- Compression — Compression
- Noise-reduction examples at an introductory level — Noise-reduction examples at an introductory level
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Linear systems and elimination
Represent linear systems with augmented matrices
Coming soon - Matrix operations
Matrix addition and scalar multiplication
Coming soon - Inverses and determinants
Invertible matrices
Coming soon - Least squares and applications
Orthogonal projections onto column spaces (computational)
Coming soon - Eigenvalues for applications
Characteristic polynomial and eigenpairs for 2x2 and 3x3
Coming soon - SVD and PCA (intro)
Singular values and low-rank intuition
Coming soon
Notes
Undergraduate Matrix Algebra — scope drawn from Boyd & Vandenberghe Introduction to Applied Linear Algebra, Hefferon Linear Algebra, and related OER (see docs/reference/matrix-linear-algebra-books.md). Topic outline: `content/topics/undergraduate/matrix_algebra.json`.