# Alignment: Introduction to Linear Algebra • Gilbert Strang • Fifth Edition

• 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 and Weighted Least Squares

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