Understanding Linear Algebra 649HGZKE96

URI	`m2://res/649HGZKE96`
Kind	textbook
Creators	David Austin
Scope	deferred (no test decides the record, and the classifier has no model)

Expressions

e1: edition

Contents

chapter 1: Systems of equations
section 1.1: What can we expect
subsection 1.1.1: Some simple examples
subsection 1.1.2: Systems of linear equations
subsection 1.1.3: Summary
section 1.2: Finding solutions to linear systems
subsection 1.2.1: Gaussian elimination
subsection 1.2.2: Augmented matrices
subsection 1.2.3: Reduced row echelon form
subsection 1.2.4: Summary
section 1.3: Computation with Sage
subsection 1.3.1: Introduction to Sage
subsection 1.3.2: Sage and matrices
subsection 1.3.3: Computational effort
subsection 1.3.4: Summary
section 1.4: Pivots and their influence on solution spaces
subsection 1.4.1: Introduction
subsection 1.4.2: The existence of solutions
subsection 1.4.3: The uniqueness of solutions
subsection 1.4.4: Summary
chapter 2: Vectors, matrices, and linear combinations
section 2.1: Vectors and linear combinations
subsection 2.1.1: Vectors
subsection 2.1.2: Linear combinations
subsection 2.1.3: Summary
section 2.2: Matrix multiplication and linear combinations
subsection 2.2.1: Scalar multiplication and addition of matrices
subsection 2.2.2: Matrix-vector multiplication and linear combinations
subsection 2.2.3: Matrix-vector multiplication and linear systems
subsection 2.2.4: Matrix-matrix products
subsection 2.2.5: Summary
section 2.3: The span of a set of vectors
subsection 2.3.1: Introduction
subsection 2.3.2: The span of a set of vectors
subsection 2.3.3: Pivot positions and span
subsection 2.3.4: Summary
section 2.4: Linear independence
subsection 2.4.1: Introduction
subsection 2.4.2: Linear dependence
subsection 2.4.3: How to recognize linear dependence
subsection 2.4.4: Homogeneous equations
subsection 2.4.5: Summary
section 2.5: Matrix transformations
subsection 2.5.1: Introduction
subsection 2.5.2: Matrix transformations
subsection 2.5.3: Composing matrix transformations
subsection 2.5.4: Discrete Dynamical Systems
subsection 2.5.5: Summary
section 2.6: The geometry of matrix transformations
subsection 2.6.1: Introduction
subsection 2.6.2: The geometry of $2\times2$ matrix transformations
subsection 2.6.3: Matrix transformations and computer animation
subsection 2.6.4: Summary
chapter 3: Invertibility, bases, and coordinate systems
section 3.1: Invertibility
subsection 3.1.1: Introduction
subsection 3.1.2: Invertible matrices
subsection 3.1.3: Solving equations with an inverse
subsection 3.1.4: Triangular matrices and Gaussian elimination
subsection 3.1.5: Summary
section 3.2: Bases and coordinate systems
subsection 3.2.1: Introduction
subsection 3.2.2: Bases
subsection 3.2.3: Coordinate systems
subsection 3.2.4: Examples of bases
subsection 3.2.5: Summary
section 3.3: Image compression
subsection 3.3.1: Introduction
subsection 3.3.2: Color models
subsection 3.3.3: The JPEG compression algorithm
subsection 3.3.4: Summary
section 3.4: Determinants
subsection 3.4.1: Introduction
subsection 3.4.2: Determinants of $2\times2$ matrices
subsection 3.4.3: Determinants and invertibility
subsection 3.4.4: Cofactor expansions
subsection 3.4.5: Summary
section 3.5: Subspaces
subsection 3.5.1: Introduction
subsection 3.5.2: Subspaces
subsection 3.5.3: The column space of $A$
subsection 3.5.4: The null space of $A$
subsection 3.5.5: Summary
chapter 4: Eigenvalues and eigenvectors
section 4.1: An introduction to eigenvalues and eigenvectors
subsection 4.1.1: Introduction
subsection 4.1.2: A few examples
subsection 4.1.3: The usefulness of eigenvalues and eigenvectors
subsection 4.1.4: Summary
section 4.2: Finding eigenvalues and eigenvectors
subsection 4.2.1: Introduction
subsection 4.2.2: The characteristic polynomial
subsection 4.2.3: Finding eigenvectors
subsection 4.2.4: The characteristic polynomial and the dimension of eigenspaces
subsection 4.2.5: Using Sage to find eigenvalues and eigenvectors
subsection 4.2.6: Summary
section 4.3: Diagonalization, similarity, and powers of a matrix
subsection 4.3.1: Introduction
subsection 4.3.2: Diagonalization of matrices
subsection 4.3.3: Powers of a diagonalizable matrix
subsection 4.3.4: Similarity and complex eigenvalues
subsection 4.3.5: Summary
section 4.4: Dynamical systems
subsection 4.4.1: Introduction
subsection 4.4.2: A first example
subsection 4.4.3: Classifying dynamical systems
subsection 4.4.4: A $3\times3$ system
subsection 4.4.5: Summary
section 4.5: Markov chains and Google's PageRank algorithm
subsection 4.5.1: Introduction
subsection 4.5.2: A first example
subsection 4.5.3: Markov chains
subsection 4.5.4: Google's PageRank algorithm
subsection 4.5.5: Summary
chapter 5: Linear algebra and computing
section 5.1: Gaussian elimination revisited
subsection 5.1.1: Introduction
subsection 5.1.2: Partial pivoting
subsection 5.1.3: $LU$ factorizations
subsection 5.1.4: Summary
section 5.2: Finding eigenvectors numerically
subsection 5.2.1: Introduction
subsection 5.2.2: The power method
subsection 5.2.3: Finding other eigenvalues
subsection 5.2.4: Summary
chapter 6: Orthogonality and Least Squares
section 6.1: The dot product
subsection 6.1.1: Introduction
subsection 6.1.2: The geometry of the dot product
subsection 6.1.3: $k$-means clustering
subsection 6.1.4: Summary
section 6.2: Orthogonal complements and the matrix transpose
subsection 6.2.1: Introduction
subsection 6.2.2: Orthogonal complements
subsection 6.2.3: The matrix transpose
subsection 6.2.4: Properties of the matrix transpose
subsection 6.2.5: Summary
section 6.3: Orthogonal bases and projections
subsection 6.3.1: Introduction
subsection 6.3.2: Orthogonal sets
subsection 6.3.3: Orthogonal projections
subsection 6.3.4: Summary
section 6.4: Finding orthogonal bases
subsection 6.4.1: Introduction
subsection 6.4.2: Gram-Schmidt orthogonalization
subsection 6.4.3: $QR$ factorizations
subsection 6.4.4: Summary
section 6.5: Orthogonal least squares
subsection 6.5.1: Introduction
subsection 6.5.2: A first example
subsection 6.5.3: Solving least-squares problems
subsection 6.5.4: Using $QR$ factorizations
subsection 6.5.5: Polynomial Regression
subsection 6.5.6: Summary
chapter 7: Singular value decompositions
section 7.1: Symmetric matrices and variance
subsection 7.1.1: Introduction
subsection 7.1.2: Symmetric matrices and orthogonal diagonalization
subsection 7.1.3: Variance
subsection 7.1.4: Summary
section 7.2: Quadratic forms
subsection 7.2.1: Introduction
subsection 7.2.2: Quadratic forms
subsection 7.2.3: Definite symmetric matrices
subsection 7.2.4: Summary
section 7.3: Principal Component Analysis
subsection 7.3.1: Introduction
subsection 7.3.2: Principal Component Analysis
subsection 7.3.3: Using Principal Component Analysis
subsection 7.3.4: Summary
section 7.4: Singular Value Decompositions
subsection 7.4.1: Introduction
subsection 7.4.2: Finding singular value decompositions
subsection 7.4.3: The structure of singular value decompositions
subsection 7.4.4: Reduced singular value decompositions
subsection 7.4.5: Summary
section 7.5: Using Singular Value Decompositions
subsection 7.5.1: Introduction
subsection 7.5.2: Least-squares problems
subsection 7.5.3: Rank $k$ approximations
subsection 7.5.4: Principal component analysis
subsection 7.5.5: Image compressing and denoising
subsection 7.5.6: Analyzing Supreme Court cases
subsection 7.5.7: Summary
appendix A: Sage Reference

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