Оглавление
Michael L. O'Leary. Linear Algebra
Table of Contents
List of Illustrations
Guide
Pages
Linear Algebra
Preface
Acknowledgments
About the Companion Website
CHAPTER 1 Logic and Set Theory. 1.1 Statements
Connectives
Definition 1.1.1
Definition 1.1.2
Definition 1.1.3
Example 1.1.4
Logical Equivalence
Example 1.1.5
Example 1.1.6
Example 1.1.7
Example 1.1.8
Exercises
1.2 Sets and Quantification
Universal Quantifiers
Definition 1.2.1
Example 1.2.2
Example 1.2.3
Existential Quantifiers
Definition 1.2.4
Example 1.2.5
Example 1.2.6
Negating Quantifiers
Theorem 1.2.7
Example 1.2.8
Example 1.2.9
Example 1.2.10
Set‐Builder Notation
Definition 1.2.11
Example 1.2.12
Example 1.2.13
Example 1.2.14
Set Operations
Definition 1.2.15
Example 1.2.16
Definition 1.2.17
Example 1.2.18
Families of Sets
Definition 1.2.19
Definition 1.2.20
Example 1.2.21
Exercises
1.3 Sets and Proofs
Definition 1.3.1
Definition 1.3.2
Definition 1.3.3
Direct Proof
Theorem 1.3.4 [Direct Proof]
Example 1.3.5
Example 1.3.6
Example 1.3.7
Example 1.3.8
Subsets
Definition 1.3.9
Example 1.3.10
Example 1.3.11
Theorem 1.3.12
Proof
Example 1.3.13
Set Equality
Definition 1.3.14
Example 1.3.15
Example 1.3.16
Example 1.3.17
Indirect Proof
Theorem 1.3.18 [Indirect Proof]
Example 1.3.19
Example 1.3.20
Example 1.3.21
Mathematical Induction
Axiom 1.3.22 [Mathematical Induction]
Theorem 1.3.23
Proof
Example 1.3.24
Exercises
1.4 Functions
Definition 1.4.1
Example 1.4.2
Example 1.4.3
Example 1.4.4
Example 1.4.5
Theorem 1.4.6
Proof
Example 1.4.7
Example 1.4.8
Definition 1.4.9
Example 1.4.10
Injections
Definition 1.4.11
Example 1.4.12
Example 1.4.13
Theorem 1.4.14
Proof
Surjections
Definition 1.4.15
Example 1.4.16
Example 1.4.17
Example 1.4.18
Example 1.4.19
Theorem 1.4.20
Example 1.4.21
Theorem 1.4.22
Proof
Bijections and Inverses
Definition 1.4.23
Definition 1.4.24
Theorem 1.4.25
Proof
Example 1.4.26
Example 1.4.27
Theorem 1.4.28
Proof
Example 1.4.29
Theorem 1.4.30
Corollary 1.4.31
Images and Inverse Images
Definition 1.4.32
Example 1.4.33
Operations
Definition 1.4.34
Definition 1.4.35
Example 1.4.36
Example 1.4.37
Definition 1.4.38
Theorem 1.4.39
Proof
Example 1.4.40
Definition 1.4.41
Definition 1.4.42
Example 1.4.43
Exercises
CHAPTER 2 Euclidean Space
2.1 Vectors
Definition 2.1.1
Example 2.1.2
Vector Operations
Definition 2.1.3
Definition 2.1.4
Theorem 2.1.5
Proof
Example 2.1.6
Theorem 2.1.7
Proof
Corollary 2.1.8 [Cancellation]
Distance and Length
Definition 2.1.9
Example 2.1.10
Theorem 2.1.11 [Cauchy‐Schwartz Inequality]
Proof
Theorem 2.1.12
Proof
Definition 2.1.13
Theorem 2.1.14
Theorem 2.1.15
Proof
Definition 2.1.16
Theorem 2.1.17
Proof
Lines and Planes
Definition 2.1.18
Definition 2.1.19
Definition 2.1.20
Definition 2.1.21
Example 2.1.22
Example 2.1.23
Example 2.1.24
Definition 2.1.25
Example 2.1.26
Example 2.1.27
Example 2.1.28
Exercises
2.2 Dot Product
Definition 2.2.1
Definition 2.2.2
Example 2.2.3
Example 2.2.4
Example 2.2.5
Theorem 2.2.6
Proof
Definition 2.2.7
Theorem 2.2.8
Proof
Lines and Planes
Definition 2.2.9
Theorem 2.2.10
Example 2.2.11
Example 2.2.12
Theorem 2.2.13
Proof
Example 2.2.14
Example 2.2.15
Definition 2.2.16
Orthogonal Projection
Theorem 2.2.17
Proof
Definition 2.2.18
Example 2.2.19
Theorem 2.2.20
Proof
Example 2.2.21
Exercises
2.3 Cross Product
Definition 2.3.1
Theorem 2.3.2
Example 2.3.3
Properties
Theorem 2.3.4
Proof
Theorem 2.3.5 [Lagrange’s Identity]
Proof
Areas and Volumes
Theorem 2.3.6
Proof
Corollary 2.3.7
Example 2.3.8
Example 2.3.9
Exercises
CHAPTER 3 Transformations and Matrices. 3.1 Linear Transformations
Definition 3.1.1
Example 3.1.2
Example 3.1.3
Example 3.1.4
Example 3.1.5
Properties
Theorem 3.1.6
Proof
Theorem 3.1.7
Proof
Theorem 3.1.8
Example 3.1.9
Matrices
Definition 3.1.10
Definition 3.1.11
Example 3.1.12
Definition 3.1.13
Example 3.1.14
Definition 3.1.15 [Matrix Multiplication ‐ Preliminary Form]
Definition 3.1.16
Theorem 3.1.17
Proof
Theorem 3.1.18
Proof
Theorem 3.1.19
Proof
Definition 3.1.20
Example 3.1.21
Theorem 3.1.22
Proof
Exercises
3.2 Matrix Algebra
Addition, Subtraction, and Scalar Multiplication
Definition 3.2.1
Example 3.2.2
Theorem 3.2.3
Proof
Example 3.2.4
Definition 3.2.5
Theorem 3.2.6
Proof
Properties
Definition 3.2.7
Definition 3.2.8
Theorem 3.2.9
Proof
Theorem 3.2.10
Proof
Example 3.2.11
Multiplication
Theorem 3.2.12
Proof
Definition 3.2.13 [Matrix Multiplication]
Theorem 3.2.14
Proof
Example 3.2.15
Example 3.2.16
Theorem 3.2.17
Proof
Theorem 3.2.18
Proof
Identity Matrix
Definition 3.2.19
Lemma 3.2.20
Proof
Theorem 3.2.21
Proof
Corollary 3.2.22
Example 3.2.23
Distributive Law
Theorem 3.2.24 [Distributive Law]
Proof
Matrices and Polynomials
Definition 3.2.25
Theorem 3.2.26
Proof
Exercises
3.3 Linear Operators
Reflections
Definition 3.3.1
Example 3.3.2
Example 3.3.3
Theorem 3.3.4
Proof
Example 3.3.5
Definition 3.3.6
Example 3.3.7
Rotations
Definition 3.3.8
Example 3.3.9
Isometries
Definition 3.3.10
Theorem 3.3.11
Proof
Theorem 3.3.12
Proof
Theorem 3.3.13
Theorem 3.3.14
Proof
Theorem 3.3.15
Definition 3.3.16
Theorem 3.3.17
Proof
Contractions, Dilations, and Shears
Definition 3.3.18
Theorem 3.3.19
Proof
Definition 3.3.20
Exercises
3.4 Injections and Surjections
Kernel
Definition 3.4.1
Theorem 3.4.2
Proof
Theorem 3.4.3
Proof
Example 3.4.4
Range
Theorem 3.4.5
Example 3.4.6
Example 3.4.7
Example 3.4.8
Exercises
3.5 Gauss‐Jordan Elimination
Definition 3.5.1
Example 3.5.2
Definition 3.5.3
Elementary Row Operations
Definition 3.5.4
Example 3.5.5
Definition 3.5.6
Theorem 3.5.7
Definition 3.5.8
Example 3.5.9
Square Matrices
Definition 3.5.10
Example 3.5.11
Example 3.5.12
Example 3.5.13
Example 3.5.14
Example 3.5.15
Theorem 3.5.16
Proof
Nonsquare Matrices
Example 3.5.17
Example 3.5.18
Theorem 3.5.19
Proof
Example 3.5.20
Theorem 3.5.21
Proof
Gaussian Elimination
Definition 3.5.22
Example 3.5.23
Example 3.5.24
Exercises
CHAPTER 4 Invertibility. 4.1 Invertible Matrices
Theorem 4.1.1
Proof
Definition 4.1.2
Theorem 4.1.3
Example 4.1.4
Example 4.1.5
Example 4.1.6
Elementary Matrices
Definition 4.1.7
Example 4.1.8
Theorem 4.1.9
Proof
Corollary 4.1.10
Example 4.1.11
Example 4.1.12
Theorem 4.1.13
Finding the Inverse of a Matrix
Theorem 4.1.14
Proof
Theorem 4.1.15
Proof
Corollary 4.1.16
Example 4.1.17
Systems of Linear Equations
Theorem 4.1.18
Example 4.1.19
Exercises
4.2 Determinants
Definition 4.2.1
Definition 4.2.2
Example 4.2.3
Example 4.2.4
Example 4.2.5
Definition 4.2.6
Theorem 4.2.7
Proof
Example 4.2.8
Theorem 4.2.9
Proof
Multiplying a Row by a Scalar
Example 4.2.10
Adding a Multiple of a Row to Another Row
Lemma 4.2.11
Proof
Definition 4.2.12
Definition 4.2.13
Theorem 4.2.14
Proof
Example 4.2.15
Switching Rows
Theorem 4.2.16
Corollary 4.2.17
Proof
Corollary 4.2.18
Proof
Corollary 4.2.19
Proof
Example 4.2.20
Exercises
4.3 Inverses and Determinants
Uniqueness of the Determinant
Definition 4.3.1
Theorem 4.3.2
Proof
Theorem 4.3.3
Lemma 4.3.4
Proof
Theorem 4.3.5
Proof
Theorem 4.3.6
Proof
Corollary 4.3.7
Example 4.3.8
Equivalents to Invertibility
Theorem 4.3.9
Proof
Example 4.3.10
Theorem 4.3.11
Proof
Example 4.3.12
Theorem 4.3.13
Proof
Example 4.3.14
Theorem 4.3.15
Proof
Products
Theorem 4.3.16
Proof
Theorem 4.3.17
Proof
Lemma 4.3.18
Proof
Theorem 4.3.19
Proof
Theorem 4.3.20
Proof
Exercises
4.4 Applications
Definition 4.4.1
Definition 4.4.2
The Classical Adjoint
Definition 4.4.3
Definition 4.4.4
Theorem 4.4.5
Proof
Symmetric and Orthogonal Matrices
Definition 4.4.6
Example 4.4.7
Theorem 4.4.8
Proof
Corollary 4.4.9
Example 4.4.10
Theorem 4.4.11
Proof
Theorem 4.4.12
Definition 4.4.13
Theorem 4.4.14
Theorem 4.4.15
Proof
Theorem 4.4.16
Example 4.4.17
Theorem 4.4.18
Proof
Corollary 4.4.19
Cramer’s Rule
Theorem 4.4.20 [Cramer’s Rule]
Proof
Example 4.4.21
LU Factorization
Theorem 4.4.22 [LU Factorization]
Example 4.4.23
Example 4.4.24
Area and Volume
Example 4.4.25
Theorem 4.4.26
Theorem 4.4.27
Exercises
CHAPTER 5 Abstract Vectors
5.1 Vector Spaces
Definition 5.1.1
Theorem 5.1.2
Proof
Examples of Vector Spaces
Example 5.1.3
Example 5.1.4
Example 5.1.5
Definition 5.1.6
Example 5.1.7
Example 5.1.8
Example 5.1.9
Linear Transformations
Definition 5.1.10
Example 5.1.11
Example 5.1.12
Example 5.1.13
Theorem 5.1.14
Example 5.1.15
Example 5.1.16
Definition 5.1.17
Theorem 5.1.18
Exercises
5.2 Subspaces
Definition 5.2.1
Examples of Subspaces
Example 5.2.2
Example 5.2.3
Example 5.2.4
Properties
Theorem 5.2.5
Proof
Example 5.2.6
Example 5.2.7
Theorem 5.2.8
Definition 5.2.9
Theorem 5.2.10
Proof
Example 5.2.11
Spanning Sets
Definition 5.2.12
Definition 5.2.13
Example 5.2.14
Theorem 5.2.15
Proof
Example 5.2.16
Example 5.2.17
Kernel and Range
Definition 5.2.18
Theorem 5.2.19
Theorem 5.2.20
Example 5.2.21
Example 5.2.22
Definition 5.2.23
Theorem 5.2.24
Proof
Example 5.2.25
Exercises
5.3 Linear Independence
Definition 5.3.1
Theorem 5.3.2
Proof
Euclidean Examples
Example 5.3.3
Example 5.3.4
Example 5.3.5
Theorem 5.3.6
Proof
Abstract Vector Space Examples
Example 5.3.7
Example 5.3.8
Example 5.3.9
Theorem 5.3.10
Proof
Exercises
5.4 Basis and Dimension
Basis
Definition 5.4.1
Example 5.4.2
Example 5.4.3
Example 5.4.4
Example 5.4.5
Lemma 5.4.6
Proof
Theorem 5.4.7
Proof
Zorn’s Lemma
Definition 5.4.8
Example 5.4.9
Example 5.4.10
Definition 5.4.11
Example 5.4.12
Example 5.4.13
Axiom 5.4.14 [Zorn’s Lemma]
Theorem 5.4.15
Proof
Dimension
Definition 5.4.16
Example 5.4.17
Example 5.4.18
Theorem 5.4.19
Proof
Example 5.4.20
Expansions and Reductions
Theorem 5.4.21
Proof
Theorem 5.4.22
Proof
Example 5.4.23
Theorem 5.4.24
Proof
Corollary 5.4.25
Example 5.4.26
Theorem 5.4.27
Proof
Exercises
5.5 Rank and Nullity
Definition 5.5.1
Rank‐Nullity Theorem
Theorem 5.5.2 [Rank‐Nullity Theorem]
Proof
Example 5.5.3
Corollary 5.5.4
Proof
Example 5.5.5
Example 5.5.6
Fundamental Subspaces
Definition 5.5.7
Theorem 5.5.8
Proof
Rank and Nullity of a Matrix
Definition 5.5.9
Theorem 5.5.10
Definition 5.5.11
Example 5.5.12
Theorem 5.5.13 [Rank‐Nullity Theorem for Matrices]
Proof
Theorem 5.5.14
Corollary 5.5.15
Exercises
5.6 Isomorphism
Definition 5.6.1
Definition 5.6.2
Example 5.6.3
Example 5.6.4
Lemma 5.6.5
Proof
Theorem 5.6.6
Proof
Lemma 5.6.7
Proof
Theorem 5.6.8
Proof
Corollary 5.6.9
Example 5.6.10
Example 5.6.11
Coordinates
Theorem 5.6.12
Proof
Definition 5.6.13
Example 5.6.14
Example 5.6.15
Definition 5.6.16
Example 5.6.17
Theorem 5.6.18
Proof
Corollary 5.6.19
Corollary 5.6.20
Proof
Change of Basis
Theorem 5.6.21
Proof
Theorem 5.6.22
Example 5.6.23
Example 5.6.24
Matrix of a Linear Transformation
Theorem 5.6.25
Proof
Example 5.6.26
Example 5.6.27
Definition 5.6.28
Example 5.6.29
Example 5.6.30
Example 5.6.31
Exercises
CHAPTER 6 Inner Product Spaces. 6.1 Inner Products
Definition 6.1.1
Theorem 6.1.2
Example 6.1.3
Example 6.1.4
Example 6.1.5
Example 6.1.6
Theorem 6.1.7 [Cauchy‐Schwartz Inequality]
Norms
Definition 6.1.8
Definition 6.1.9
Metrics
Definition 6.1.10
Definition 6.1.11
Theorem 6.1.12
Example 6.1.13
Example 6.1.14
Angles
Definition 6.1.15
Example 6.1.16
Definition 6.1.17
Example 6.1.18
Theorem 6.1.19 [Pythagoras]
Proof
Orthogonal Projection
Definition 6.1.20
Theorem 6.1.21
Proof
Example 6.1.22
Example 6.1.23
Exercises
6.2 Orthonormal Bases
Definition 6.2.1
Example 6.2.2
Example 6.2.3
Definition 6.2.4
Example 6.2.5
Theorem 6.2.6
Proof
Corollary 6.2.7
Orthogonal Complement
Definition 6.2.8
Theorem 6.2.9
Proof
Example 6.2.10
Theorem 6.2.11
Example 6.2.12
Example 6.2.13
Direct Sum
Definition 6.2.14
Example 6.2.15
Theorem 6.2.16
Proof
Example 6.2.17
Theorem 6.2.18
Proof
Gram‐Schmidt Process
Definition 6.2.19
Theorem 6.2.20
Proof
Theorem 6.2.21 [Gram‐Schmidt]
Corollary 6.2.22
Corollary 6.2.23
Example 6.2.24
Theorem 6.2.25
Proof
Example 6.2.26
QR Factorization
Theorem 6.2.27 [QR Factorization]
Example 6.2.28
Exercises
CHAPTER 7 Matrix Theory. 7.1 Eigenvectors and Eigenvalues
Definition 7.1.1
Example 7.1.2
Eigenspaces
Definition 7.1.3
Theorem 7.1.4
Proof
Theorem 7.1.5
Example 7.1.6
Characteristic Polynomial
Definition 7.1.7
Example 7.1.8
Example 7.1.9
Theorem 7.1.10
Proof
Example 7.1.11
Example 7.1.12
Theorem 7.1.13
Definition 7.1.14
Example 7.1.15
Cayley–Hamilton Theorem
Theorem [Cayley–Hamilton] 7.1.16
Proof
Exercises
7.2 Minimal Polynomial
Example 7.2.1
Example 7.2.2
Theorem 7.2.3
Proof
Corollary 7.2.4
Theorem 7.2.5
Proof
Invariant Subspaces
Definition 7.2.6
Theorem 7.2.7
Proof
Corollary 7.2.8
Example 7.2.9
Example 7.2.10
Generalized Eigenvectors
Definition 7.2.11
Theorem 7.2.12
Proof
Corollary 7.2.13
Corollary 7.2.14
Primary Decomposition Theorem
Definition 7.2.15
Lemma 7.2.16
Proof
Example 7.2.17
Lemma 7.2.18 [Bézout’s Identity]
Proof
Theorem 7.2.19 [Primary Decomposition Theorem]
Proof
Lemma 7.2.20
Theorem 7.2.21
Example 7.2.22
Example 7.2.23
Exercises
7.3 Similar Matrices
Definition 7.3.1
Theorem 7.3.2
Proof
Theorem 7.3.3
Proof
Corollary 7.3.4
Proof
Schur’s Lemma
Lemma 7.3.5 [Schur]
Proof
Example 7.3.6
Block Diagonal Form
Lemma 7.3.7
Proof
Theorem 7.3.8
Proof
Example 7.3.9
Nilpotent Matrices
Definition 7.3.10
Definition 7.3.11
Lemma 7.3.12
Proof
Example 7.3.13
Jordan Canonical Form
Definition 7.3.14
Definition 7.3.15
Lemma 7.3.16
Proof
Example 7.3.17
Example 7.3.18
Theorem 7.3.19
Proof
Corollary 7.3.20
Example 7.3.21
Exercises
7.4 Diagonalization
Definition 7.4.1
Theorem 7.4.2
Proof
Example 7.4.3
Theorem 7.4.4
Proof
Example 7.4.5
Theorem 7.4.6
Proof
Example 7.4.7
Orthogonal Diagonalization
Theorem 7.4.8
Proof
Definition 7.4.9
Theorem 7.4.10
Proof
Theorem 7.4.11
Proof
Simultaneous Diagonalization
Lemma 7.4.12
Proof
Theorem 7.4.13
Proof
Example 7.4.14
Quadratic Forms
Definition 7.4.15
Theorem 7.4.16 [Principal Axis Theorem]
Proof
Example 7.4.17
Example 7.4.18
Exercises
Further Reading
Index
WILEY END USER LICENSE AGREEMENT
Отрывок из книги
Michael L. O’Leary
In addition to the focus on proofs, linear transformations play a central role. For this reason, functions are introduced early, and once the important sets of ℝn are defined in the second chapter, linear transformations are described in the third chapter and motivate the introduction of matrices and their operations. From there, invertible linear transformations and invertible matrices are encountered in the fourth chapter followed by a complete generalization of all previous topics in the fifth with the definition of abstract vector spaces. Geometries are added to the abstractions in the sixth chapter, and the book concludes with nice matrix representations. Therefore, the book’s structure is as follows.
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As with any textbook, where the course is taught influences how the book is used. Many universities and colleges have an introduction to proof course. Because such courses serve as a prerequisite for any proof‐intensive mathematics course, the first chapter of this book can be passed over at these institutions and used only as a reference. If there is no such prerequisite, the first chapter serves as a detailed introduction to proof‐writing that is short enough not to infringe too much on the time spent on purely linear algebra topics. Wherever the book finds itself, the course outline can easily be adjusted with any excluded topics serving as bonus reading for the eager student.
Now for some technical comments. Theorems, definitions, and examples are numbered sequentially as a group in the now common chapter.section.number format. Although some proofs find their way into the text, most start with Proof, end with ■, and are indented. Examples, on the other hand, are simply indented. Some equations are numbered as (chapter.number) and are referred to simply using (chapter.number). Most if not all of the mathematical notation should be clear. Itwas decided to represent vectors as columns. This leads to some interesting type‐setting, but the clarity and consistency probably more than makes up for any formatting issues. Vectors are boldface, such as u and v, and scalars are not. Most sums are written like u1 + u2 + ⋯ + uk. There is a similar notation for products. However, there are times when summation and product notation must be used. Therefore, if u1, u2, …, uk are vectors,
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