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by Steve Leon, Lisette de Pillis
Linear Algebra with Applications, 10th Edition
Linear Algebra with Applications
Dedication
Contents
Preface
What’s New in the Tenth Edition?
Overview of Text
Suggested Course Outlines
Computer Exercises
Chapter 1 Matrices and Systems of Equations
1.1 Systems of Linear Equations
2×2 Systems
Equivalent Systems
n×n Systems
Section 1.1 Exercises
1.2 Row Echelon Form
Overdetermined Systems
Underdetermined Systems
Reduced Row Echelon Form
Homogeneous Systems
Section 1.2 Exercises
1.3 Matrix Arithmetic
Matrix Notation
Vectors
Equality
Scalar Multiplication
Matrix Addition
Matrix Multiplication and Linear Systems
Matrix Multiplication
Notational Rules
The Transpose of a Matrix
Section 1.3 Exercises
1.4 Matrix Algebra
Algebraic Rules
Notation
The Identity Matrix
Matrix Inversion
Note
Algebraic Rules for Transposes
Symmetric Matrices and Networks
Section 1.4 Exercises
1.5 Elementary Matrices
Equivalent Systems
Elementary Matrices
Diagonal and Triangular Matrices
Triangular Factorization
Section 1.5 Exercises
1.6 Partitioned Matrices
Block Multiplication
Outer Product Expansions
Section 1.6 Exercises
Chapter 1 Exercises
MATLAB Exercises
Chapter Test A True or False
Chapter Test B
Chapter 2 Determinants
2.1 The Determinant of a Matrix
Section 2.1 Exercises
2.2 Properties of Determinants
Row Operation I
Row Operation II
Row Operation III
Note
Main Results
Section 2.2 Exercises
2.3 Additional Topics and Applications
The Adjoint of a Matrix
Cramer’s Rule
Reference
The Cross Product
Section 2.3 Exercises
Chapter 2 Exercises
MATLAB Exercises
Chapter Test A True or False
Chapter Test B
Chapter 3 Vector Spaces
3.1 Definition and Examples
Euclidean Vector Spaces
The Vector Space ℝm×n
Vector Space Axioms
The Vector Space C[a,b]
The Vector Space Pn
Additional Properties of Vector Spaces
Section 3.1 Exercises
3.2 Subspaces
The Null Space of a Matrix
The Span of a Set of Vectors
Spanning Set for a Vector Space
Linear Systems Revisited
Section 3.2 Exercises
3.3 Linear Independence
Geometric Interpretation
Theorems and Examples
Vector Spaces of Functions
The Vector Space Pn
The Vector Space C(n−1)[a,b]
Section 3.3 Exercises
3.4 Basis and Dimension
Standard Bases
Section 3.4 Exercises
3.5 Change of Basis
Changing Coordinates in ℝ2
Changing Coordinates
Change of Basis for a General Vector Space
Section 3.5 Exercises
3.6 Row Space and Column Space
Linear Systems
The Column Space
Note
Section 3.6 Exercises
Chapter 3 Exercises
MATLAB Exercises
Chapter Test A True or False
Chapter Test B
Chapter 4 Linear Transformations
4.1 Definition and Examples
Notation
Linear Operators on ℝ2
Linear Transformations from ℝn to ℝm
Linear Transformations from V to W
The Image and Kernel
Section 4.1 Exercises
4.2 Matrix Representations of Linear Transformations
Section 4.2 Exercises
4.3 Similarity
Section 4.3 Exercises
Chapter 4 Exercises
MATLAB Exercises
Chapter Test A True or False
Chapter Test B
Chapter 5 Orthogonality
5.1 The Scalar Product in ℝn
The Scalar Product in ℝ2 and ℝ3
Scalar and Vector Projections
Notation
Orthogonality in ℝn
Section 5.1Exercises
5.2 Orthogonal Subspaces
Note
Remarks
Fundamental Subspaces
Section 5.2 Exercises
5.3 Least Squares Problems
Least Squares Solutions of Overdetermined Systems
Section 5.3 Exercises
5.4 Inner Product Spaces
Definition and Examples
The Vector Space ℝn
The Vector Space ℝm×n
The Vector Space C[a, b]
The Vector Space Pn
Basic Properties of Inner Product Spaces
Observations
Norms
Section 5.4 Exercises
5.5 Orthonormal Sets
Orthogonal Matrices
Permutation Matrices
Orthonormal Sets and Least Squares
Approximation of Functions
Approximation by Trigonometric Polynomials
Section 5.5 Exercises
5.6 The Gram–Schmidt Orthogonalization ProcessThe Gram–Schmidt Orthogonalization Process
The Modified Gram–Schmidt Process
Section 5.6 Exercises
5.7 Orthogonal Polynomials
Orthogonal Sequences
Classical Orthogonal Polynomials
Legendre Polynomials
Chebyshev Polynomials
Jacobi Polynomials
Hermite Polynomials
Laguerre Polynomials
Section 5.7 Exercises
Chapter 5 Exercises
MATLAB Exercises
Chapter Test A True or False
Chapter Test B
Chapter 6 Eigenvalues
6.1 Eigenvalues and Eigenvectors
Geometric Visualization of Eigenvalues and Eigenvectors
Finding Eigenvalues and Eigenvectors
Complex Eigenvalues
The Product and Sum of the Eigenvalues
Similar Matrices
Section 6.1 Exercises
6.2 Systems of Linear Differential Equations
Complex Eigenvalues
Higher-Order Systems
Section 6.2 Exercises
6.3 Diagonalization
Remarks
The Exponential of a Matrix
Section 6.3 Exercises
6.4 Hermitian Matrices
Complex Inner Products
Hermitian Matrices
The Real Schur Decomposition
Normal Matrices
Section 6.4 Exercises
6.5 The Singular Value Decomposition
Observations
Visualizing the SVD
Numerical Rank and Lower Rank Approximations
Section 6.5 Exercises
6.6 Quadratic Forms
Conic Sections
Optimization: An Application to the Calculus
Section 6.6 Exercises
6.7 Positive Definite Matrices
Section 6.7 Exercises
6.8 Nonnegative Matrices
Section 6.8 Exercises
Chapter 6 Exercises
MATLAB Exercises
Critical Loads for a Beam
Diagonalizable and Defective Matrices
Application: Sex-Linked Genes
Similarity
Hermitian Matrices
Optimization
Positive Definite Matrices
Chapter Test A True or False
Chapter Test B
Chapter 7 Numerical Linear Algebra
7.1 Floating-Point Numbers
The IEEE Standard 754 Floating-Point Representation
Loss of Accuracy and Instability
Section 7.1 Exercises
7.2 Gaussian Elimination
Gaussian Elimination without Interchanges
Using the Triangular Factorization to Solve Ax=b
Operation Count
Section 7.2 Exercises
7.3 Pivoting Strategies
Gaussian Elimination with Interchanges
Remarks
Partial Pivoting
Section 7.3 Exercises
7.4 Matrix Norms and Condition Numbers
Matrix Norms
Subordinate Matrix Norms
Condition Numbers
Section 7.4 Exercises
7.5 Orthogonal Transformations
Elementary Orthogonal Transformations
Householder Transformations
Remarks
Operation Count
Rotations and Reflections
Operation Count
The QR Factorization for Solving General Linear Systems
Section 7.5 Exercises
7.6 The Eigenvalue Problem
The Power Method
Deflation
Reduction to Hessenberg Form
QR Algorithm
Remarks
Section 7.6 Exercises
7.7 Least Squares Problems
Normal Equations
Modified Gram–Schmidt Method for Solving Least Squares Problems
The Householder QR Factorization
The Pseudoinverse
Bidiagonalization
The Golub–Reinsch Algorithm
Section 7.7 Exercises
7.8 Iterative Methods
Matrix Splittings
Jacobi Iteration
Gauss–Seidel Iteration
Section 7.8 Exercises
Chapter 7 Exercises
MATLAB Exercises
Sensitivity of Linear Systems
Sensitivity of Eigenvalues
Householder Transformations
Rotations and Reflections
Singular Value Decomposition
Gerschgorin Circles
Distribution of Condition Numbers and Eigenvalues of Random Matrices
Chapter Test A True or False
Chapter Test B
Chapter 8 Canonical Forms
8.1 Nilpotent Operators
Section 8.1 Exercises
8.2 The Jordan Canonical Form
Section 8.2 Exercises
Appendix: Matlab
The MATLAB Desktop Display
Basic Data Elements
Submatrices
Generating Matrices
Matrix Arithmetic
Addition and Multiplication of Matrices
Backslash or Matrix Left Division
Exponentiation
MATLAB Functions
Programming Features
M-files
Script Files
Function Files
The MATLAB Path
Relational and Logical Operators
Columnwise Array Operators
Graphics
Symbolic Toolbox
Help Facility
Conclusions
Bibliography
A Linear Algebra and Matrix Theory
B Applied and Numerical Linear Algebra
C Books of Related Interest
Answers to Selected Exercises
Chapter 1
1.1
1.2
1.3
1.4
1.5
1.6
Chapter Test A
Chapter Test B
Chapter 2
2.1
2.2
2.3
Chapter Test A
Chapter Test B
Chapter 3
3.1
3.2
3.3
3.4
3.5
3.6
Chapter Test A
Chapter Test B
Chapter 4
4.1
4.2
4.3
Chapter Test A
Chapter Test B
Chapter 5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
MATLAB Exercises
Chapter Test A
Chapter Test B
Chapter 6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
MATLAB Exercises
Chapter Test A
Chapter Test B
Chapter 7
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
MATLAB Exercises
Chapter Test A
Chapter Test B
Chapter 8
8.1
8.2
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Y
Z
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