Differential Equations and Linear Algebra

by Steven Zook

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This text is intended for engineering, science, and math majors who have completed a three semester calculus sequence.

Description from LibreTexts.

Publisher
Reedley College
Subjects
Mathematics
Licence
CC BY-NC-SA 4.0, as LibreTexts states it. Non-commercial use only.
Who can read
Tweens profiles and older, judged from its title and subjects

Contents

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  1. Front Matter
  2. TitlePage
  3. InfoPage
  4. Table of Contents
  5. Licensing
  6. 1: Systems of Equations
  7. 1.1: Systems of Equations, Geometry
  8. 1.2: Systems of Equations, Algebraic Procedures
  9. 1.3: Gaussian Elimination
  10. 1.4: Uniqueness of the Reduced Row-Echelon Form
  11. 1.5: Rank and Homogeneous Systems
  12. 1.6: Exercises
  13. 2: Matrices
  14. 2.1: Matrix Arithmetic
  15. 2.2: Multiplication of Matrices
  16. 2.3: The ijth Entry of a Product
  17. 2.4: Properties of Matrix Multiplication
  18. 2.5: The Transpose
  19. 2.6: The Identity and Inverses
  20. 2.7: Finding the Inverse of a Matrix
  21. 2.8: Elementary Matrices
  22. 2.9: More on Matrix Inverses
  23. 2.10: LU Factorization
  24. 2.11: Exercises
  25. 3: Determinants
  26. 3.1: Basic Techniques
  27. 3.2: Properties of Determinants
  28. 3.3: Finding Determinants using Row Operations
  29. 3.4: Applications of the Determinant
  30. 3.5: Exercises
  31. 4: Rⁿ
  32. 4.1: Vectors in Rⁿ
  33. 4.2: Vector Algebra
  34. 4.3: Geometric Meaning of Vector Addition
  35. 4.4: Length of a Vector
  36. 4.5: Geometric Meaning of Scalar Multiplication
  37. 4.6: Parametric Lines
  38. 4.7: Spanning, Linear Independence and Basis in Rⁿ
  39. 4.8: Orthogonality
  40. 4.9: Applications
  41. 4.10: Exercises
  42. 5: Linear Transformations
  43. 5.1: Linear Transformations
  44. 5.2: The Matrix of a Linear Transformation I
  45. 5.3: Properties of Linear Transformations
  46. 5.4: Special Linear Transformations in R²
  47. 5.5: One-to-One and Onto Transformations
  48. 5.6: The Kernel and Image of A Linear Map
  49. 5.7: The Matrix of a Linear Transformation II
  50. 5.8: The General Solution of a Linear System
  51. 5.9: Exercises
  52. 6: Spectral Theory
  53. 6.1: Eigenvalues and Eigenvectors of a Matrix
  54. 6.2: Diagonalization
  55. 6.3: Applications of Spectral Theory
  56. 6.4: Orthogonality
  57. 6.5: Exercises
  58. 7: What are Differential Equations?
  59. 7.1: Applications Leading to Differential Equations
  60. 7.2: Basic Concepts
  61. 7.2.1: Basic Concepts (Exercises)
  62. 7.3: Direction Fields for First Order Equations
  63. 7.3.1: Direction Fields for First Order Equations (Exercises)
  64. 8: First Order Equations
  65. 8.1: Linear First Order Equations
  66. 8.1.1: Linear First Order Equations (Exercises)
  67. 8.2: Separable Equations
  68. 8.2.1: Separable Equations (Exercises)
  69. 8.3: Existence and Uniqueness of Solutions of Nonlinear Equations
  70. 8.3.1: Existence and Uniqueness of Solutions of Nonlinear Equations (Exercises)
  71. 8.4: Transformation of Nonlinear Equations into Separable Equations
  72. 8.4.1: Transformation of Nonlinear Equations into Separable Equations (Exercises)
  73. 8.5: Exact Equations
  74. 8.5.1: Exact Equations (Exercises)
  75. 8.6: Integrating Factors
  76. 8.6.1: Integrating Factors (Exercises)
  77. 9: Applications of First Order Equations
  78. 9.1: Growth and Decay
  79. 9.1.1: Growth and Decay (Exercises)
  80. 9.2: Cooling and Mixing
  81. 9.2.1: Cooling and Mixing (Exercises)
  82. 9.3: Elementary Mechanics
  83. 9.3.1: Elementary Mechanics (Exercises)
  84. 10: Linear Second Order Equations
  85. 10.1: Homogeneous Linear Equations
  86. 10.1.1: Homogeneous Linear Equations (Exercises)
  87. 10.2: Constant Coefficient Homogeneous Equations
  88. 10.2.1: Constant Coefficient Homogeneous Equations (Exercises)
  89. 10.3: Nonhomogeneous Linear Equations
  90. 10.3.1: Nonhomgeneous Linear Equations (Exercises)
  91. 10.4: The Method of Undetermined Coefficients I
  92. 10.4.1: The Method of Undetermined Coefficients I (Exercises)
  93. 10.5: The Method of Undetermined Coefficients II
  94. 10.5.1: The Method of Undetermined Coefficients II (Exercises)
  95. 10.6: Reduction of Order
  96. 10.6.1: Reduction of Order (Exercises)
  97. 11: Applications of Linear Second Order Equations
  98. 11.1: Spring Problems I
  99. 11.1.1: Spring Problems I (Exercises)
  100. 11.2: Spring Problems II
  101. 11.2.1: Spring Problems II (Exercises)
  102. 12: Series Solutions of Linear Second Order Equations
  103. 12.1: Prelude to Series Solutions of Linear Second Order Equations
  104. 12.2: Review of Power Series
  105. 12.2.1: Review of Power Series (Exercises)
  106. 12.3: Series Solutions Near an Ordinary Point I
  107. 12.3.1: Series Solutions Near an Ordinary Point I (Exercises)
  108. 13: Laplace Transforms
  109. 13.1: Introduction to the Laplace Transform
  110. 13.1.1: Introduction to the Laplace Transform (Exercises)
  111. 13.2: The Inverse Laplace Transform
  112. 13.2.1: The Inverse Laplace Transform (Exercises)
  113. 13.3: Solution of Initial Value Problems
  114. 13.3.1: Solution of Initial Value Problems (Exercises)
  115. 13.4: The Unit Step Function
  116. 13.4.1: The Unit Step Function (Exercises)
  117. 13.5: Constant Coefficient Equations with Piecewise Continuous Forcing Functions
  118. 13.5.1: Constant Coefficient Equations with Piecewise Continuous Forcing Functions (Exercises)
  119. 13.6: A Brief Table of Laplace Transforms
  120. 14: Linear Higher Order Differential Equations
  121. 14.1: Introduction to Linear Higher Order Equations
  122. 14.1.1: Introduction to Linear Higher Order Equations (Exercises)
  123. 14.2: Higher Order Constant Coefficient Homogeneous Equations
  124. 14.2.1: Higher Order Constant Coefficient Homogeneous Equations (Exercises)
  125. 15: Linear Systems of Differential Equations
  126. 15.1: Introduction to Systems of Differential Equations
  127. 15.1.1: Introduction to Systems of Differential Equations (Exercises)
  128. 15.2: Linear Systems of Differential Equations
  129. 15.2.1: Linear Systems of Differential Equations (Exercises)
  130. 15.3: Basic Theory of Homogeneous Linear Systems
  131. 15.3.1: Basic Theory of Homogeneous Linear Systems (Exercises)
  132. 15.4: Constant Coefficient Homogeneous Systems I
  133. 15.4.1: Constant Coefficient Homogeneous Systems I (Exercises)
  134. 16: Appendices
  135. 16.1: Trigonometric Identities
  136. 16.2: Review of Derivative Rules
  137. 16.3: Table of Derivatives
  138. 16.4: Table of Integrals
  139. Back Matter
  140. Detailed Licensing
  141. Index
  142. Glossary
  143. Detailed Licensing