Table of Contents for Introduction to Linear Algebra (5th edition 2016)

• 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: A = LU
  • 2.7 Transposes and Permutations
• 3 Vector Spaces and Subspaces
  • 3.1 Spaces of Vectors
  • 3.2 The Nullspace of A: Solving Ax = 0 and Rx = 0
  • 3.3 The Complete Solution to Ax = b
  • 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 Gram-Schmidt
• 5 Determinants
  • 5.1 The Properties of Determinants
  • 5.2 Permutations and Cofactors
  • 5.3 Cramer’s 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 andWeighted Least Squares
• Matrix Factorizations
• Index
• Six Great Theorems / Linear Algebra in a Nutshell

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