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Coordinated Linear Algebra
Anna Davis, Paul Zachlin, Allan Donsig, Levi Heath, Mikkel Munkholm
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Front Matter
Colophon
Acknowledgements
1
Systems of Linear Equations
1.1
Introduction to Systems of Linear Equations
1.1.1
Algebra of Linear Systems
1.1.2
Augmented Matrix Notation
1.1.3
Geometry of Linear Systems in Two Variables
1.1.4
Geometry of Linear Systems in Three Variables
1.1.5
General Systems of Linear Equations
1.1.6
Exercises
1.2
Row Echelon Forms
1.2.1
Row Echelon Form and Reduced Row Echelon Form
1.2.2
Gaussian and Gauss-Jordan Elimination
1.2.3
Solutions to Systems of Equations
1.2.4
Exercises
1.2.5
References
1.3
Extra Topic: Applications of Systems of Linear Equations
1.3.1
Network Flows
1.3.2
Electrical Networks
1.3.3
Chemical Equations
1.3.4
Exercises
2
Vectors
2.1
Introduction to Vectors
2.1.1
A Brief Introduction to
\(\R^n\)
2.1.2
Distance in
\(\R^n\)
2.1.3
What is a Vector?
2.1.4
Vectors in Standard Position
2.1.5
Vectors in
\(\R^3\)
2.1.6
Vectors in
\(\R^n\)
2.1.7
Length of a Vector
2.1.8
Geometry of Scalar Multiplication
2.1.9
Algebra of Scalar Multiplication
2.1.10
Standard Unit Vectors in
\(\R^2\)
and
\(\R^3\)
2.1.11
A Vector as a Linear Combination of Standard Unit Vectors
2.1.12
Standard Unit Vectors in
\(\R^n\)
2.1.13
Unit Vector in the Direction of a Given Vector
2.1.14
Geometry of Vector Addition
2.1.14.1
“Head-to-Tail” Addition Method
2.1.14.2
Parallelogram Addition Method
2.1.15
Algebra of Vector Addition
2.1.16
Geometry of Vector Addition in
\(\R^3\)
2.1.17
Vector Subtraction
2.1.18
Properties of Vector Addition and Scalar Multiplication
2.1.19
Exercises
2.2
Linear Combination Equations and Span
2.2.1
Visualizing Linear Combinations in
\(\R^2\)
and
\(\R^3\)
2.2.2
Solving Linear Combination Equations
2.2.3
The Linear Span
2.2.4
Exercises
2.3
Matrix Equations
2.3.1
Matrix-Vector Multiplication
2.3.2
Matrix Equations
2.3.3
Singular and Nonsingular Matrices
2.3.4
Connection to Linear Combination Equations
2.3.5
Exercises
2.4
Homogeneous Linear Systems
2.4.1
General and Particular Solutions
2.4.2
Linear Independence
2.4.3
Geometry of Linearly Dependent and Linearly Independent Vectors
2.4.3.1
A Set of Two Vectors
2.4.3.2
A Set of Three Vectors
2.4.4
Exercises
2.5
Matrix Transformations
2.5.1
Functions from
\(\R^n\)
into
\(\R^m\)
2.5.2
Examples of Matrix Transformations
2.5.3
Linearity of Matrix Transformations
2.5.4
Columns and the standard basis
2.5.5
Exercises
2.6
Linear Transformations
2.6.1
Understanding Linearity
2.6.2
Standard Matrix of a Linear Transformation from
\(\R^n\)
to
\(\R^m\)
2.6.3
Injective and Surjective Linear Transformations
2.6.4
Exercises
3
Matrices
3.1
Matrix Operations
3.1.1
Addition and Scalar Multiplication of Matrices
3.1.2
Matrix Multiplication
3.1.2.1
Matrix-Matrix Multiplication
3.1.2.2
Properties of Matrix Multiplication
3.1.3
Transpose of a Matrix
3.1.4
Exercises
3.2
The Inverse of a Matrix
3.2.1
Definition and Properties of the Inverse
3.2.2
Computing the Inverse
3.2.3
Inverse of a
\(2\times 2\)
Matrix
3.2.4
Exercises
3.3
Extra Topic: Elementary Matrices
3.3.1
Elmentary Row Operations as Matrices
3.3.2
Inverses of Elementary Matrices
3.3.3
Exercises
3.4
Extra Topic:
\(LU\)
Factorization
3.4.1
Motivation and Definition of the Factorization
3.4.2
Finding an
\(LU\)
factorization by the Multiplier Method
3.4.3
Exercises
3.5
Extra Topic: Application to Markov Chains
3.5.1
Definition of a Markov Chain
3.5.2
Steady State Vector
3.5.3
Exercises
4
Vector Spaces
4.1
Introduction to Vector Spaces
4.1.1
Closure
4.1.2
Properties of Vector Spaces
4.1.3
Definition of a Vector Space
4.1.4
Subspaces
4.1.5
Linear Combinations and Span
4.1.6
Exercises
4.2
Bases and Dimension
4.2.1
Coordinate Vectors for Vector Spaces
4.2.2
What Constitutes a Basis?
4.2.3
Definition of a Basis
4.2.4
Exploring Dimension
4.2.5
Finite-Dimensional Vector Spaces
4.2.6
Coordinate Vectors Revisited
4.2.7
Exercises
4.3
Subspaces Associated with Matrices
4.3.1
Row Space of a Matrix
4.3.2
Column Space of a Matrix
4.3.3
The Null Space
4.3.4
Rank and Nullity Theorem
4.3.5
Subspaces Associated with Matrix Transformations
4.3.5.1
The Image of a Matrix Transformation
4.3.5.2
The Kernel of a Linear Transformation
4.3.5.3
Rank-Nullity Theorem for Linear Transformations
4.3.6
Exercises
4.4
Linear Transformations of Vector Spaces
4.4.1
Defining Linear Transformations Again
4.4.2
Linear Transformations and Bases
4.4.3
Coordinate Vectors
4.4.4
Matrix Representation of a Linear Transformation
4.4.5
Exercises
4.5
Extra Topic: Isomorphic Vector Spaces
4.5.1
Inverses of Linear Transformations
4.5.2
When are two vector spaces the same?
4.5.3
The Coordinate Vector Isomorphism
4.5.4
Properties of Isomorphic Vector Spaces and Isomorphisms
4.5.5
Proofs of Isomorphism Properties
4.5.6
Finite-dimensional Vector Spaces
4.5.7
Exercises
5
Determinants
5.1
Finding the Determinant
5.1.1
Cofactor Expansion Along the Top Row
5.1.2
Cofactor Expansion Along the First Column
5.1.3
Cofactor Expansion Along Any Row or Column
5.1.4
A Note on Equivalency
5.1.5
Determinants of Some Special Matrices
5.1.6
Exercises
5.2
Properties of the Determinant
5.2.1
The Effects of Elementary Row Operations on the Determinant
5.2.2
Computing the Determinant Using Elementary Row Operations
5.2.3
Properties of the Determinant
5.2.4
Exercises
5.3
Extra Topic: Cramer’s Rule
5.3.1
Cramer’s Rule
5.3.2
Adjugate Formula for the Inverse of a Matrix
5.3.3
Exercises
5.4
Extra Topic: Determinants, Areas, and Volumes
5.4.1
\(2\times 2\)
Determinant and the Area of a Parallelogram
5.4.2
\(3\times 3\)
Determinant and the Volume of a Parallelepiped
5.4.3
Determinants and Linear Transformations
5.4.4
Exercises
6
Eigenvalues and Eigenvectors
6.1
Eigenvalues and Eigenvectors
6.1.1
Defining Eigenvectors and Eigenvalues
6.1.2
The Characteristic Equation
6.1.3
Eigenvalues
6.1.4
Eigenvectors
6.1.5
Exercises
6.2
Similar and Diagonalizable Matrices
6.2.1
Similar Matrices and their Properties
6.2.2
Diagonalizable Matrices and Multiplicity
6.2.3
Exercises
7
Orthogonality
7.1
The Dot Product
7.1.1
The Dot Product
7.1.2
Orthogonal Projections
7.1.3
Distance from a Point to a Line
7.1.4
Exercises
7.2
Orthogonality and Projections
7.2.1
Orthogonal and Orthonormal Sets
7.2.2
Orthogonal and Orthonormal Bases
7.2.3
Orthogonal Projection onto a Subspace
7.2.4
Orthogonal Decomposition of
\(\mathbf{x}\)
7.2.5
Exercises
7.3
Gram-Schmidt Orthogonalization
7.3.1
A Visual Guide to Creating an Orthogonal Basis
7.3.2
Gram-Schmidt Orthogonalization Algorithm
7.3.3
Exercises
7.4
Orthogonal Complements and Decompositions
7.4.1
Orthogonal Complements
7.4.2
Orthogonal Decomposition Theorem
7.4.3
Exercises
7.5
Orthogonal Matrices and Symmetric Matrices
7.5.1
Orthogonal Matrices
7.5.2
Symmetric Matrices
7.5.3
Exercises
7.6
Extra Topic:
\(QR\)
Factorization and Least Square Approximations
7.6.1
Definition of QR Factorization
7.6.2
QR-Algorithm for approximating eigenvalues
7.6.3
Least-Squares Approximation
7.6.4
Application of Least Squares to Curve Fitting
7.6.5
\(QR\)
-Factorization: A Quicker Way to do Least Squares
7.6.6
Exercises
7.7
Extra Topic: Singular Value Decomposition
7.8
Extra Topic: Curve Fitting
7.8.1
Basics of Curve Fitting
7.8.2
On the Dangers of Overfitting
7.8.3
Exercises
Backmatter
Colophon
Colophon
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