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A First Journey Through Linear Algebra

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A First Journey Through Linear Algebra

Mathematics
Publisher
LibreTexts
Source
LibreTexts
Language
EN
Category
Basic Sciences
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Contents· 138 sections
  1. 01: Systems of Equations
  2. 1.01: Systems of Equations Geometry
  3. 1.02: Systems of Equations Algebraic Procedures
  4. 1.03: Gaussian Elimination
  5. 1.04: Uniqueness of the Reduced Row-Echelon Form
  6. 1.05: Rank and Homogeneous Systems
  7. Homogeneous Equations GeoGebra Demo by Kevin Hopkins
  8. 1.06: Balancing Chemical Reactions
  9. 1.07: Dimensionless Variables
  10. 1.08: An Application to Resistor Networks
  11. 1.09: Exercises
  12. 1.10: Supplementary Notes - Homogeneous Systems of Equations
  13. 1.10.01: Homogeneous Equations
  14. 1.10.1E: Homogeneous Equations
  15. 02: Matrices
  16. 2.01: Matrix Arithmetic
  17. 2.02: Multiplication of Matrices
  18. Matrix Operations GeoGebra Demo by Kevin Hopkins
  19. 2.03: The ijth Entry of a Product
  20. 2.04: Properties of Matrix Multiplication
  21. 2.05: The Transpose
  22. 2.06: The Identity and Inverses
  23. 2.07: Finding the Inverse of a Matrix
  24. 2.08: Elementary Matrices
  25. 2.09: More on Matrix Inverses
  26. 2.10: LU Factorization
  27. 2.E: Exercises
  28. 03: Determinants
  29. 3.01: Basic Techniques
  30. 3.02: Properties of Determinants
  31. 3.03: Finding Determinants using Row Operations
  32. 3.04: Applications of the Determinant
  33. Geometric Interpretation of Determinants GeoGebra Demo GeoGebra Team
  34. 3.05: Supplementary Notes
  35. 3.5.01: The Determinant Formula
  36. 3.5.02: Elementary Matrices and Determinants
  37. 3.E: Exercises
  38. 04: R
  39. 4.01: Vectors in R
  40. 4.02: Vector Algebra
  41. 4.03: Geometric Meaning of Vector Addition
  42. 4.04: Length of a Vector
  43. 4.05: Geometric Meaning of Scalar Multiplication
  44. 4.06: Parametric Lines
  45. 4.07: The Dot Product
  46. 4.08: Planes in R
  47. 4.09: The Cross Product
  48. 4.10: Spanning Linear Independence and Basis in R
  49. 4.11: Orthogonality
  50. Vector Projection GeoGebra Demo SKKU - Matrix - Lab
  51. 4.12: Applications
  52. 4.E: Exercises
  53. 05: Vector Spaces
  54. 5.01: Algebraic Considerations
  55. 5.02: Spanning Sets
  56. Is a Vector in Span GeoGebra Demo by Kevin Hopkins
  57. 5.03: Linear Independence
  58. 5.04: Subspaces and Basis
  59. 5.05: Sums and Intersections
  60. 5.06: Linear Transformations
  61. 5.07: Isomorphisms
  62. 5.08: The Kernel and Image of a Linear Map
  63. 5.09: The Matrix of a Linear Transformation
  64. 5.10: Supplementary Notes - The Fundamental Subspaces
  65. 5.10.01: The Fundamental Subspaces
  66. 5.10.1.01: Column Space
  67. 5.10.1.02: Null Space
  68. 5.10.1.03: The Null and Column Spaces- An Example
  69. 5.10.1.04: Left Null Space
  70. 5.10.1.05: Row Space
  71. 5.10.1.06: Exercises- Columns and Null Spaces
  72. Spaces Associated with a Matrix Geogebra Demo by Kevin Hopkins
  73. 5.11: Supplementary Notes - A More In-Depth Look at Vector Spaces
  74. 5.11.01: Vector Spaces
  75. 5.11.1.01: Examples and Basic Properties
  76. 5.11.1.1E: Examples and Basic Properties Exercises
  77. 5.11.1.02: Subspaces and Spanning Sets
  78. 5.11.1.2E: Subspaces and Spanning Sets Exercises
  79. 5.11.1.03: Linear Independence and Dimension
  80. 5.11.1.3E: Linear Independence and Dimension Exercises
  81. 5.11.1.04: Finite Dimensional Spaces
  82. 5.11.1.4E: Finite Dimensional Spaces Exercises
  83. 5.11.1.05: An Application to Polynomials
  84. 5.11.1.5E: An Application to Polynomials Exercises
  85. 5.11.1.06: Supplementary Exercises for Chapter 6
  86. 5.E: Exercises
  87. 06: Linear Transformations
  88. 6.01: Linear Transformations
  89. 6.02: The Matrix of a Linear Transformation I
  90. 6.03: Properties of Linear Transformations
  91. 6.04: Special Linear Transformations in R
  92. Geometric Linear Transformations in 2D GeoGebra Demo Ku Yin Bin (Albert)
  93. 6.05: One-to-One and Onto Transformations
  94. 6.06: Isomorphisms
  95. 6.07: The Kernel and Image of A Linear Map
  96. 6.08: The Matrix of a Linear Transformation II
  97. 6.09: The General Solution of a Linear System
  98. 6.10: Supplementary Notes - More on Linear Transformations
  99. 6.10.01: Examples and Elementary Properties
  100. 6.10.1E: Examples and Elementary Properties Exercises
  101. 6.10.02: Kernel and Image of a Linear Transformation
  102. 6.10.2E: Kernel and Image of a Linear Transformation Exercises
  103. 6.10.03: Isomorphisms and Composition
  104. 6.10.3E: Isomorphisms and Composition Exercises
  105. 6.10.04: A Theorem about Differential Equations
  106. 6.11: Supplementary Notes - Nullity Rank and Dimension Theorem
  107. 6.11.01: Kernel Range Nullity Rank
  108. 6.11.1.01: Summary
  109. 6.11.1.02: Review Problems
  110. 6.E: Exercises
  111. 07: Inner Product Spaces
  112. 7.01: Inner Products and Norms
  113. 7.1E: Inner Products and Norms Exercises
  114. 7.02: Orthogonal Sets of Vectors
  115. 7.2E: Orthogonal Sets of Vectors Exercises
  116. Gram Schmidt Orthonormalization Process GeoGebra Demo by SKKU-Matrix-Lab
  117. 7.03: Orthogonal Diagonalization
  118. 7.3E: Orthogonal Diagonalization Exercises
  119. 7.04: Isometries
  120. 7.4E: Isometries Exercises
  121. 7.05: An Application to Fourier Approximation
  122. 7.5E: An Application to Fourier Approximation Exercises
  123. 08: Spectral Theory
  124. 8.01: Eigenvalues and Eigenvectors of a Matrix
  125. Matrix Transformations and Eigenvalues GeoGebra Demo by SKKU-Matrix-Lab
  126. 8.02: Diagonalization
  127. 8.03: Applications of Spectral Theory
  128. 8.04: Orthogonality
  129. 8.05: Supplemental Notes - More on Eigenvalues and Intro to Eigenspaces
  130. 8.5.01: The Eigenvalue-Eigenvector Equation
  131. 8.5.02: Eigenspaces
  132. 8.5.03: Eigenvalues and Eigenvectors - Visualizations
  133. 8.E: Exercises
  134. 09: Appendices
  135. 9.01: Sets and Set Notation
  136. 9.02: A Complex Numbers
  137. 9.03: Proofs
  138. 9.04: Well Ordering and Induction