Linear Algebra (Textbooks in Mathematics)

✍ Scribed by James R. Kirkwood, Bessie H. Kirkwood

• Tongue: English
• Leaves: 429
• Category: Library

✦ Synopsis

Linear Algebra, James R. Kirkwood and Bessie H. Kirkwood, 978-1-4987-7685-1, K29751

Shelving Guide: Mathematics

This text has a major focus on demonstrating facts and techniques of linear systems that will be invaluable in higher mathematics and related fields. A linear algebra course has two major audiences that it must satisfy. It provides an important theoretical and computational tool for nearly every discipline that uses mathematics. It also provides an introduction to abstract mathematics.

This book has two parts. Chapters 1–7 are written as an introduction. Two primary goals of these chapters are to enable students to become adept at computations and to develop an understanding of the theory of basic topics including linear transformations. Important applications are presented.

Part two, which consists of Chapters 8–14, is at a higher level. It includes topics not usually taught in a first course, such as a detailed justification of the Jordan canonical form, properties of the determinant derived from axioms, the Perron–Frobenius theorem and bilinear and quadratic forms.

Though users will want to make use of technology for many of the computations, topics are explained in the text in a way that will enable students to do these computations by hand if that is desired.

Key features include:

• Chapters 1–7 may be used for a first course relying on applications

• Chapters 8–14 offer a more advanced, theoretical course

• Definitions are highlighted throughout

• MATLAB® and R Project tutorials in the appendices

• Exercises span a range from simple computations to fairly direct abstract exercises

• Historical notes motivate the presentation

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Table of Contents
Preface
A Note about Mathematical Proofs
Chapter 1 Matrices
Section 1.1 Matrix Arithmetic
Matrix Arithmetic
Matrix Addition
Scalar Multiplication
Matrix Multiplication
Historical Note
Exercises
Section 1.2 The Algebra of Matrices
Properties of Matrix Addition, Scalar Multiplication, and Matrix Multiplication
The Identity Matrix
The Inverse of a Square Matrix
Determinants
Elementary matrices
Matrices that Interchange Two Rows of a Matrix
Multiplying a Row of a Matrix by a Constant
Adding a Multiple of One Row to Another Row
Computing the Inverse of a Matrix
The Transpose of a Matrix
Historical Note
Exercises
Historical Note
Section 1.3 The LU Decomposition of a Square Matrix (Optional)
Exercises
Chapter 2 Systems of Linear Equations
Historical Note
Section 2.1 Basic Definitions
Exercises
Section 2.2 Solving Systems of Linear Equations (Gaussian Elimination)
Solving Systems of Linear Equations
Using Technology to Accomplish Gaussian Elimination
Exercises
Section 2.3 Equivalent Systems of Linear Equations
Row Reduced Form of a Matrix
Exercises
Section 2.4 Expressing the Solution of a System of Linear Equations
Systems of Linear Equations That Have No Solutions
Systems of Linear Equations That Have Exactly One Solution
Systems of Linear Equations That Have Infinitely Many Solutions
Application of Linear Systems to Curve Fitting
Exercises
Section 2.5 Expressing Systems of Linear Equations in Other Forms
Representing a System of Linear Equations as a Vector Equation
Equivalence of a System of Linear Equations and a Matrix Equation
Exercises
Section 2.6 Applications
Flow Problems
Balancing Chemical Equations Using Linear Algebra
Exercises
Markov Chains
Exercises
Chapter 3 Vector Spaces
Historical Note
Section 3.1 Vector Spaces in
Normalizing a Vector
Exercises
Section 3.2 Axioms and Examples of Vector Spaces
Some Examples of Sets That Are Not Vector Spaces
Additional Properties of Vector Spaces
Exercises
Section 3.3 Subspaces of a Vector Space
Exercises
Section 3.4 Spanning Sets, Linearly independent Sets, and Bases
Exercises
Section 3.5 Converting a Set of Vectors to a Basis
A Synopsis of Sections 3.4 and 3.5
Exercises
Section 3.6 Change of Bases
A Vector Versus the Representation of a Vector.
Changing the Representation of a Vector From One Basis to Another
Converting Between Two Non-Standard Bases
Exercises
Section 3.7 The Null Space, Row Space, and Column Space of a Matrix
Exercises
Section 3.8 Sums and Direct Sums of Vector Spaces (Optional)
Exercises
Chapter 4 Linear Transformations
Historical Note
Section 4.1 Properties of a Linear Transformation
Null Space and Range (Image) of a Linear Transformation
Exercises
Section 4.2 Representing a Linear Transformation
The Representation of a Linear Transformation in the Usual Basis
Exercises
Historical Note
Hermann Grassmann (born 15 April 1809, died 26 September 1877)
Section 4.3 Finding the Representation of a Linear Operator with Respect to Different Bases
Exercises
Section 4.4 Composition (Multiplication) of Linear Transformations
Exercises
Chapter 5 Eigenvalues and Eigenvectors
Section 5.1 Determining Eigenvalues and Eigenvectors
Finding the Eigenvectors after the Eigenvalues Have Been Found
Exercises
Section 5.2 Diagonalizing a Matrix
The Algebraic and Geometric Multiplicities of an Eigenvalue
Diagonalizing a Matrix
Exercises
Section 5.3 Similar Matrices
Exercises
Section 5.4 Eigenvalues and Eigenvectors in Systems of Differential Equations
An Algorithm to Solve the Matrix Equation when A Can Be Diagonalized
Exercises
Chapter 6 Inner Product Spaces
Section 6.1 Some Facts About Complex Numbers
Exercises
Section 6.2 Inner Product Spaces
Historical Note
Hermann Schwarz (born 25 January 1843, died 30 November 1921)
Exercises
Section 6.3 Orthogonality
Exercises
Section 6.4 The Gram–Schmidt Process
Historical Note
Erhardt Schmidt (born 13 January 1876, died 6 December 1959)
Jurgen Gram (born 27 June 1850, died 29 April 1916)
Algorithm for the Gram–Schmidt Process
Exercises
Section 6.5 Representation of a Linear Transformation on Inner Product Spaces (Optional)
Exercises
Section 6.6 Orthogonal Complement
Exercises
Section 6.7 Four Subspaces Associated with a Matrix (Optional)
Section 6.8 Projections
The Projection Matrix
Exercises
Section 6.9 Least Squares Estimates in Statistics (Optional)
Summary
Exercise
Section 6.10 Weighted Inner Products (Optional)
Chapter 7 Linear Functionals, Dual Spaces, and Adjoint Operators
Section 7.1 Linear Functionals
The Second Dual of a Vector Space (Optional)
Exercises
Section 7.2 The Adjoint of a Linear Operator
The Adjoint Operator
Adjoint on Weighted Inner Product Spaces (Optional)
Exercises
Section 7.3 The Spectral Theorem
Exercises
Chapter 8 Two Decompositions of a Matrix
Section 8.1 Polar Decomposition of a Matrix
Exercises
Section 8.2 Singular Value Decomposition of a Matrix
Exercises
Chapter 9 Determinants
Introduction
Determinants from a Geometrical Point of View
Three Postulates of the Determinant
The Determinant Based on the Three Postulates
Formula for Computing the Determinant
Case of a 2 × 2 Matrix
The Case of a 3 × 3 Matrix
The Case of an n × n Matrix
Permutations
Further Properties of the Determinant
Computing a Determinant
Exercise
Chapter 10 Jordan Canonical Form
Historical Note
Camille Jordan (born 5 January 1838, died 20 January 1922)
Introduction
Section 10.1 Finding the Jordan Canonical Form and the Associated Basis
Section 10.2 Constructing the Basis That Gives the Jordan Canonical Form for the Matrix
Section 10.3 The Theory That Justifies the Algorithms We Have Developed
Section 10.4 Determining the Block Structure of the Jordan Canonical Form
An Algorithm for Determining the Jordan Blocks for a Particular Eigenvalue λ
Exercises
Chapter 11 Applications of the Jordan Canonical Form
Section 11.1 Finding the Square Root of a Matrix
Exercises
Section 11.2 The Cayley–Hamilton Theorem, Minimal Polynomials, Characteristic Polynomials, and the Jordan Canonical Form.
Historical Note
Arthur Cayley (born16 August 1821, died 26 January 1895)
Exercises
Chapter 12 The Perron–Frobenius Theorem
Theorem 12.1: Perron–Frobenius Theorem
An Application of the Perron–Frobenius Theorem
The Perron–Frobenius Theorem in Population Dynamics
Exercises
Chapter 13 Bilinear Forms
Section 13.1 Bilinear Forms
Change of Basis
Exercises
Section 13.2 Orthogonality on Bilinear Forms
Section 13.3 Quadratic Forms
Classifying Quadratic Forms
Singular Values
Signature of Quadratic Forms
Exercises
Chapter 14 Introduction to Tensor Product
Construction 1
Construction 2
Example of a Particular Tensor Product
Tensor Product of Matrices
Appendix I: A Brief Guide to MATLAB
Appendix II: R for Linear Algebra
Answers to Selected Exercises
Index