Linear Algebra with Applications

by W. Keith Nicholson

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The aim of the textbook is to achieve a balance among computational skills, theory, and applications of linear algebra. It is a relatively advanced introduction to the ideas and techniques of linear algebra targeted for science and engineering students who need to understand not only how to use these methods but also gain insight into why they work. The contents have enough flexibility to present a traditional introduction to the subject, or to allow for a more applied course.

Description from LibreTexts.

Publisher
University of Calgary
Subjects
Mathematics, Linear Algebra
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. Forward
  7. Preface
  8. 1: Systems of Linear Equations
  9. 1.0: Prelude to Systems of Linear Equations
  10. 1.1: Solutions and Elementary Operations
  11. 1.1E: Exercises for Section 1.1
  12. 1.2: Gaussian Elimination
  13. 1.2E: Solutions and Elementary Operations Exercises
  14. 1.3: Homogeneous Equations
  15. 1.3E: Homogeneous Equations
  16. 1.4: An Application to Network Flow
  17. 1.4E: An Application to Network Flow Exercises
  18. 1.5: An Application to Electrical Networks
  19. 1.5E: An Application to Electrical Networks Exercises
  20. 1.6: An Application to Chemical Reactions
  21. 1.6E: An Application to Chemical Reactions Exercises
  22. 1.E: Supplementary Exercises for Chapter 1
  23. 2: Matrix Algebra
  24. 2.0: Prelude to Matrix Algebra
  25. 2.1: Matrix Addition, Scalar Multiplication, and Transposition
  26. 2.1E: Matrix Addition, Scalar Multiplication, and Transposition Exercises
  27. 2.2: Equations, Matricies and Transformations
  28. 2.2E: Matrix-Vector Multiplication Exercises
  29. 2.3: Matrix Multiplication
  30. 2.3E: Matrix Multiplication Exercises
  31. 2.4: Matrix Inverses
  32. 2.4E: Matrix Inverses Exercises
  33. 2.5: Elementary Matrices
  34. 2.5E: Elementary Matrices Exercises
  35. 2.6: Linear Transformations
  36. 2.6E: Linear Transformations Exercises
  37. 2.7: LU-Factorization
  38. 2.7E: LU-Factorization Exercises
  39. 2.8: An Application to Input-Output Economic Models
  40. 2.8E: An Application to Input-Output Economic Models Exercises
  41. 2.9: An Application to Markov Chains
  42. 2.9E: An Application to Markov Chains Exercises
  43. 2.E: Supplementary Exercises for Chapter 2
  44. 3: Determinants and Diagonalization
  45. 3.0: Prelude to Determinants and Diagonalization
  46. 3.1: The Cofactor Expansion
  47. 3.1E: The Cofactor Expansion Exercises
  48. 3.2: Determinants and Matrix Inverses
  49. 3.2E: Determinants and Matrix Inverses Exercises
  50. 3.3: Eigenvalues and Eigenvectors
  51. 3.3E: Eigenvalues and Eigenvectors Exercises
  52. 3.4: Diagonalization
  53. 3.4E: Diagonalization Exercises
  54. 3.5: Linear Dynamical Systems
  55. 3.5E: Linear Dynamical Systems Exercises
  56. 3.6: An Application to Linear Recurrences
  57. 3.6E: An Application to Linear Recurrences Exercises
  58. 3.7: An Application to Systems of Differential Equations
  59. 3.7E: An Application to Systems of Differential Equations Exercises
  60. 3.8: Proof of the Cofactor Expansion Theorem
  61. 3.8E: Proof of the Cofactor Expansion Theorem Exercises
  62. 3.E: Supplementary Exercises for Chapter 3
  63. 4: Vector Geometry
  64. 4.0: Prelude to Vector Geometry
  65. 4.1: Vectors and Lines
  66. 4.1E: Vectors and Lines Exercises
  67. 4.2: Projections and Planes
  68. 4.2E: Projections and Planes Exercises
  69. 4.3: More on the Cross Product
  70. 4.3E: More on the Cross Product Exercises
  71. 4.4: Linear Operators on R³
  72. 4.4E: Linear Operators on R³ Exercises
  73. 4.5: An Application to Computer Graphics
  74. 4.5E: An Application to Computer Graphics Exercises
  75. 4.E: Supplementary Exercises for Chapter 4
  76. 5: Vector Space Rⁿ
  77. 5.0: Prelude to Vector Space Rⁿ
  78. 5.1: Subspaces and Spanning
  79. 5.1E: Subspaces and Spanning Exercises
  80. 5.2: Independence and Dimension
  81. 5.2E: Independence and Dimension Exercises
  82. 5.3: Orthogonality
  83. 5.3E: Orthogonality Exercises
  84. 5.4: Rank of a Matrix
  85. 5.4E: Rank of a Matrix Exercises
  86. 5.5: Similarity and Diagonalization
  87. 5.5E: Similarity and Diagonalization Exercises
  88. 5.6: Best Approximation and Least Squares
  89. 5.6E: Best Approximation and Least Squares Exercises
  90. 5.7: An Application to Correlation and Variance
  91. 5.7E: An Application to Correlation and Variance Exercises
  92. 5.E: Supplementary Exercises for Chapter 5
  93. 6: Vector Spaces
  94. 6.0: Prelude to Vector Spaces
  95. 6.1: Examples and Basic Properties
  96. 6.1E: Examples and Basic Properties Exercises
  97. 6.2: Subspaces and Spanning Sets
  98. 6.2E: Subspaces and Spanning Sets Exercises
  99. 6.3: Linear Independence and Dimension
  100. 6.3E: Linear Independence and Dimension Exercises
  101. 6.4: Finite Dimensional Spaces
  102. 6.4E: Finite Dimensional Spaces Exercises
  103. 6.5: An Application to Polynomials
  104. 6.5E: An Application to Polynomials Exercises
  105. 6.6: An Application to Differential Equations
  106. 6.6E: An Application to Differential Equations Exercises
  107. 6.E: Supplementary Exercises for Chapter 6
  108. 7: Linear Transformations
  109. 7.0: Prelude to Linear Transformations
  110. 7.1: Examples and Elementary Properties
  111. 7.1E: Examples and Elementary Properties Exercises
  112. 7.2: Kernel and Image of a Linear Transformation
  113. 7.2E: Kernel and Image of a Linear Transformation Exercises
  114. 7.3: Isomorphisms and Composition
  115. 7.3E: Isomorphisms and Composition Exercises
  116. 7.4: A Theorem about Differential Equations
  117. 7.5: More on Linear Recurrences
  118. 7.5E: Exercises for Linear Recurrences
  119. 8: Orthogonality
  120. 8.0: Prelude to Orthogonality
  121. 8.1: Orthogonal Complements and Projections
  122. 8.1E: Orthogonal Complements and Projections Exercises
  123. 8.2: Orthogonal Diagonalization
  124. 8.2E: Orthogonal Diagonalization Exercises
  125. 8.3: Positive Definite Matrices
  126. 8.3E: Positive Definite Matrices Exercises
  127. 8.4: QR-Factorization
  128. 8.4E: QR-Factorization Exercises
  129. 8.5: Computing Eigenvalues
  130. 8.5E: Computing Eigenvalues Exercises
  131. 8.6: The Singular Value Decomposition
  132. 8.6E: The Singular Value Decomposition Exercises
  133. 8.7: Complex Matrices
  134. 8.7E: Complex Matrices Exercises
  135. 8.8: An Application to Linear Codes over Finite Fields
  136. 8.8E: An Application to Linear Codes over Finite Fields Exercises
  137. 8.9: An Application to Quadratic Forms
  138. 8.9E: An Application to Quadratic Forms Exercises
  139. 8.10: An Application to Constrained Optimization
  140. 8.11: An Application to Statistical Principal Component Analysis
  141. 9: Change of Basis
  142. 9.0: Prelude to Change of Basis
  143. 9.1: The Matrix of a Linear Transformation
  144. 9.1E: The Matrix of a Linear Transformation Exercises
  145. 9.2: Operators and Similarity
  146. 9.2E: Operators and Similarity Exercises
  147. 9.3: Invariant Subspaces and Direct Sums
  148. 9.3E: Invariant Subspaces and Direct Sums Exercises
  149. 10: Inner Product Spaces
  150. 10.0: Prelude to Inner Product Spaces
  151. 10.1: Inner Products and Norms
  152. 10.1E: Inner Products and Norms Exercises
  153. 10.2: Orthogonal Sets of Vectors
  154. 10.2E: Orthogonal Sets of Vectors Exercises
  155. 10.3: Orthogonal Diagonalization
  156. 10.3E: Orthogonal Diagonalization Exercises
  157. 10.4: Isometries
  158. 10.4E: Isometries Exercises
  159. 10.5: An Application to Fourier Approximation
  160. 10.5E: An Application to Fourier Approximation Exercises
  161. 11: Canonical Forms
  162. 11.0: Prelude to Canonical Forms
  163. 11.1: Block Triangular Form
  164. 11.1E: Block Triangular Form Exercises
  165. 11.2: The Jordan Canonical Form
  166. 11.2E: The Jordan Canonical Form Exercises
  167. 12: Appendices
  168. 12.A: Complex Numbers
  169. 12.A.E: Exercises for Complex Numbers
  170. 12.B: Proofs
  171. 12.B.E: Exercises for Proofs
  172. 12.C: Mathematical Induction
  173. 12.C.E: Exercises for Mathematical Induction
  174. 12.D: Polynomials
  175. Back Matter
  176. Index
  177. Glossary
  178. Detailed Licensing