Оглавление
Richard C. Penney. Linear Algebra
Table of Contents
List of Tables
List of Illustrations
Guide
Pages
LINEAR ALGEBRA. Ideas and Applications
PREFACE
FEATURES OF THE TEXT
ACKNOWLEDGMENTS
ABOUT THE COMPANION WEBSITE
CHAPTER 1 SYSTEMS OF LINEAR EQUATIONS
1.1 THE VECTOR SPACE OF m × n MATRICES
The Space
Example 1.1
Linear Combinations and Linear Dependence
Example 1.2
Example 1.3
What Is a Vector Space?
Example 1.4
Example 1.5
Example 1.6
Why Prove Anything?
True‐False Questions: Justify your answers
EXERCISES
1.1.1 Computer Projects/Exercises/Exercises
EXERCISES
1.1.2 Applications to Graph Theory I
Self‐Study Questions
EXERCISES
1.2 SYSTEMS
Rank: The Maximum Number of Linearly Independent Equations
True‐False Questions: Justify your answers
EXERCISES
1.2.1 Computer Projects/Exercises. The Translation Theorem. EXERCISES
1.2.2 Applications to Circuit Theory
Example 1.7
Example 1.8
Self‐Study Questions
EXERCISES
1.3 GAUSSIAN ELIMINATION
Example 1.9
Example 1.10
Example 1.11
Example 1.12
Spanning in Polynomial Spaces. Example 1.13
Example 1.14
Computational Issues: Pivoting
True‐False Questions: Justify your answers
EXERCISES
1.3.1 Using tolerances in MATLAB's rref and rank. Using Tolerances in rref and Rank
EXERCISES
1.3.2 Applications to Traffic Flow
Self‐Study Questions
EXERCISES
1.4 COLUMN SPACE AND NULLSPACE
Example 1.15
Example 1.16
Subspaces
Example 1.17
Example 1.18
True‐False Questions: Justify your answers
EXERCISES
1.4.1 Computer Projects/Exercises
EXERCISES
CHAPTER SUMMARY
Notes
CHAPTER 2 LINEAR INDEPENDENCE AND DIMENSION
2.1 THE TEST FOR LINEAR INDEPENDENCE
EXAMPLE 2.1
EXAMPLE 2.2
EXAMPLE 2.3
Bases for the Column Space
EXAMPLE 2.4
Testing Functions for Independence
True‐False Questions: Justify your answers
EXERCISES
2.1.1 Computer Projects/Exercises. Changing Pivot Columns. EXERCISES
2.2 DIMENSION
EXAMPLE 2.5
EXAMPLE 2.6
Example 2.7
Example 2.8
Example 2.9
Example 2.10
EXAMPLE 2.11
True‐False Questions: Justify your answers
EXERCISES
2.2.1 Computer Projects/Exercises
EXERCISES
2.2.2 Applications to Differential Equations
EXAMPLE 2.12
EXAMPLE 2.13
Self‐Study Questions
EXERCISES
2.3 ROW SPACE AND THE RANK‐NULLITY THEOREM
Example 2.14
Bases for the Row Space
EXAMPLE 2.15
EXAMPLE 2.16
Example 2.17
Example 2.18
Computational Issues: Computing Rank
True‐False Questions: Justify your answers
EXERCISES
2.3.1 Computer Projects/Exercises. Random Matrices of a Given Rank. EXERCISES
CHAPTER SUMMARY
Notes
CHAPTER 3 LINEAR TRANSFORMATIONS
3.1 THE LINEARITY PROPERTIES
EXAMPLE 3.1
Example 3.2
Example 3.3
EXERCISES
3.1.1 Computer Projects/Exercises
EXERCISES
3.2 MATRIX MULTIPLICATION (COMPOSITION)
Example 3.4
EXAMPLE 3.5
Partitioned Matrices
EXAMPLE 3.6
Computational Issues: Parallel Computing
True‐False Questions: Justify your answers
EXERCISES
3.2.1 Computer Projects/Exercises. 3‐D Computer Graphics. EXERCISES
3.2.2 Applications to Graph Theory II
Self‐Study Questions
EXERCISES
3.2.3 Computer Projects/Exercises. Google's Page Rank Algorithm
EXAMPLE 3.7
EXERCISES
3.3 INVERSES
EXAMPLE 3.8
EXAMPLE 3.9
Computational Issues: Reduction versus Inverses
True‐False Questions: Justify your answers
EXERCISES
3.3.1 Computer Projects/Exercises. Ill‐Conditioned Systems
EXERCISES
3.3.2 Applications to Economics: The Leontief Open Model
EXAMPLE 3.10
EXAMPLE 3.11
Self‐Study Questions
EXERCISES
3.4 The LU Factorization
EXAMPLE 3.12
Example 3.13
Example 3.14
EXERCISES
3.4.1 Computer Projects/Exercises. Row Exchanges in theFactorization. EXERCISES
3.5 THE MATRIX OF A LINEAR TRANSFORMATION. Coordinates
EXAMPLE 3.15
Example 3.16
EXAMPLE 3.17
Example 3.18
Example 3.19
Example 3.20
Application to Differential Equations. EXAMPLE 3.21
EXAMPLE 3.22
EXAMPLE 3.23
Isomorphism
Invertible Linear Transformations
True‐False Questions: Justify your answers
EXERCISES
3.5.1 Computer Projects/Exercises. Graphing in Skewed‐Coordinates
EXERCISES
3.5.2 Computer Projects/Exercises. Pricing Long Term Health Care Insurance
EXAMPLE 3.24
EXAMPLE 3.25
EXERCISES
CHAPTER SUMMARY
Note
CHAPTER 4 DETERMINANTS
4.1 DEFINITION OF THE DETERMINANT
EXAMPLE 4.1
Example 4.2
Example 4.3
Example 4.4
Example 4.5
Example 4.6
Example 4.7
4.1.1 The Rest of the Proofs
EXERCISES
4.1.2 Computer Projects/Exercises
4.2 REDUCTION AND DETERMINANTS
Example 4.8
Example 4.9
Uniqueness of the Determinant
EXERCISES
4.2.1 Volume
EXERCISES
4.3 A FORMULA FOR INVERSES
EXAMPLE 4.10
Example 4.11
Example 4.12
EXERCISES
CHAPTER SUMMARY
CHAPTER 5 EIGENVECTORS AND EIGENVALUES
5.1 EIGENVECTORS
EXAMPLE 5.1
EXAMPLE 5.2
EXAMPLE 5.3
EXAMPLE 5.4
EXERCISES
5.1.1 Computer Projects/Exercises. Computing Roots of Polynomials
EXERCISES
5.1.2 Application to Markov Chains
EXAMPLE 5.5
EXERCISES
5.2 DIAGONALIZATION
Powers of Matrices. EXAMPLE 5.6
True‐False Questions: Justify your answers
EXERCISES
5.2.1 Application to Systems of Differential Equations
EXAMPLE 5.7
EXAMPLE 5.8
Self‐Study Questions
EXERCISES
5.3 COMPLEX EIGENVECTORS
EXAMPLE 5.9
EXAMPLE 5.10
Complex Vector Spaces
EXERCISES
5.3.1 Computer Projects/Exercises. Complex Eigenvalues. EXERCISES
CHAPTER SUMMARY
CHAPTER 6 ORTHOGONALITY
6.1 THE SCALAR PRODUCT IN
Orthogonal/Orthonormal Bases and Coordinates
EXERCISES
6.2 PROJECTIONS: THE GRAM–SCHMIDT PROCESS
The QR Decomposition
Uniqueness of the Factorization
EXERCISES
6.2.1 Computer Projects/Exercises. The Least Squares Solution
EXERCISES
6.3 FOURIER SERIES: SCALAR PRODUCT SPACES
EXERCISES
6.3.1 Computer Projects/Exercises. Plotting Fourier Series
EXERCISES
6.4 ORTHOGONAL MATRICES
Householder Matrices
EXERCISES
6.4.1 Computer Projects/Exercises
EXERCISES
6.5 LEAST SQUARES
EXERCISES
6.5.1 Computer Projects/Exercises. Finding the Orbit of an Asteroid. EXERCISES
6.6 QUADRATIC FORMS: ORTHOGONAL DIAGONALIZATION
The Spectral Theorem
The Principal Axis Theorem
True‐False Questions: Justify your answers
EXERCISES
6.6.1 Computer Projects/Exercises. The Principal Axis Theorem
EXERCISES
6.7 THE SINGULAR VALUE DECOMPOSITION (SVD)
Application of the SVD to Least‐Squares Problems
EXERCISES
Computing the SVD Using Householder Matrices
Diagonalizing Matrices Using Householder Matrices
6.8 HERMITIAN SYMMETRIC AND UNITARY MATRICES
EXERCISES
CHAPTER SUMMARY
Note
CHAPTER 7 GENERALIZED EIGENVECTORS
7.1 GENERALIZED EIGENVECTORS
EXAMPLE 7.1
EXAMPLE 7.2
EXERCISES
7.2 CHAIN BASES
Jordan Form
EXAMPLE 7.7
EXERCISES
The Cayley–Hamilton Theorem
CHAPTER SUMMARY
Note
CHAPTER 8 NUMERICAL TECHNIQUES
8.1 CONDITION NUMBER
Condition Number
Example 8.1
Least Squares
EXERCISES
8.2 COMPUTING EIGENVALUES
Iteration
Example 8.2
Example 8.3
The Method
Example 8.4
EXERCISES
CHAPTER SUMMARY
Notes
Index
ANSWERS AND HINTS. Section 1.1 on page 16
Section 1.2 on page 37
Section 1.2.2 on page 44
Section 1.3 on page 60
Section 1.3.2 on page 70
Section 1.4 on page 82
Section 2.1 on page 104
Section 2.2 on page 118
Section 2.2.2 on page 128
Section 2.3 page 140
Section 3.1 on page 155
Section 3.2 on page 171
Section 3.2.2 on page 179
Section 3.3 on page 192
Section 3.3.2 on page 204
Section 3.4 on page 213
Section 3.5 on page 231
Section 4.1 on page 256
Section 4.2 on page 266
Section 4.3 on page 275
Section 5.1 on page 288
Section 5.1.2 on page 294
Section 5.2 on page 299
Section 5.3 on page 312
Section 6.1 page 326
Section 6.2 on page 338
Section 6.3 on page 350
Section 6.4 on page 364
Section 6.5 on page 377
Section 6.6 on page 392
Section 6.7 on page 404
Section 6.8 on page 416
Section 7.1 on page 429
Section 7.2 on page 443
Section 8.1 on page 453
WILEY END USER LICENSE AGREEMENT
