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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
Preliminaries
1.1
Introduction to Vectors
1.1.1
A Brief Introduction to
\(\R^n\)
1.1.2
Distance in
\(\R^n\)
1.1.3
What is a Vector?
1.1.4
Vectors in Standard Position
1.1.5
Vectors in
\(\R^3\)
1.1.6
Vectors in
\(\R^n\)
1.1.7
Length of a Vector
1.1.8
Geometry of Scalar Multiplication
1.1.9
Algebra of Scalar Multiplication
1.1.10
Geometry of Vector Addition
1.1.10.1
“Head-to-Tail” Addition Method
1.1.10.2
Parallelogram Addition Method
1.1.11
Algebra of Vector Addition
1.1.12
Geometry of Vector Addition in
\(\R^3\)
1.1.13
Vector Subtraction
1.1.14
Properties of Vector Addition and Scalar Multiplication
1.1.15
Exercises
1.2
Unit Vectors, the Dot Product, and Orthogonal Projection
1.2.1
Standard Unit Vectors in
\(\R^2\)
and
\(\R^3\)
1.2.2
A Vector as a Linear Combination of Standard Unit Vectors
1.2.3
Standard Unit Vectors in
\(\R^n\)
1.2.4
Unit Vector in the Direction of a Given Vector
1.2.5
The Dot Product
1.2.6
Orthogonal Projections
1.2.7
Distance from a Point to a Line
1.2.8
Exercises
1.3
Lines and Planes
1.3.1
Parametric Lines in
\(\R^2\)
1.3.2
Parametric Lines in
\(\R^3\)
1.3.3
Parametric Lines in
\(\R^n\)
1.3.4
From Parametric Equations to Vector Equations
1.3.5
Planes in
\(\R^3\)
1.3.6
Linear Equations and their Graphs: From Lines to Hyperplanes
1.3.7
Exercises
1.4
Extra Topic: Dot Product and Angle
1.4.1
Orthogonal Vectors
1.4.2
Exercises
1.5
Extra Topic: Cross Product and its Properties
1.5.1
Preliminaries
1.5.2
Definition of the Cross Product
1.5.3
Properties of the Cross Product
1.5.4
Orthogonality Property
1.5.5
Cross Product and the Angle between Vectors
1.5.6
Exercises
2
Systems of Linear Equations
2.1
Introduction to Systems of Linear Equations
2.1.1
Algebra of Linear Systems
2.1.2
Geometry of Linear Systems in Two Variables
2.1.3
Geometry of Linear Systems in Three Variables
2.1.4
General Systems of Linear Equations
2.1.5
Exercises
2.2
Augmented Matrix Notation and Elementary Row Operations
2.2.1
Augmented Matrix Notation
2.2.2
Row-Echelon and Reduced Row-Echelon Forms
2.2.3
Exercises
2.3
Gaussian Elimination and Rank
2.3.1
Row Echelon and Reduced Row Echelon Forms
2.3.2
Gaussian and Gauss-Jordan Elimination
2.3.3
Rank
2.3.4
Exercises
2.3.5
References
2.4
Extra Topic: Applications of Systems of Linear Equations
2.4.1
Network Flows
2.4.2
Electrical Networks
2.4.3
Chemical Equations
2.4.4
Exercises
3
Big Ideas about Vectors
3.1
Linear Combinations of Vectors and their Span
3.1.1
Visualizing Linear Combinations in
\(\R^2\)
and
\(\R^3\)
3.1.2
Geometry of Linear Combinations
3.1.3
The Linear Span
3.1.4
Exercises
3.2
Linear Independence
3.2.1
Redundant Vectors
3.2.2
Linear Independence
3.2.3
Geometry of Linearly Dependent and Linearly Independent Vectors
3.2.3.1
A Set of Two Vectors
3.2.3.2
A Set of Three Vectors
3.2.4
Exercises
4
Matrices
4.1
Matrix Operations
4.1.1
Addition - and Scalar multiplication of Matrices
4.1.2
Matrix Multiplication
4.1.1
Matrix-Vector Multiplication
4.1.2
Matrix-Matrix Multiplication
4.1.3
Properties of Matrix Multiplication
4.1.4
Transpose of a Matrix
4.1.7
Exercises
4.2
Linear Systems as Matrix and Linear Combination Equations
4.2.1
Singular and Nonsingular Matrices
4.2.2
A Linear System as a Linear Combination Equation
4.2.3
Exercises
4.3
Homogeneous Linear Systems
4.3.1
General and Particular Solutions
4.3.2
Exercises
4.4
The Inverse of a Matrix
4.4.1
Computing the Inverse
4.4.2
Inverse of a
\(2\times 2\)
Matrix
4.4.3
Exercises
4.5
Extra Topic: Elementary Matrices
4.5.1
Inverses of Elementary Matrices
4.5.2
Exercises
4.6
Extra Topic:
\(LU\)
Factorization
4.6.1
Finding an
\(LU\)
factorization by the Multiplier Method
4.6.2
Exercises
4.7
Extra Topic: Application to Markov Chains
4.7.1
Steady State Vector
4.7.2
Exercises
5
Subspaces of
\(\R^n\)
5.1
\(\mathbb{R}^n\)
and subspaces of
\(\R^n\)
5.1.1
Closure
5.1.2
\(\R^n\)
as a Vector Space
5.1.3
Subspaces of
\(\R^n\)
5.1.4
Exercises
5.2
Bases and Dimension
5.2.1
Coordinate Vectors
5.2.2
What Constitutes a Basis?
5.2.3
Definition of a Basis
5.2.4
Exploring Dimension
5.2.5
Exercises
5.3
Subspaces of
\(\R^n\)
Associated with Matrices
5.3.1
Row Space of a Matrix
5.3.2
Column Space of a Matrix
5.3.3
The Null Space
5.3.4
Rank and Nullity Theorem
5.3.5
Exercises
6
Linear Transformations
6.1
Matrix Transformations
6.1.1
Functions from
\(\R^n\)
into
\(\R^m\)
6.1.2
Examples of Matrix Transformations
6.1.3
Linearity of Matrix Transformations
6.1.4
Where did
\(\mathbf{i}\)
Go?
6.1.5
Exercises
6.2
Linear Transformations
6.2.1
Linear Transformations Induced by Matrices
6.2.2
Linear Transformations of Subspaces of
\(\R^n\)
6.2.3
Standard Matrix of a Linear Transformation from
\(\R^n\)
to
\(\R^m\)
6.2.4
The Image
6.2.5
The Kernel of a Linear Transformation
6.2.6
Rank-Nullity Theorem for Linear Transformations
6.2.7
Exercises
6.3
Extra Topic: Composition and Inverses
6.3.1
Composition and Matrix Multiplication
6.3.2
Inverses of Linear Transformations
6.3.3
Linear Transformations of
\(\R^n\)
and the Standard Matrix of the Inverse Transformation
6.3.4
Exercises
6.4
Extra Topic: Geometric Transformations of the Plane
6.4.1
Horizontal and Vertical Scaling
6.4.2
Horizontal and Vertical Shears
6.4.3
Rotations about the Origin
6.4.4
Reflections about Lines of the Form
\(y=mx\)
6.4.5
Composition of Linear Transformations
6.4.6
Exercises
6.5
Extra Topic: Application to Computer Graphics
6.5
Exercises
7
The Determinant
7.1
Finding the Determinant
7.1.1
Cofactor Expansion Along the Top Row
7.1.2
Cofactor Expansion Along the First Column
7.1.3
Cofactor Expansion Along Any Row or Column
7.1.4
A Note on Equivalency
7.1.5
Determinants of Some Special Matrices
7.1.6
Exercises
7.2
Properties of the Determinant
7.2.1
The Effects of Elementary Row Operations on the Determinant
7.2.2
Computing the Determinant Using Elementary Row Operations
7.2.3
Properties of the Determinant
7.2.4
Exercises
7.3
Extra Topic: Cramer’s Rule
7.3.1
Cramer’s Rule
7.3.2
Adjugate Formula for the Inverse of a Matrix
7.3.3
Exercises
7.4
Extra Topic: Determinants, Areas, and Volumes
7.4.1
\(2\times 2\)
Determinant and the Area of a Parallelogram
7.4.2
\(3\times 3\)
Determinant and the Volume of a Parallelepiped
7.4.3
Determinants and Linear Transformations
7.4.4
Exercises
8
Eigenvalues and Eigenvectors
8.1
Eigenvalues and Eigenvectors
8.1.1
The Characteristic Equation
8.1.2
Eigenvalues
8.1.3
Eigenvectors
8.1.4
Exercises
8.2
Similar and Diagonalizable Matrices
8.2.1
Diagonalizable Matrices and Multiplicity
8.2.2
Exercises
9
Vector Spaces
9.1
Abstract Vector Spaces
9.1.1
Properties of Vector Spaces
9.1.2
Definition of a Vector Space
9.1.3
Subspaces
9.1.4
Linear Combinations and Span
9.1.5
Bases and Dimension of Abstract Vector Spaces
9.1.6
Bases and Dimension
9.1.7
Finite-Dimensional Vector Spaces
9.1.8
Coordinate Vectors
9.1.9
Exercises
9.2
Linear Transformations of Abstract Vector Spaces
9.2.1
Linear Transformations and Bases
9.2.2
Coordinate Vectors
9.2.3
Inverses of a Linear Transformations
9.2.4
One-to-one and Onto Linear Transformations
9.2.5
Existence and Uniqueness of Inverses
9.2.6
Exercises
9.3
Extra Topic: Isomorphic Vector Spaces
9.3.1
The Coordinate Vector Isomorphism
9.3.2
Properties of Isomorphic Vector Spaces and Isomorphisms
9.3.3
Proofs of Isomorphism Properties
9.3.4
Finite-dimensional Vector Spaces
9.3.5
Exercises
9.4
Extra Topic: Inner Product Spaces
9.4.1
Norm and Distance
9.4.2
Exercises
10
Orthogonality
10.1
Orthogonality and Projections
10.1.1
Orthogonal and Orthonormal Sets
10.1.2
Orthogonal and Orthonormal Bases
10.1.3
Orthogonal Projection onto a Subspace
10.1.4
Orthogonal Decomposition of
\(\mathbf{x}\)
10.1.5
Exercises
10.2
Gram-Schmidt Orthogonalization
10.2.1
A Visual Guide to Creating an Orthogonal Basis
10.2.2
Gram-Schmidt Orthogonalization Algorithm
10.2.3
Exercises
10.3
Orthogonal Complements and Decompositions
10.3.1
Orthogonal Complements
10.3.2
Orthogonal Decomposition Theorem
10.3.3
Exercises
10.4
Orthogonal Matrices and Symmetric Matrices
10.4.1
Orthogonal Matrices
10.4.2
Symmetric Matrices
10.4.3
Exercises
10.5
\(QR\)
Factorization and Least Square Approximations
10.5.1
QR-Algorithm for approximating eigenvalues
10.5.2
Least-Squares Approximation
10.5.3
Application of Least Squares to Curve Fitting
10.5.4
\(QR\)
-Factorization: A Quicker Way to do Least Squares
10.5.5
Exercises
10.6
SVD Decomposition
10.7
Extra Topic: Positive Definite Matrices
10.7
Exercises
10.8
Extra Topic: Curve Fitting
10.8.1
On the Dangers of Overfitting
10.8.2
Exercises
Backmatter
Colophon
Colophon
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