Khan Academy

Khan Academy

Linear Algebra

Matrices, vectors, vector spaces, transformations. Covers all topics in a first year college linear algebra course. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.

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. Linear Algebra: Exploring the solution set of Ax=b
73. Linear Algebra: Matrix condition for one-to-one trans
74. Linear Algebra: Simplifying conditions for invertibility
75. Linear Algebra: Showing that Inverses are Linear
76. Linear Algebra: Deriving a method for determining inverses
77. Linear Algebra: Example of Finding Matrix Inverse
78. Linear Algebra: Formula for 2x2 inverse
79. Linear Algebra: 3x3 Determinant
80. Linear Algebra: nxn Determinant
81. Linear Algebra: Determinants along other rows/cols
82. Linear Algebra: Rule of Sarrus of Determinants
83. Linear Algebra: Determinant when row multiplied by scalar
84. Linear Algebra: (correction) scalar muliplication of row
85. Linear Algebra: Determinant when row is added
86. Linear Algebra: Duplicate Row Determinant
87. Linear Algebra: Determinant after row operations
88. Linear Algebra: Upper Triangular Determinant
89. Linear Algebra: Simpler 4x4 determinant
90. Linear Algebra: Determinant and area of a parallelogram
91. Linear Algebra: Determinant as Scaling Factor
92. Linear Algebra: Transpose of a Matrix
93. Linear Algebra: Determinant of Transpose
94. Linear Algebra: Transposes of sums and inverses
95. Linear Algebra: Transpose of a Vector
96. Linear Algebra: Rowspace and Left Nullspace
97. Lin Alg: Visualizations of Left Nullspace and Rowspace
98. Linear Algebra: Orthogonal Complements
99. Linear Algebra: Rank(A) = Rank(transpose of A)
100. Linear Algebra: dim(V) + dim(orthogonoal complelent of V)=n
101. Lin Alg: Representing vectors in Rn using subspace members
102. Lin Alg: Orthogonal Complement of the Orthogonal Complement
103. Lin Alg: Orthogonal Complement of the Nullspace
104. Lin Alg: Unique rowspace solution to Ax=b
105. Linear Alg: Rowspace Solution to Ax=b example
106. Lin Alg: Showing that A-transpose x A is invertible
107. Linear Algebra: Projections onto Subspaces
108. Linear Alg: Visualizing a projection onto a plane
109. Lin Alg: A Projection onto a Subspace is a Linear Transforma
110. Linear Algebra: Subspace Projection Matrix Example
111. Lin Alg: Another Example of a Projection Matrix
112. Linear Alg: Projection is closest vector in subspace
113. Linear Algebra: Least Squares Approximation
114. Linear Algebra: Least Squares Examples
115. Linear Algebra: Another Least Squares Example
116. Linear Algebra: Coordinates with Respect to a Basis
117. Linear Algebra: Change of Basis Matrix
118. Lin Alg: Invertible Change of Basis Matrix
119. Lin Alg: Transformation Matrix with Respect to a Basis
120. Lin Alg: Alternate Basis Tranformation Matrix Example
121. Lin Alg: Alternate Basis Tranformation Matrix Example Part 2
122. Lin Alg: Changing coordinate systems to help find a transformation matrix
123. Linear Algebra: Introduction to Orthonormal Bases
124. Linear Algebra: Coordinates with respect to orthonormal bases
125. Lin Alg: Projections onto subspaces with orthonormal bases
126. Lin Alg: Finding projection onto subspace with orthonormal basis example
127. Lin Alg: Example using orthogonal change-of-basis matrix to find transformation matrix
128. Lin Alg: Orthogonal matrices preserve angles and lengths
129. Linear Algebra: The Gram-Schmidt Process
130. Linear Algebra: Gram-Schmidt Process Example
131. Linear Algebra: Gram-Schmidt example with 3 basis vectors
132. Linear Algebra: Introduction to Eigenvalues and Eigenvectors
133. Linear Algebra: Proof of formula for determining Eigenvalues
134. Linear Algebra: Example solving for the eigenvalues of a 2x2 matrix
135. Linear Algebra: Finding Eigenvectors and Eigenspaces example
136. Linear Algebra: Eigenvalues of a 3x3 matrix
137. Linear Algebra: Eigenvectors and Eigenspaces for a 3x3 matrix
138. Linear Algebra: 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
142. Distance Between Planes

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