2.1 Linear Algebra

Introduction to Linear Algebra, Sixth Edition

Created Date: 2025-05-10

"Introduction to Linear Algebra, Sixth Edition" primarily by Gilbert Strang, is a widely adopted textbook that offers a comprehensive and accessible guide to the study of linear algebra. This edition aims to combine rigorous mathematical concepts with a clear and gentle teaching approach, making it suitable for students across various quantitative fields, including engineering, science, economics, and business.

2.1.1 Vectors and Matrices

2.1.1.1 Vectors and Linear Combinations

2.1.1.2 Lengths and Angles from Dot Products

2.1.1.3 Matrices and Their Column Spaces

2.1.1.4 Matrix Multiplication \(AB\) and \(CR\)

2.1.2 Solving Linear Equations

2.1.2.1 Elimination and Back Substitution

2.1.2.2 Elimination Matrices and Inverse Matrices

2.1.2.3 Matrix Computations and \(A = LU\)

2.1.2.4 Permutations and Transposes

2.1.3 The Four Fundamental Subspaces

2.1.3.1 Vector Spaces and Subspaces

2.1.3.2 Computing the Nullspace by Elimination: \(A = CR\)

2.1.3.3 The Complete Solution to \(Ax = b\)

2.1.3.4 Independence, Basis, and Dimension

2.1.3.5 Dimensions of the Four Subspaces

2.1.4 Orthogonality

2.1.4.1 Orthogonality of Vectors and Subspaces

2.1.4.2 Projections onto Lines and Subspaces

2.1.4.3 Least Squares Approximations

2.1.4.4 Orthonormal Bases and Gram-Schmidt

2.1.4.5 The Pseudoinverse of a Matrix

2.1.5 Determinants

2.1.5.1 3 by 3 Determinants and Cofactors

2.1.5.2 Computing and Using Determinants

2.1.5.3 Areas and Volumes by Determinants

2.1.6 Eigenvalues and Eigenvectors

2.1.6.1 Introduction to Eigenvalues : \(Ax = λx\)

2.1.6.2 Diagonalizing a Matrix

2.1.6.3 Symmetric Positive Definite Matrices

2.1.6.4 Complex Numbers and Vectors and Matrices

2.1.6.5 Solving Linear Differential Equations

2.1.7 The Singular Value Decomposition (SVD)

2.1.7.1 Singular Values and Singular Vectors

2.1.7.2 Image Processing by Linear Algebra

2.1.7.3 Principal Component Analysis (PCA by the SVD)

2.1.8 Linear Transformations

2.1.8.1 The Idea of a Linear Transformation

2.1.8.2 The Matrix of a Linear Transformation

2.1.8.3 The Search for a Good Basis

2.1.9 Linear Algebra in Optimization

2.1.9.1 Minimizing a Multivariable Function

2.1.9.2 Backpropagation and Stochastic Gradient Descent

2.1.9.3 Constraints, Lagrange Multipliers, Minimum Norms

2.1.9.4 Linear Programming, Game Theory, and Duality

2.1.10 Learning from Data

2.1.10.1 Piecewise Linear Learning Functions

2.1.10.2 Creating and Experimenting

2.1.10.3 Mean, Variance, and Covariance