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An Inquiry-Based Introduction to Linear Algebra and Applications
Feryal Alayont, Steven Schlicker
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Contents
Index
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Contents
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Front Matter
Preface
I
Systems of Linear Equations
1
Introduction to Systems of Linear Equations
Application: Electrical Circuits
Introduction
Notation and Terminology
Solving Systems of Linear Equations
The Geometry of Solution Sets of Linear Systems
Examples
Summary
Exercises
Project: Modeling an Electrical Circuit and the Wheatstone Bridge Circuit
2
The Matrix Representation of a Linear System
Application: Approximating Area Under a Curve
Introduction
Simplifying Linear Systems Represented in Matrix Form
Linear Systems with Infinitely Many Solutions
Linear Systems with No Solutions
Examples
Summary
Exercises
Project: Polynomial Interpolation to Approximate the Area Under a Curve
3
Row Echelon Forms
Application: Balancing Chemical Reactions
Introduction
The Echelon Forms of a Matrix
Determining the Number of Solutions of a Linear System
Producing the Echelon Forms
Examples
Summary
Exercises
Project: Modeling a Chemical Reaction
4
Vector Representation
Application: The Knight's Tour
Introduction
Vectors and Vector Operations
Geometric Representation of Vectors and Vector Operations
Linear Combinations of Vectors
The Span of a Set of Vectors
Examples
Summary
Exercises
Project: Analyzing Knight Moves
5
The Matrix-Vector Form of a Linear System
Application: Modeling an Economy
Introduction
The Matrix-Vector Product
The Matrix-Vector Form of a Linear System
Homogeneous and Nonhomogeneous Systems
The Geometry of Solutions to the Homogeneous System
Examples
Summary
Exercises
Project: Input-Output Models
6
Linear Dependence and Independence
Application: Bézier Curves
Introduction
Linear Independence
Determining Linear Independence
Minimal Spanning Sets
Examples
Summary
Exercises
Project: Generating Bézier Curves
7
Matrix Transformations
Application: Computer Graphics
Introduction
Properties of Matrix Transformations
Onto and One-to-One Transformations
Examples
Summary
Exercises
Project: The Geometry of Matrix Transformations
II
Matrices
8
Matrix Operations
Application: Algorithms for Matrix Multiplication
Introduction
Properties of Matrix Addition and Multiplication by Scalars
A Matrix Product
The Transpose of a Matrix
Properties of the Matrix Transpose
Examples
Summary
Exercises
Project: Strassen's Algorithm and Partitioned Matrices
9
Introduction to Eigenvalues and Eigenvectors
Application: The Google PageRank Algorithm
Introduction
Eigenvalues and Eigenvectors
Dynamical Systems
Examples
Summary
Exercises
Project: Understanding the PageRank Algorithm
10
The Inverse of a Matrix
Application: Modeling an Arms Race
Introduction
Invertible Matrices
Finding the Inverse of a Matrix
Properties of the Matrix Inverse
Examples
Summary
Exercises
Project: The Richardson Arms Race Model
11
The Invertible Matrix Theorem
Introduction
The Invertible Matrix Theorem
Examples
Summary
Exercises
III
The Vector Space
R
n
12
The Structure of
R
n
Application: Connecting GDP and Consumption in Romania
Introduction
Vector Spaces
The Subspace Spanned by a Set of Vectors
Examples
Summary
Exercises
Project: Least Squares Linear Approximation
13
The Null Space and Column Space of a Matrix
Application: The Lights Out Game
Introduction
The Null Space of a Matrix and the Kernel of a Matrix Transformation
The Column Space of a Matrix and the Range of a Matrix Transformation
The Row Space of a Matrix
Bases for
Nul
Nul
A
and
Col
Col
A
Examples
Summary
Exercises
Project: Solving the Lights Out Game
14
Eigenspaces of a Matrix
Application: Population Dynamics
Introduction
Eigenspaces of Matrix
Linearly Independent Eigenvectors
Examples
Summary
Exercises
Project: Modeling Population Migration
15
Bases and Dimension
Application: Lattice Based Cryptography
Introduction
The Dimension of a Subspace of
R
n
Conditions for a Basis of a Subspace of
R
n
Finding a Basis for a Subspace
Rank of a Matrix
Examples
Summary
Exercises
Project: The GGH Cryptosystem
16
Coordinate Vectors and Change of Basis
Application: Describing Orbits of Planets
Introduction
Bases as Coordinate Systems in
R
n
Change of Basis in
R
n
The Change of Basis Matrix in
R
n
Properties of the Change of Basis Matrix
Examples
Summary
Exercises
Project: Planetary Orbits and Change of Basis
IV
Eigenvalues and Eigenvectors
17
The Determinant
Application: Area and Volume
Introduction
The Determinant of a Square Matrix
Cofactors
The Determinant of a
3
×
3
Matrix
Two Devices for Remembering Determinants
Examples
Summary
Exercises
Project: Area and Volume Using Determinants
18
The Characteristic Equation
Application: Modeling the Second Law of Thermodynamics
Introduction
The Characteristic Equation
Eigenspaces, A Geometric Example
Dimensions of Eigenspaces
Examples
Summary
Exercises
Project: The Ehrenfest Model
19
Diagonalization
Application: The Fibonacci Numbers
Introduction
Diagonalization
Similar Matrices
Similarity and Matrix Transformations
Diagonalization in General
Examples
Summary
Exercises
Project: Binet's Formula for the Fibonacci Numbers
20
Approximating Eigenvalues and Eigenvectors
Application: Leslie Matrices and Population Modeling
Introduction
The Power Method
The Inverse Power Method
Examples
Summary
Exercises
Project: Managing a Sheep Herd
21
Complex Eigenvalues
Application: The Gershgorin Disk Theorem
Introduction
Complex Eigenvalues
Rotation and Scaling Matrices
Matrices with Complex Eigenvalues
Examples
Summary
Exercises
Project: Understanding the Gershgorin Disk Theorem
22
Properties of Determinants
Introduction
Elementary Row Operations and Their Effects on the Determinant
Elementary Matrices
Geometric Interpretation of the Determinant
An Explicit Formula for the Inverse and Cramer's Rule
The Determinant of the Transpose
Row Swaps and Determinants
Cofactor Expansions
The LU Factorization of a Matrix
Examples
Summary
Exercises
V
Orthogonality
23
The Dot Product in
R
n
Application: Hidden Figures in Computer Graphics
Introduction
The Distance Between Vectors
The Angle Between Two Vectors
Orthogonal Projections
Orthogonal Complements
Examples
Summary
Exercises
Project: Back-Face Culling
24
Orthogonal and Orthonormal Bases in
R
n
Application: Rotations in 3D
Introduction
Orthogonal Sets
Properties of Orthogonal Bases
Orthonormal Bases
Orthogonal Matrices
Examples
Summary
Exercises
Project: Understanding Rotations in 3-Space
25
Projections onto Subspaces and the Gram-Schmidt Process in
R
n
Application: MIMO Systems
Introduction
Projections onto Subspaces and Orthogonal Projections
Best Approximations
The Gram-Schmidt Process
The QR Factorization of a Matrix
Examples
Summary
Exercises
Project: MIMO Systems and Householder Transformations
26
Least Squares Approximations
Application: Fitting Functions to Data
Introduction
Least Squares Approximations
Examples
Summary
Exercises
Project: Other Least Squares Approximations
VI
Applications of Orthogonality
27
Orthogonal Diagonalization
Application: The Multivariable Second Derivative Test
Introduction
Symmetric Matrices
The Spectral Decomposition of a Symmetric Matrix
A
Examples
Summary
Exercises
Project: The Second Derivative Test for Functions of Two Variables
28
Quadratic Forms and the Principal Axis Theorem
Application: The Tennis Racket Effect
Introduction
Equations Involving Quadratic Forms in
R
2
Classifying Quadratic Forms
Inner Products
Examples
Summary
Exercises
Project: The Tennis Racket Theorem
29
The Singular Value Decomposition
Application: Search Engines and Semantics
Introduction
The Operator Norm of a Matrix
The SVD
SVD and the Null, Column, and Row Spaces of a Matrix
Examples
Summary
Exercises
Project: Latent Semantic Indexing
30
Using the Singular Value Decomposition
Application: Global Positioning System
Introduction
Image Compression
Calculating the Error in Approximating an Image
The Condition Number of a Matrix
Pseudoinverses
Least Squares Approximations
Examples
Summary
Exercises
Project: GPS and Least Squares
VII
Vector Spaces
31
Vector Spaces
Application: The Hat Puzzle
Introduction
Spaces with Similar Structure to
R
n
Vector Spaces
Subspaces
Examples
Summary
Exercises
Project: Hamming Codes and the Hat Puzzle
32
Bases for Vector Spaces
Application: Image Compression
Introduction
Linear Independence
Bases
Finding a Basis for a Vector Space
Examples
Summary
Exercises
Project: Image Compression with Wavelets
33
The Dimension of a Vector Space
Application: Principal Component Analysis
Introduction
Finite Dimensional Vector Spaces
The Dimension of a Subspace
Conditions for a Basis of a Vector Space
Examples
Summary
Exercises
Project: Understanding Principal Component Analysis
34
Coordinate Vectors and Coordinate Transformations
Application: Calculating Sums
Introduction
The Coordinate Transformation
Examples
Summary
Exercises
Project: Finding Formulas for Sums of Powers
35
Inner Product Spaces
Application: Fourier Series
Introduction
Inner Product Spaces
The Length of a Vector
Orthogonality in Inner Product Spaces
Orthogonal and Orthonormal Bases in Inner Product Spaces
Orthogonal Projections onto Subspaces
Best Approximations in Inner Product Spaces
Orthogonal Complements
Examples
Summary
Exercises
Project: Fourier Series and Musical Tones
36
The Gram-Schmidt Process in Inner Product Spaces
Application: Gaussian Quadrature
Introduction
The Gram-Schmidt Process using Inner Products
Examples
Summary
Exercises
Project: Gaussian Quadrature and Legendre Polynomials
VIII
Linear Transformations
37
Linear Transformations
Application: Fractals
Introduction
Onto and One-to-One Transformations
The Kernel and Range of Linear Transformation
Isomorphisms
Examples
Summary
Exercises
Project: Fractals via Iterated Function Systems
38
The Matrix of a Linear Transformation
Application: Secret Sharing Algorithms
Introduction
Linear Transformations from
R
n
to
R
m
The Matrix of a Linear Transformation
A Connection between
Ker
Ker
(
T
)
and a Matrix Representation of
T
Examples
Summary
Exercises
Project: Shamir's Secret Sharing and Lagrange Polynomials
39
Eigenvalues of Linear Transformations
Application: Linear Differential Equations
Introduction
Finding Eigenvalues and Eigenvectors of Linear Transformations
Diagonalization
Examples
Summary
Exercises
Project: Linear Transformations and Differential Equations
40
The Jordan Canonical Form
Application: The Bailey Model of an Epidemic
Introduction
When an Eigenvalue Decomposition Does Not Exist
Generalized Eigenvectors and the Jordan Canonical Form
Geometry of Matrix Transformations using the Jordan Canonical Form
Proof of the Existence of the Jordan Canonical Form
Nilpotent Matrices and Invariant Subspaces
The Jordan Canonical Form
Examples
Summary
Exercises
Project: Modeling an Epidemic
Back Matter
A
Complex Numbers
Complex Numbers
Conjugates and Modulus
Complex Vectors
B
Answers and Hints for Selected Exercises
Index
Authored in PreTeXt
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Chapter
I
Systems of Linear Equations
1
Introduction to Systems of Linear Equations
2
The Matrix Representation of a Linear System
3
Row Echelon Forms
4
Vector Representation
5
The Matrix-Vector Form of a Linear System
6
Linear Dependence and Independence
7
Matrix Transformations
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