Math 220: Linear Algebra
Winter 2016
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About Linear Algebra
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Topical Videos Videos by Topic (from the Khan Academy) 1. Introduction to matrices 2. Matrix multiplication (part 1) 3. Matrix multiplication (part 2) 4. Inverse Matrix (part 1) 5. Inverting matrices (part 2) 6. Inverting Matrices (part 3) 7. Matrices to solve a system of equations 8. Matrices to solve a vector combination problem 9. Singular Matrices 10. 3-variable linear equations (part 1) 11. Solving 3 Equations with 3 Unknowns 12. Linear Algebra: Introduction to Vectors 13. Linear Algebra: Vector Examples 14. Linear Algebra: Parametric Representations of Lines 15. Linear Combinations and Span 16. Linear Algebra: Introduction to Linear Independence 17. More on linear independence 18. Span and Linear Independence Example 19. Linear Subspaces 20. Linear Algebra: Basis of a Subspace 21. Vector Dot Product and Vector Length 22. Proving Vector Dot Product Properties 23. Proof of the Cauchy-Schwarz Inequality 24. Linear Algebra: Vector Triangle Inequality 25. Defining the angle between vectors 26. Defining a plane in R3 with a point and normal vector 27. Linear Algebra: Cross Product Introduction 28. Proof: Relationship between cross product and sin of angle 29. Dot and Cross Product Comparison/Intuition 30. Matrices: Reduced Row Echelon Form 1 31. Matrices: Reduced Row Echelon Form 2 32. Matrices: Reduced Row Echelon Form 3 33. Matrix Vector Products 34. Introduction to the Null Space of a Matrix 35. Null Space 2: Calculating the null space of a matrix 36. Null Space 3: Relation to Linear Independence 37. Column Space of a Matrix 38. Null Space and Column Space Basis 39. Visualizing a Column Space as a Plane in R3 40. Proof: Any subspace basis has same number of elements 41. Dimension of the Null Space or Nullity 42. Dimension of the Column Space or Rank 43. Showing relation between basis cols and pivot cols 44. Showing that the candidate basis does span C(A) 45. A more formal understanding of functions 46. Vector Transformations 47. Linear Transformations 48. Matrix Vector Products as Linear Transformations 49. Linear Transformations as Matrix Vector Products 50. Image of a subset under a transformation 51. im(T): Image of a Transformation 52. Preimage of a set 53. Preimage and Kernel Example 54. Sums and Scalar Multiples of Linear Transformations 55. More on Matrix Addition and Scalar Multiplication 56. Linear Transformation Examples: Scaling and Reflections 57. Linear Transformation Examples: Rotations in R2 58. Rotation in R3 around the X-axis 59. Unit Vectors 60. Introduction to Projections 61. Expressing a Projection on to a line as a Matrix Vector prod 62. Compositions of Linear Transformations 1 63. Compositions of Linear Transformations 2 64. Linear Algebra: Matrix Product Examples 65. Matrix Product Associativity 66. Distributive Property of Matrix Products 67. Linear Algebra: Introduction to the inverse of a function 68. Proof: Invertibility implies a unique solution to f(x)=y 69. Surjective (onto) and Injective (one-to-one) functions 70. Relating invertibility to being onto and one-to-one 71. Determining whether a transformation is onto 72. Exploring the solution set of Ax=b 73. Matrix condition for one-to-one trans 74. Simplifying conditions for invertibility 75. Showing that Inverses are Linear 76. Deriving a method for determining inverses 77. Example of Finding Matrix Inverse 78. Formula for 2x2 inverse 79. 3x3 Determinant 80. nxn Determinant 81. Determinants along other rows/cols 82. Rule of Sarrus of Determinants 83. Determinant when row multiplied by scalar 84. (correction) scalar muliplication of row 85. Determinant when row is added 86. Duplicate Row Determinant 87. Determinant after row operations 88. Upper Triangular Determinant 89. Simpler 4x4 determinant 90. Determinant and area of a parallelogram 91. Determinant as Scaling Factor 92. Transpose of a Matrix 93. Determinant of Transpose 94. Transposes of sums and inverses 95. Transpose of a Vector 96. Rowspace and Left Nullspace 97. Visualizations of Left Nullspace and Rowspace 98. Orthogonal Complements 99. Rank(A) = Rank(transpose of A) 100. dim(V) + dim(orthogonoal complelent of V)=n 101. Representing vectors in Rn using subspace members 102. Orthogonal Complement of the Orthogonal Complement 103. Orthogonal Complement of the Nullspace 104. Unique rowspace solution to Ax=b 105. Rowspace Solution to Ax=b example 106. Showing that A-transpose x A is invertible 107. Projections onto Subspaces 108. Visualizing a projection onto a plane 109. A Projection onto a Subspace is a Linear Transforma 110. Subspace Projection Matrix Example 111. Another Example of a Projection Matrix 112. Projection is closest vector in subspace 113. Least Squares Approximation 114. Least Squares Examples 115. Another Least Squares Example 116. Linear Algebra: Coordinates with Respect to a Basis 117. Change of Basis Matrix 118. Invertible Change of Basis Matrix 119. Transformation Matrix with Respect to a Basis 120. Alternate Basis Tranformation Matrix Example 121. Alternate Basis Tranformation Matrix Example Part 2 122. Changing coordinate systems to help find a transformation matrix 123. Introduction to Orthonormal Bases 124. Coordinates with respect to orthonormal bases 125. Projections onto subspaces with orthonormal bases 126. Finding projection onto subspace with orthonormal basis example 127. Example using orthogonal change-of-basis matrix to find transformation matrix 128. Orthogonal matrices preserve angles and lengths 129. The Gram-Schmidt Process 130. Gram-Schmidt Process Example 131. Gram-Schmidt example with 3 basis vectors 132. Introduction to Eigenvalues and Eigenvectors 133. Proof of formula for determining Eigenvalues 134 Example solving for the eigenvalues of a 2x2 matrix 135. Finding Eigenvectors and Eigenspaces example 136. Eigenvalues of a 3x3 matrix 137. Eigenvectors and Eigenspaces for a 3x3 matrix 138. Showing that an eigenbasis makes for good coordinate systems 139. Vector Triple Product Expansion (very optional) 140. Normal vector from plane equation 141. Point distance to plane and Distance Between Planes |
Full Lectures Video Lectures from MIT: Additional supplemental materials (including these video lectures) may be found on the MIT OpenCourseWare site 1. The Geometry of Linear Equations 2. Elimination with Matrices 3. Multiplication and Inverse Matrices 4. Factorization into A = LU 5. Transposes, Permutations, Spaces R^n 6. Column Space and Nullspace 7. Solving Ax = 0: Pivot Variables, Special Solutions 8. Solving Ax = b: Row Reduced Form R 9. Independence, Basis, and Dimension 10. The Four Fundamental Subspaces 11. Matrix Spaces; Rank 1; Small World Graphs 12. Graphs, Networks, Incidence Matrices 13. Quiz 1 Review 14. Orthogonal Vectors and Subspaces 15. Projections onto Subspaces 16. Projection Matrices and Least Squares 17. Orthogonal Matrices and Gram-Schmidt 18. Properties of Determinants 19. Determinant Formulas and Cofactors 20. Cramer's Rule, Inverse Matrix, and Volume 21. Eigenvalues and Eigenvectors 22. Diagonalization and Powers of A 23. Differential Equations and exp(At) 24. Markov Matrices; Fourier Series 24b. Quiz 2 Review 25. Symmetric Matrices and Positive Definiteness 26. Complex Matrices; Fast Fourier Transform 27. Positive Definite Matrices and Minima 28. Similar Matrices and Jordan Form 29. Singular Value Decomposition 30. Linear Transformations and Their Matrices 31. Change of Basis; Image Compression 32. Quiz 3 Review 33. Left and Right Inverses; Pseudoinverse 34. Final Course Review |