Cover image for Linear Algebra with Applications, 10th Edition

Linear Algebra with Applications, 10th Edition

Author Steve Leon , Lisette de Pillis
Release Date: 2019/06/01
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Book Description

A thorough and accessible introduction to linear algebra, delivered digitally.

The new 10th Edition of Linear Algebra with Applications continues to encourage a challenging and broad understanding of the subject.

For this edition, Steve Leon – one of the leading figures in the use of technology for linear algebra – is joined by new co-author Lisette de Pillis of Harvey Mudd College, who brings her passion for teaching and solving real-world problems to this revision.

Key to the 10th Edition was transforming from a primarily print-based resource to a digital learning tool. The eText is packed with content and tools, such as interactive figures, that help bring course content to life for students and augment instruction. This change in format supports the authors’ hallmark – using modern practical application to make key concepts tangible and demonstrating how mathematics is used in the real world.

Each chapter contains integrated worked examples, practical applications, computer exercises, and chapter tests. The important roles played by geometry and visualization in understanding linear algebra are emphasized.

For sophomore-level or junior/senior-level first courses in linear algebra; assumes calculus as a prerequisite.

Pearson eText allows educators to easily share their own notes with students so they see the connection between their reading and what they learn in class – motivating them to keep reading, and keep learning. Portable access lets students study on the go, even offline. And, student usage analytics offer insight into how students use the eText, helping educators tailor their instruction.

NOTE: This ISBN is for the Pearson eText access card. For students purchasing this product from an online retailer, Pearson eText is a fully digital delivery of Pearson content and should only be purchased when required by your instructor. In addition to your purchase, you will need a course invite link, provided by your instructor, to register for and use Pearson eText.

Table of Contents

  1. Linear Algebra with Applications
  2. Dedication
  3. Contents
  4. Preface
    1. What’s New in the Tenth Edition?
    2. Overview of Text
    3. Suggested Course Outlines
    4. Computer Exercises
  5. Chapter 1 Matrices and Systems of Equations
    1. 1.1 Systems of Linear Equations
      1. 2×2 Systems
      2. Equivalent Systems
      3. n×n Systems
    2. Section 1.1 Exercises
    3. 1.2 Row Echelon Form
      1. Overdetermined Systems
      2. Underdetermined Systems
      3. Reduced Row Echelon Form
      4. Homogeneous Systems
    4. Section 1.2 Exercises
    5. 1.3 Matrix Arithmetic
      1. Matrix Notation
      2. Vectors
      3. Equality
      4. Scalar Multiplication
      5. Matrix Addition
      6. Matrix Multiplication and Linear Systems
      7. Matrix Multiplication
      8. Notational Rules
      9. The Transpose of a Matrix
    6. Section 1.3 Exercises
    7. 1.4 Matrix Algebra
      1. Algebraic Rules
        1. Notation
      2. The Identity Matrix
      3. Matrix Inversion
        1. Note
      4. Algebraic Rules for Transposes
      5. Symmetric Matrices and Networks
    8. Section 1.4 Exercises
    9. 1.5 Elementary Matrices
      1. Equivalent Systems
      2. Elementary Matrices
      3. Diagonal and Triangular Matrices
      4. Triangular Factorization
    10. Section 1.5 Exercises
    11. 1.6 Partitioned Matrices
      1. Block Multiplication
      2. Outer Product Expansions
    12. Section 1.6 Exercises
    13. Chapter 1 Exercises
      1. MATLAB Exercises
      2. Chapter Test A True or False
      3. Chapter Test B
  6. Chapter 2 Determinants
    1. 2.1 The Determinant of a Matrix
    2. Section 2.1 Exercises
    3. 2.2 Properties of Determinants
      1. Row Operation I
      2. Row Operation II
      3. Row Operation III
        1. Note
      4. Main Results
    4. Section 2.2 Exercises
    5. 2.3 Additional Topics and Applications
      1. The Adjoint of a Matrix
      2. Cramer’s Rule
      3. Reference
      4. The Cross Product
    6. Section 2.3 Exercises
    7. Chapter 2 Exercises
      1. MATLAB Exercises
      2. Chapter Test A True or False
      3. Chapter Test B
  7. Chapter 3 Vector Spaces
    1. 3.1 Definition and Examples
      1. Euclidean Vector Spaces
      2. The Vector Space ℝm×n
      3. Vector Space Axioms
      4. The Vector Space C[a,b]
      5. The Vector Space Pn
      6. Additional Properties of Vector Spaces
    2. Section 3.1 Exercises
    3. 3.2 Subspaces
      1. The Null Space of a Matrix
      2. The Span of a Set of Vectors
      3. Spanning Set for a Vector Space
      4. Linear Systems Revisited
    4. Section 3.2 Exercises
    5. 3.3 Linear Independence
      1. Geometric Interpretation
      2. Theorems and Examples
      3. Vector Spaces of Functions
        1. The Vector Space Pn
        2. The Vector Space C(n−1)[a,b]
    6. Section 3.3 Exercises
    7. 3.4 Basis and Dimension
      1. Standard Bases
    8. Section 3.4 Exercises
    9. 3.5 Change of Basis
      1. Changing Coordinates in ℝ2
      2. Changing Coordinates
      3. Change of Basis for a General Vector Space
    10. Section 3.5 Exercises
    11. 3.6 Row Space and Column Space
      1. Linear Systems
      2. The Column Space
        1. Note
    12. Section 3.6 Exercises
    13. Chapter 3 Exercises
      1. MATLAB Exercises
      2. Chapter Test A True or False
      3. Chapter Test B
  8. Chapter 4 Linear Transformations
    1. 4.1 Definition and Examples
      1. Notation
      2. Linear Operators on ℝ2
      3. Linear Transformations from ℝn to ℝm
      4. Linear Transformations from V to W
      5. The Image and Kernel
    2. Section 4.1 Exercises
    3. 4.2 Matrix Representations of Linear Transformations
    4. Section 4.2 Exercises
    5. 4.3 Similarity
    6. Section 4.3 Exercises
    7. Chapter 4 Exercises
      1. MATLAB Exercises
      2. Chapter Test A True or False
      3. Chapter Test B
  9. Chapter 5 Orthogonality
    1. 5.1 The Scalar Product in ℝn
      1. The Scalar Product in ℝ2 and ℝ3
        1. Scalar and Vector Projections
          1. Notation
        2. Orthogonality in ℝn
    2. Section 5.1Exercises
    3. 5.2 Orthogonal Subspaces
      1. Note
      2. Remarks
      3. Fundamental Subspaces
    4. Section 5.2 Exercises
    5. 5.3 Least Squares Problems
      1. Least Squares Solutions of Overdetermined Systems
    6. Section 5.3 Exercises
    7. 5.4 Inner Product Spaces
      1. Definition and Examples
        1. The Vector Space ℝn
        2. The Vector Space ℝm×n
        3. The Vector Space C[a, b]
        4. The Vector Space Pn
      2. Basic Properties of Inner Product Spaces
        1. Observations
        2. Norms
    8. Section 5.4 Exercises
    9. 5.5 Orthonormal Sets
      1. Orthogonal Matrices
      2. Permutation Matrices
      3. Orthonormal Sets and Least Squares
      4. Approximation of Functions
      5. Approximation by Trigonometric Polynomials
    10. Section 5.5 Exercises
    11. 5.6 The Gram–Schmidt Orthogonalization ProcessThe Gram–Schmidt Orthogonalization Process
      1. The Modified Gram–Schmidt Process
    12. Section 5.6 Exercises
    13. 5.7 Orthogonal Polynomials
      1. Orthogonal Sequences
      2. Classical Orthogonal Polynomials
        1. Legendre Polynomials
        2. Chebyshev Polynomials
        3. Jacobi Polynomials
        4. Hermite Polynomials
        5. Laguerre Polynomials
    14. Section 5.7 Exercises
    15. Chapter 5 Exercises
      1. MATLAB Exercises
      2. Chapter Test A True or False
      3. Chapter Test B
  10. Chapter 6 Eigenvalues
    1. 6.1 Eigenvalues and Eigenvectors
      1. Geometric Visualization of Eigenvalues and Eigenvectors
      2. Finding Eigenvalues and Eigenvectors
      3. Complex Eigenvalues
      4. The Product and Sum of the Eigenvalues
      5. Similar Matrices
    2. Section 6.1 Exercises
    3. 6.2 Systems of Linear Differential Equations
      1. Complex Eigenvalues
      2. Higher-Order Systems
    4. Section 6.2 Exercises
    5. 6.3 Diagonalization
      1. Remarks
      2. The Exponential of a Matrix
    6. Section 6.3 Exercises
    7. 6.4 Hermitian Matrices
      1. Complex Inner Products
      2. Hermitian Matrices
      3. The Real Schur Decomposition
      4. Normal Matrices
    8. Section 6.4 Exercises
    9. 6.5 The Singular Value Decomposition
      1. Observations
      2. Visualizing the SVD
      3. Numerical Rank and Lower Rank Approximations
    10. Section 6.5 Exercises
    11. 6.6 Quadratic Forms
      1. Conic Sections
      2. Optimization: An Application to the Calculus
    12. Section 6.6 Exercises
    13. 6.7 Positive Definite Matrices
    14. Section 6.7 Exercises
    15. 6.8 Nonnegative Matrices
    16. Section 6.8 Exercises
    17. Chapter 6 Exercises
      1. MATLAB Exercises
        1. Critical Loads for a Beam
        2. Diagonalizable and Defective Matrices
        3. Application: Sex-Linked Genes
        4. Similarity
        5. Hermitian Matrices
        6. Optimization
        7. Positive Definite Matrices
      2. Chapter Test A True or False
      3. Chapter Test B
  11. Chapter 7 Numerical Linear Algebra
    1. 7.1 Floating-Point Numbers
      1. The IEEE Standard 754 Floating-Point Representation
      2. Loss of Accuracy and Instability
    2. Section 7.1 Exercises
    3. 7.2 Gaussian Elimination
      1. Gaussian Elimination without Interchanges
      2. Using the Triangular Factorization to Solve Ax=b
        1. Operation Count
    4. Section 7.2 Exercises
    5. 7.3 Pivoting Strategies
      1. Gaussian Elimination with Interchanges
      2. Remarks
      3. Partial Pivoting
    6. Section 7.3 Exercises
    7. 7.4 Matrix Norms and Condition Numbers
      1. Matrix Norms
      2. Subordinate Matrix Norms
      3. Condition Numbers
    8. Section 7.4 Exercises
    9. 7.5 Orthogonal Transformations
      1. Elementary Orthogonal Transformations
      2. Householder Transformations
      3. Remarks
        1. Operation Count
      4. Rotations and Reflections
        1. Operation Count
      5. The QR Factorization for Solving General Linear Systems
    10. Section 7.5 Exercises
    11. 7.6 The Eigenvalue Problem
      1. The Power Method
      2. Deflation
      3. Reduction to Hessenberg Form
      4. QR Algorithm
      5. Remarks
    12. Section 7.6 Exercises
    13. 7.7 Least Squares Problems
      1. Normal Equations
      2. Modified Gram–Schmidt Method for Solving Least Squares Problems
      3. The Householder QR Factorization
      4. The Pseudoinverse
      5. Bidiagonalization
      6. The Golub–Reinsch Algorithm
    14. Section 7.7 Exercises
    15. 7.8 Iterative Methods
      1. Matrix Splittings
      2. Jacobi Iteration
      3. Gauss–Seidel Iteration
    16. Section 7.8 Exercises
    17. Chapter 7 Exercises
      1. MATLAB Exercises
        1. Sensitivity of Linear Systems
        2. Sensitivity of Eigenvalues
        3. Householder Transformations
        4. Rotations and Reflections
        5. Singular Value Decomposition
        6. Gerschgorin Circles
        7. Distribution of Condition Numbers and Eigenvalues of Random Matrices
      2. Chapter Test A True or False
      3. Chapter Test B
  12. Chapter 8 Canonical Forms
    1. 8.1 Nilpotent Operators
    2. Section 8.1 Exercises
    3. 8.2 The Jordan Canonical Form
    4. Section 8.2 Exercises
  13. Appendix: Matlab
    1. The MATLAB Desktop Display
    2. Basic Data Elements
    3. Submatrices
    4. Generating Matrices
    5. Matrix Arithmetic
      1. Addition and Multiplication of Matrices
      2. Backslash or Matrix Left Division
      3. Exponentiation
    6. MATLAB Functions
    7. Programming Features
    8. M-files
      1. Script Files
      2. Function Files
      3. The MATLAB Path
    9. Relational and Logical Operators
    10. Columnwise Array Operators
    11. Graphics
    12. Symbolic Toolbox
    13. Help Facility
    14. Conclusions
  14. Bibliography
    1. A Linear Algebra and Matrix Theory
    2. B Applied and Numerical Linear Algebra
    3. C Books of Related Interest
  15. Answers to Selected Exercises
    1. Chapter 1
      1. 1.1
      2. 1.2
      3. 1.3
      4. 1.4
      5. 1.5
      6. 1.6
        1. Chapter Test A
          1. Chapter Test B
    2. Chapter 2
      1. 2.1
      2. 2.2
      3. 2.3
        1. Chapter Test A
        2. Chapter Test B
    3. Chapter 3
      1. 3.1
      2. 3.2
      3. 3.3
      4. 3.4
      5. 3.5
      6. 3.6
        1. Chapter Test A
        2. Chapter Test B
    4. Chapter 4
      1. 4.1
      2. 4.2
      3. 4.3
        1. Chapter Test A
        2. Chapter Test B
    5. Chapter 5
      1. 5.1
      2. 5.2
      3. 5.3
      4. 5.4
      5. 5.5
      6. 5.6
      7. 5.7
        1. MATLAB Exercises
        2. Chapter Test A
          1. Chapter Test B
    6. Chapter 6
      1. 6.1
      2. 6.2
      3. 6.3
      4. 6.4
      5. 6.5
      6. 6.6
      7. 6.7
      8. 6.8
        1. MATLAB Exercises
        2. Chapter Test A
        3. Chapter Test B
    7. Chapter 7
      1. 7.1
      2. 7.2
      3. 7.3
      4. 7.4
      5. 7.5
      6. 7.6
      7. 7.7
      8. 7.8
      9. MATLAB Exercises
        1. Chapter Test A
          1. Chapter Test B
    8. Chapter 8
      1. 8.1
      2. 8.2
  16. Index
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    9. I
    10. J
    11. K
    12. L
    13. M
    14. N
    15. O
    16. P
    17. Q
    18. R
    19. S
    20. T
    21. U
    22. V
    23. W
    24. Y
    25. Z
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