Table of Contents

1 Introduction to Vectors
 1.1 Vectors and Linear Combinations
 1.2 Lengths and Dot Products
 1.3 Matrices

2 Solving Linear Equations
 2.1 Vectors and Linear Equations
 2.2 The Idea of Elimination
 2.3 Elimination Using Matrices
 2.4 Rules for Matrix Operations
 2.5 Inverse Matrices
 2.6 Elimination = Factorization: π΄ = πΏπ
 2.7 Transposes and Permutations

3 Vector Spaces and Subspaces
 3.1 Spaces of Vectors
 3.2 The Nullspace of π΄: Solving π΄π₯ = 0 and π π₯ = 0
 3.3 The Complete Solution to π΄π₯ = π
 3.4 Independence, Basis and Dimension
 3.5 Dimensions of the Four Subspaces

4 Orthogonality
 4.1 Orthogonality of the Four Subspaces
 4.2 Projections
 4.3 Least Squares Approximations
 4.4 Orthonormal Bases and GramSchmidt

5 Determinants
 5.1 The Properties of Determinants
 5.2 Permutations and Cofactors
 5.3 Cramerβs Rule, Inverses, and Volumes

6 Eigenvalues and Eigenvectors
 6.1 Introduction to Eigenvalues
 6.2 Diagonalizing a Matrix
 6.3 Systems of Differential Equations
 6.4 Symmetric Matrices
 6.5 Positive Definite Matrices

7 The Singular Value Decomposition (SVD)
 7.1 Image Processing by Linear Algebra
 7.2 Bases and Matrices in the SVD
 7.3 Principal Component Analysis (PCA by the SVD)
 7.4 The Geometry of the SVD

8 Linear Transformations
 8.1 The Idea of a Linear Transformation
 8.2 The Matrix of a Linear Transformation
 8.3 The Search for a Good Basis

9 Complex Vectors and Matrices
 9.1 Complex Numbers
 9.2 Hermitian and Unitary Matrices
 9.3 The Fast Fourier Transform

10 Applications
 10.1 Graphs and Networks
 10.2 Matrices in Engineering
 10.3 Markov Matrices, Population, and Economics
 10.4 Linear Programming
 10.5 Fourier Series: Linear Algebra for Functions
 10.6 Computer Graphics
 10.7 Linear Algebra for Cryptography

11 Numerical Linear Algebra
 11.1 Gaussian Elimination in Practice
 11.2 Norms and Condition Numbers
 11.3 Iterative Methods and Preconditioners

12 Linear Algebra in Probability & Statistics
 12.1 Mean, Variance, and Probability
 12.2 Covariance Matrices and Joint Probabilities
 12.3 Multivariate Gaussian and Weighted Least Squares