Table of Contents

Preliminaries

A Brief Introduction to ℝn

Introduction to Vectors

Length of a Vector

Vector Arithmetic

Standard Unit Vectors in ℝn

Unit Vector in the Direction of a Given Vector

Dot Product and its Properties

Dot Product and Angle

Orthogonal Projections

Cross Product and its Properties

Parametric Equations of Lines

Planes in ℝ3

Systems of Linear Equations

Introduction to Systems of Linear Equations

Augmented Matrix Notation and Elementary Row Operations

Gaussian Elimination and Rank

Iterative Methods for Solving Linear Systems

Big Ideas about Vectors

Linear Combinations of Vectors

Span

Linear Independence

Matrices

Addition and Scalar Multiplication of Matrices

Matrix Multiplication

Block Matrix Multiplication

Transpose of a Matrix

Linear Systems as Matrix and Linear Combination Equations

Homogeneous Linear Systems

The Inverse of a Matrix

Elementary Matrices

LU Factorization

Subspaces of ℝn

ℝn and Subspaces of ℝn

Introduction to Bases

Bases and Dimension

Subspaces of ℝn Associated with Matrices

Linear Transformations

Matrix Transformations

Geometric Transformations of the Plane

Introduction to Linear Transformations

Standard Matrix of a Linear Transformation from ℝn to ℝm

Composition and Inverses of Linear Transformations

Image and Kernel of a Linear Transformation

The Determinant

Finding the Determinant

Determinants, Areas, and Volumes

Elementary Row Operations and the Determinant

Properties of the Determinant

Tedious Proofs Concerning Determinants

Determinants and Inverses of Nonsingular Matrices

Eigenvalues and Eigenvectors

Describing Eigenvalues and Eigenvectors

The Characteristic Equation

Similar Matrices and Their Properties

Diagonalizable Matrices and Multiplicity

Gershgorin’s Theorem

The Power Method and the Dominant Eigenvalue

Orthogonality

Orthogonality and Projections

Least-Squares Approximation

Gram-Schmidt Orthogonalization

Orthogonal Complements and Decompositions

Orthogonal Matrices and Symmetric Matrices

Positive Definite Matrices

QR Factorization

SVD Decomposition

Vector Spaces

Abstract Vector Spaces

Bases and Dimension of Abstract Vector Spaces

Linear Transformations of Abstract Vector Spaces

Existence of the Inverse of a Linear Transformation

Isomorphic Vector Spaces

Matrices of Linear Transformations with Respect to Arbitrary Bases

Applications

Application to Network Flow

Application to Electrical Networks

Application to Chemical Equations

Application to Input-Output Economic Models

Application to Markov Chains

Curve Fitting

Application to Computer Graphics

Appendix

Triangle Inequality

Complex Numbers

Complex Matrices

Inner Product Spaces

INDEX

Index of GeoGebra Interactives

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