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Linear Algebra with Applications

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.

Linear AlgebraMathematics
Author
W. Keith Nicholson
Publisher
University of Calgary
Source
LibreTexts
License
CC BY-NC-SA
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 167 sections
  1. 01: Systems of Linear Equations
  2. 1.00: Prelude to Systems of Linear Equations
  3. 1.01: Solutions and Elementary Operations
  4. 1.1E: Exercises for Section 1.1
  5. 1.02: Gaussian Elimination
  6. 1.2E: Solutions and Elementary Operations Exercises
  7. 1.03: Homogeneous Equations
  8. 1.3E: Homogeneous Equations
  9. 1.04: An Application to Network Flow
  10. 1.4E: An Application to Network Flow Exercises
  11. 1.05: An Application to Electrical Networks
  12. 1.5E: An Application to Electrical Networks Exercises
  13. 1.06: An Application to Chemical Reactions
  14. 1.6E: An Application to Chemical Reactions Exercises
  15. 1.E: Supplementary Exercises for Chapter 1
  16. 02: Matrix Algebra
  17. 2.00: Prelude to Matrix Algebra
  18. 2.01: Matrix Addition Scalar Multiplication and Transposition
  19. 2.1E: Matrix Addition Scalar Multiplication and Transposition Exercises
  20. 2.02: Equations Matricies and Transformations
  21. 2.2E: Matrix-Vector Multiplication Exercises
  22. 2.03: Matrix Multiplication
  23. 2.3E: Matrix Multiplication Exercises
  24. 2.04: Matrix Inverses
  25. 2.4E: Matrix Inverses Exercises
  26. 2.05: Elementary Matrices
  27. 2.5E: Elementary Matrices Exercises
  28. 2.06: Linear Transformations
  29. 2.6E: Linear Transformations Exercises
  30. 2.07: LU-Factorization
  31. 2.7E: LU-Factorization Exercises
  32. 2.08: An Application to Input-Output Economic Models
  33. 2.8E: An Application to Input-Output Economic Models Exercises
  34. 2.09: An Application to Markov Chains
  35. 2.9E: An Application to Markov Chains Exercises
  36. 2.E: Supplementary Exercises for Chapter 2
  37. 03: Determinants and Diagonalization
  38. 3.00: Prelude to Determinants and Diagonalization
  39. 3.01: The Cofactor Expansion
  40. 3.1E: The Cofactor Expansion Exercises
  41. 3.02: Determinants and Matrix Inverses
  42. 3.2E: Determinants and Matrix Inverses Exercises
  43. 3.03: Eigenvalues and Eigenvectors
  44. 3.3E: Eigenvalues and Eigenvectors Exercises
  45. 3.04: Diagonalization
  46. 3.4E: Diagonalization Exercises
  47. 3.05: Linear Dynamical Systems
  48. 3.5E: Linear Dynamical Systems Exercises
  49. 3.06: An Application to Linear Recurrences
  50. 3.6E: An Application to Linear Recurrences Exercises
  51. 3.07: An Application to Systems of Differential Equations
  52. 3.7E: An Application to Systems of Differential Equations Exercises
  53. 3.08: Proof of the Cofactor Expansion Theorem
  54. 3.8E: Proof of the Cofactor Expansion Theorem Exercises
  55. 3.E: Supplementary Exercises for Chapter 3
  56. 04: Vector Geometry
  57. 4.00: Prelude to Vector Geometry
  58. 4.01: Vectors and Lines
  59. 4.1E: Vectors and Lines Exercises
  60. 4.02: Projections and Planes
  61. 4.2E: Projections and Planes Exercises
  62. 4.03: More on the Cross Product
  63. 4.3E: More on the Cross Product Exercises
  64. 4.04: Linear Operators on R
  65. 4.4E: Linear Operators on R Exercises
  66. 4.05: An Application to Computer Graphics
  67. 4.5E: An Application to Computer Graphics Exercises
  68. 4.E: Supplementary Exercises for Chapter 4
  69. 05: Vector Space R
  70. 5.00: Prelude to Vector Space R
  71. 5.01: Subspaces and Spanning
  72. 5.1E: Subspaces and Spanning Exercises
  73. 5.02: Independence and Dimension
  74. 5.2E: Independence and Dimension Exercises
  75. 5.03: Orthogonality
  76. 5.3E: Orthogonality Exercises
  77. 5.04: Rank of a Matrix
  78. 5.4E: Rank of a Matrix Exercises
  79. 5.05: Similarity and Diagonalization
  80. 5.5E: Similarity and Diagonalization Exercises
  81. 5.06: Best Approximation and Least Squares
  82. 5.6E: Best Approximation and Least Squares Exercises
  83. 5.07: An Application to Correlation and Variance
  84. 5.7E: An Application to Correlation and Variance Exercises
  85. 5.E: Supplementary Exercises for Chapter 5
  86. 06: Vector Spaces
  87. 6.00: Prelude to Vector Spaces
  88. 6.01: Examples and Basic Properties
  89. 6.1E: Examples and Basic Properties Exercises
  90. 6.02: Subspaces and Spanning Sets
  91. 6.2E: Subspaces and Spanning Sets Exercises
  92. 6.03: Linear Independence and Dimension
  93. 6.3E: Linear Independence and Dimension Exercises
  94. 6.04: Finite Dimensional Spaces
  95. 6.4E: Finite Dimensional Spaces Exercises
  96. 6.05: An Application to Polynomials
  97. 6.5E: An Application to Polynomials Exercises
  98. 6.06: An Application to Differential Equations
  99. 6.6E: An Application to Differential Equations Exercises
  100. 6.E: Supplementary Exercises for Chapter 6
  101. 07: Linear Transformations
  102. 7.00: Prelude to Linear Transformations
  103. 7.01: Examples and Elementary Properties
  104. 7.1E: Examples and Elementary Properties Exercises
  105. 7.02: Kernel and Image of a Linear Transformation
  106. 7.2E: Kernel and Image of a Linear Transformation Exercises
  107. 7.03: Isomorphisms and Composition
  108. 7.3E: Isomorphisms and Composition Exercises
  109. 7.04: A Theorem about Differential Equations
  110. 7.05: More on Linear Recurrences
  111. 7.5E: Exercises for Linear Recurrences
  112. 08: Orthogonality
  113. 8.00: Prelude to Orthogonality
  114. 8.01: Orthogonal Complements and Projections
  115. 8.1E: Orthogonal Complements and Projections Exercises
  116. 8.02: Orthogonal Diagonalization
  117. 8.2E: Orthogonal Diagonalization Exercises
  118. 8.03: Positive Definite Matrices
  119. 8.3E: Positive Definite Matrices Exercises
  120. 8.04: QR-Factorization
  121. 8.4E: QR-Factorization Exercises
  122. 8.05: Computing Eigenvalues
  123. 8.5E: Computing Eigenvalues Exercises
  124. 8.06: The Singular Value Decomposition
  125. 8.6E: The Singular Value Decomposition Exercises
  126. 8.07: Complex Matrices
  127. 8.7E: Complex Matrices Exercises
  128. 8.08: An Application to Linear Codes over Finite Fields
  129. 8.8E: An Application to Linear Codes over Finite Fields Exercises
  130. 8.09: An Application to Quadratic Forms
  131. 8.9E: An Application to Quadratic Forms Exercises
  132. 8.10: An Application to Constrained Optimization
  133. 8.11: An Application to Statistical Principal Component Analysis
  134. 09: Change of Basis
  135. 9.00: Prelude to Change of Basis
  136. 9.01: The Matrix of a Linear Transformation
  137. 9.1E: The Matrix of a Linear Transformation Exercises
  138. 9.02: Operators and Similarity
  139. 9.2E: Operators and Similarity Exercises
  140. 9.03: Invariant Subspaces and Direct Sums
  141. 9.3E: Invariant Subspaces and Direct Sums Exercises
  142. 10: Inner Product Spaces
  143. 10.00: Prelude to Inner Product Spaces
  144. 10.01: Inner Products and Norms
  145. 10.1E: Inner Products and Norms Exercises
  146. 10.02: Orthogonal Sets of Vectors
  147. 10.2E: Orthogonal Sets of Vectors Exercises
  148. 10.03: Orthogonal Diagonalization
  149. 10.3E: Orthogonal Diagonalization Exercises
  150. 10.04: Isometries
  151. 10.4E: Isometries Exercises
  152. 10.05: An Application to Fourier Approximation
  153. 10.5E: An Application to Fourier Approximation Exercises
  154. 11: Canonical Forms
  155. 11.00: Prelude to Canonical Forms
  156. 11.01: Block Triangular Form
  157. 11.1E: Block Triangular Form Exercises
  158. 11.02: The Jordan Canonical Form
  159. 11.2E: The Jordan Canonical Form Exercises
  160. 12: Appendices
  161. 12.A: Complex Numbers
  162. 12.A: Complex Numbers
  163. 12.B: Proofs
  164. 12.B: Proofs
  165. 12.C: Mathematical Induction
  166. 1.00: Exercises for Mathematical Induction
  167. 12.D: Polynomials