Library/Linear Algebra: From Linear Systems to Eigenvalues and Least Squares/Lesson 39 of 84

When a Matrix Fails to Be Invertible

8:45
Dr. Trefor Bazett
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Course Contents

  1. 1The Big Idea of Linear Algebra: Course Introduction12:58
  2. 2What Is a Solution to a Linear System?5:28
  3. 3Visualizing Solutions to Linear Systems in 2D and 3D8:19
  4. 4Rewriting a Linear System in Matrix Notation3:10
  5. 5Solving Linear Systems with Elementary Row Operations7:27
  6. 6Using Elementary Row Operations to Simplify a System9:35
  7. 7Systems with 0, 1, or Infinitely Many Solutions6:30
  8. 8Row Echelon Form and Reduced Row Echelon Form6:41
  9. 9Back Substitution with Infinitely Many Solutions11:05
  10. 10The Gaussian Algorithm Visualized13:37
  11. 11What Is a Vector? Vector Addition and Scalar Multiplication6:40
  12. 12Linear Combinations and Span9:34
  13. 13Determining Whether a Vector Is in a Span5:00
  14. 14Matrix-Vector Multiplication and the Equation Ax=b6:59
  15. 15Matrix-Vector Multiplication Example4:33
  16. 16Proving Algebraic Rules: A(b+c) = Ab + Ac8:25
  17. 17The Big Theorem, Part I14:01
  18. 18Writing Solutions to Ax=b in Vector Form4:48
  19. 19Geometric View of Solutions to Ax=b and Ax=06:21
  20. 20Three Properties of Homogeneous Systems7:53
  21. 21Linear Dependence and Independence: Geometric View8:16
  22. 22Determining Linear Independence vs Dependence6:39
  23. 23Making a Concept Map: Linear Independence7:10
  24. 24Transformations and Matrix Transformations5:16
  25. 25Three Examples of Matrix Transformations8:25
  26. 26Linear Transformations8:28
  27. 27Matrix Transformations vs Linear Transformations, Part I3:04
  28. 28The Standard Basis Vectors6:37
  29. 29Matrix Transformations Are Linear Transformations8:49
  30. 30Finding the Matrix of a Linear Transformation9:18
  31. 31One-to-One, Onto, and the Big Theorem Part II9:30
  32. 32Matrix Multiplication: Motivation and Definition10:12
  33. 33Computing Matrix Multiplication5:37
  34. 34Visualizing Composition of Linear Transformations14:00
  35. 35Elementary Matrices7:20
  36. 36Matrix Inverses and Solving Equations7:12
  37. 37Inverses of 2x2 Matrices3:04
  38. 38Finding the Inverse of a Matrix6:30
  39. 39When a Matrix Fails to Be Invertible8:45
  40. 40Visualizing Invertible Transformations8:12
  41. 41Invertible Matrices and Invertible Transformations: Proof6:38
  42. 42Determinants and Invertibility9:14
  43. 43Computing Determinants Along Any Row or Column3:43
  44. 44Vector Spaces: Definition and Examples8:11
  45. 45The Vector Space of Polynomials: Span, Independence, Basis12:50
  46. 46Subspaces: Definition and First Examples6:26
  47. 47The Span Is a Subspace5:03
  48. 48Null Space and Column Space of a Matrix10:41
  49. 49The Basis of a Subspace3:53
  50. 50Finding a Basis for Null Space and Column Space9:45
  51. 51Finding a Basis for Col(A) When A Is Not in REF2:37
  52. 52Coordinate Systems from Non-Standard Bases6:34
  53. 53Writing Vectors in a New Coordinate System: Example3:52
  54. 54What Are Grid Lines in Coordinate Systems?5:16
  55. 55The Dimension of a Subspace5:11
  56. 56Computing Dimension of Null Space and Column Space3:27
  57. 57The Dimension Theorem and Rank4:02
  58. 58Changing Between Two Bases7:55
  59. 59Visualizing Change of Basis Dynamically7:42
  60. 60Example: Writing a Vector in a New Basis8:55
  61. 61Eigenvalues and Eigenvectors Geometrically9:09
  62. 62Using Determinants to Compute Eigenvalues and Eigenvectors4:36
  63. 63Example: Computing Eigenvalues and Eigenvectors7:33
  64. 64Possibilities for Eigenvalues and Eigenvectors9:47
  65. 65Why Diagonal Matrices Are Easy6:28
  66. 66How Diagonalization Works7:30
  67. 67Computing Large Matrix Powers via Diagonalization3:01
  68. 68Full Example: Diagonalizing a Matrix10:08
  69. 69Complex Eigenvalues, Eigenvectors and Diagonalization14:10
  70. 70Visualizing Diagonalization and Eigenbases9:46
  71. 71Similar Matrices and Their Shared Properties8:47
  72. 72Similarity as Change of Basis9:59
  73. 73Dot Products and Length7:23
  74. 74Distance, Angles, Orthogonality and Pythagoras8:14
  75. 75Orthogonal Bases4:50
  76. 76Orthogonal Complements (Orthogonal Decomposition Theorem Part 1)3:57
  77. 77Orthogonal Projections: Geometric View11:07
  78. 78Orthogonal Decomposition Theorem Part 211:25
  79. 79Orthogonal Projections as Minimization7:05
  80. 80Gram-Schmidt: Orthogonalizing a Basis11:21
  81. 81Gram-Schmidt: Full Example6:18
  82. 82Rotation Matrices21:24
  83. 83Least Squares Approximations7:28
  84. 84Least Squares as Solving a System5:36
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