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Linear Algebra: A First Course

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Linear Algebra: A First Course

This book offers a comprehensive and accessible introduction to linear algebra. It begins with methods for solving linear systems, followed by an exploration of matrices and their properties, including a dedicated chapter on determinants. The text then introduces vectors, covering essential concepts such as span, linear independence, subspaces, and linear transformations. From there, it develops spectral theory, including diagonalization and orthogonal diagonalization with their applications.

Mathematics
Author
Doli Bambhania, Fatemeh Yarahmadi and Bill Wilson
Publisher
De Anza College
Source
LibreTexts
License
CC BY
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 124 sections
  1. 01: Systems of Equations
  2. 1.01: Systems of Linear Equations
  3. 1.1E: Exercises for Section 1.1
  4. 1.02: Gaussian Elimination
  5. 1.2E: Exercises for Section 1.2
  6. 1.03: Rank and Homogeneous Systems
  7. 1.3E: Exercises for Section 1.3
  8. 1.04: Balancing Chemical Reactions
  9. 1.4E: Exercises for Section 1.4
  10. 1.05: Applications to Physics
  11. 1.5E: Exercises for Section 1.5
  12. 1.06: Application to Polynomial Interpolation
  13. 1.6E: Exercises for Section 1.7
  14. 1.07: Gauss-Seidel Method
  15. 1.7E: Exercises for Section 1.8
  16. 02: Matrices
  17. 2.01: Matrix Operations
  18. 2.1E: Exercises for Section 2.1
  19. 2.02: The Inverse of a Matrix
  20. 2.2E: Exercises for Section 2.2
  21. 2.03: Elementary Matrices
  22. 2.3E: Exercises for Section 2.3
  23. 2.04: LU Factorization
  24. 2.4E: Exercises for Section 2.4
  25. 03: Determinants
  26. 3.01: Basic Techniques
  27. 3.1E: Exercises for Section 3.1
  28. 3.02: Properties of Determinants
  29. 3.2E: Exercises for Section 3.2
  30. 3.03: Application of the Determinant to Inverses Cramer's Rule
  31. 3.3E: Exercises for Section 3.3
  32. 3.04: Determinants and Geometry
  33. 3.4E: Exercises for Section 3.4
  34. 04: R
  35. 4.01: Review of Vectors
  36. 4.1E: Exercises for Section 4.1
  37. 4.02: Dot and Cross Product
  38. 4.2E: Exercises for Section 4.2
  39. 4.03: Lines and Planes
  40. 4.3E: Exercises for Section 4.3
  41. 4.04: Spanning Sets in R
  42. 4.4.E: Exercise for Section 4.4
  43. 4.05: Linear Independence
  44. 4.5.E: Exercise for Section 4.5
  45. 4.06: Subspaces and Bases
  46. 4.6.E: Exercise for Section 4.6
  47. 4.07: Row Column and Null Spaces
  48. 4.7.E: Exercise for Section 4.7
  49. 4.08: Orthogonal Vectors and Matrices
  50. 4.8.E: Exercise for Section 4.8
  51. 4.09: Gram-Schmidt Process
  52. 4.9.E: Exercises for Section 4.9
  53. 4.10: Orthogonal Projections
  54. 4.10.E: Exercise for Section 4.10
  55. 4.11: Least Squares Approximation
  56. 4.11.E: Exercises for Section 4.11
  57. 05: Linear Transformations
  58. 5.01: Linear Transformations
  59. 5.1E: Exercises for Section 5.1
  60. 5.02: The Matrix of a Linear Transformation I
  61. 5.2E: Exercises for Section 5.2
  62. 5.03: Properties of Linear Transformations
  63. 5.3E: Exercises for Section 5.3
  64. 5.04: Special Linear Transformations in Two and Three Dimensions
  65. 5.4E: Exercises for Section 5.4
  66. 5.05: One-to-One and Onto Transformations
  67. 5.5E: Exercises for Section 5.5
  68. 5.06: Isomorphisms
  69. 5.6E: Exercises for Section 5.6
  70. 5.07: The Kernel and Image of A Linear Map
  71. 5.7E: Exercises for Section 5.7
  72. 5.08: The Matrix of a Linear Transformation II
  73. 5.8E: Exercises for Section 5.8
  74. 5.09: The General Solution of a Linear System
  75. 5.9E: Exercises for Section 5.9
  76. 06: Spectral Theory
  77. 6.01: Eigenvalues and Eigenvectors of a Matrix
  78. 6.1E: Exercises for Section 6.1
  79. 6.02: Eigenvalues and Eigenvectors for Special Matrices
  80. 6.2E: Exercises for Section 6.2
  81. 6.03: Diagonalization
  82. 6.3E: Exercises for Section 6.3
  83. 6.04: Applications of Spectral Theory
  84. 6.4E: Exercises for Section 6.4
  85. 6.05: Markov Matrices
  86. 6.5E: Exercises for Section 6.5
  87. 6.06: Dynamical Systems
  88. 6.6E: Exercises for Section 6.6
  89. 6.07: Orthogonal Diagonalization
  90. 6.7E: Exercises for Section 6.7
  91. 6.08: Singular Value Decomposition
  92. 6.8E: Exercises for Section 6.8
  93. 6.09: Special Factorizations
  94. 6.9E: Exercises for Section 6.9
  95. 6.10: Quadratic Forms
  96. 6.10E: Exercises for Section 6.10
  97. 07: Vector Spaces
  98. 7.01: Vector Space - Definition
  99. 7.1.E: Exercises for Section 7.1
  100. 7.02: Vector Space - Examples
  101. 7.2.E: Exercises for Section 7.2
  102. 7.03: Spanning Sets
  103. 7.3E: Exercises for Section 7.3
  104. 7.04: Linear Independence
  105. 7.4E: Exercises for Section 7.4
  106. 7.05: Subspaces
  107. 7.5E: Exercises for Section 7.5
  108. 7.06: Basis
  109. 7.6.E: Exercises for Section 7.6
  110. 7.07: Sums and Intersections
  111. 7.7E: Exercises for Section 7.7
  112. 7.08: Linear Transformations
  113. 7.8E: Exercises for Section 7.8
  114. 7.09: Isomorphisms
  115. 7.9E: Exercises for Section 7.9
  116. 7.10: The Kernel and Image of a Linear Map
  117. 7.10E: Exercises for Section 7.10
  118. 7.11: The Matrix of a Linear Transformation
  119. 7.11E: Exercises for Section 7.11
  120. 7.12: Inner Product Spaces
  121. 7.12E: Exercises for Section 7.12
  122. 08: Appendices
  123. 8.01: Sets and Set Notation
  124. 8.02: Well Ordering and Induction