Linear AlgebraLinear Algebra

Systems of equations, row reduction, vectors and the equation Ax = b, linear independence, and linear transformations.

1. 1Systems of Linear Equations
2. 2Row Reduction and Echelon Forms
3. 3Vector Equations
4. 4The Matrix Equation Ax = b
5. 5Solution Sets of Linear Systems
6. 6Linear Independence
7. 7Linear Transformations
8. 8The Matrix of a Linear Transformation

Multiplying and inverting matrices, the Invertible Matrix Theorem, LU factorization, subspaces, rank, and the matrices of computer graphics.

1. 9Matrix Operations
2. 10The Inverse of a Matrix
3. 11The Invertible Matrix Theorem
4. 12LU Factorization
5. 13Subspaces of Rⁿ
6. 14Dimension and Rank
7. 15Matrices in Computer Graphics

One number per square matrix: how it scales area and volume, how row operations change it, and Cramer's rule.

1. 16Introduction to Determinants
2. 17Properties of Determinants
3. 18Cramer's Rule, Volume and Transformations

Linear algebra beyond arrows: polynomials, matrices and functions as vectors, bases, coordinates, dimension and change of basis.

1. 19Vector Spaces and Subspaces
2. 20Null Spaces, Column Spaces and Linear Transformations
3. 21Linearly Independent Sets and Bases
4. 22Coordinate Systems
5. 23The Dimension of a Vector Space
6. 24Change of Basis

The directions a matrix only stretches, how to find them, diagonalization, complex eigenvalues, and long-run behavior of dynamical systems.

1. 25Eigenvectors and Eigenvalues
2. 26The Characteristic Equation
3. 27Diagonalization
4. 28Eigenvectors and Linear Transformations
5. 29Complex Eigenvalues
6. 30Discrete Dynamical Systems

Perpendicular vectors, projections onto subspaces, the Gram–Schmidt process, and fitting models to data with least squares.

1. 31Inner Product, Length and Orthogonality
2. 32Orthogonal Sets
3. 33Orthogonal Projections
4. 34The Gram–Schmidt Process
5. 35Least-Squares Problems
6. 36Applications to Linear Models

The spectral theorem, quadratic forms, the singular value decomposition, and principal component analysis.

1. 37Diagonalization of Symmetric Matrices
2. 38Quadratic Forms
3. 39The Singular Value Decomposition
4. 40Principal Component Analysis