# Linear Algebra

####
Covers advanced algebra topics such as matrices, vectors, transformations plus many more. Created by Khan Academy.

Recommended that you have a good foundation in Calculus before taking this course.

###### Average Course Length

45 hours

###### Skill Level

Advanced

##### Pick a lesson

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: Vector Examples

13: Linear Algebra: Parametric Representations of Lines

14: Linear Combinations and Span

15: Linear Algebra: Introduction to Linear Independence

16: More on linear independence

17: Span and Linear Independence Example

18: Linear Subspaces

19: Linear Algebra: Basis of a Subspace

20: Vector Dot Product and Vector Length

21: Proving Vector Dot Product Properties

22: Proof of the Cauchy-Schwarz Inequality

23: Linear Algebra: Vector Triangle Inequality

24: Defining the angle between vectors

25: Defining a plane in R3 with a point and normal vector

26: Linear Algebra: Cross Product Introduction

27: Proof: Relationship between cross product and sin of angle

28: Dot and Cross Product Comparison/Intuition

29: Matrices: Reduced Row Echelon Form 1

30: Matrices: Reduced Row Echelon Form 2

31: Matrices: Reduced Row Echelon Form 3

32: Matrix Vector Products

33: Introduction to the Null Space of a Matrix

34: Null Space 2: Calculating the null space of a matrix

35: Null Space 3: Relation to Linear Independence

36: Column Space of a Matrix

37: Null Space and Column Space Basis

38: Visualizing a Column Space as a Plane in R3

39: Proof: Any subspace basis has same number of elements

40: Dimension of the Null Space or Nullity

41: Dimension of the Column Space or Rank

42: Showing relation between basis cols and pivot cols

43: Showing that the candidate basis does span C(A)

44: A more formal understanding of functions

45: Vector Transformations

46: Linear Transformations

47: Matrix Vector Products as Linear Transformations

48: Linear Transformations as Matrix Vector Products

49: Image of a subset under a transformation

50: im(T): Image of a Transformation

51: Preimage of a set

52: Preimage and Kernel Example

53: Sums and Scalar Multiples of Linear Transformations

54: More on Matrix Addition and Scalar Multiplication

55: Linear Transformation Examples: Scaling and Reflections

56: Linear Transformation Examples: Rotations in R2

57: Rotation in R3 around the X-axis

58: Unit Vectors

59: Introduction to Projections

60: Expressing a Projection on to a line as a Matrix Vector prod

61: Compositions of Linear Transformations 1

62: Compositions of Linear Transformations 2

63: Linear Algebra: Matrix Product Examples

64: Matrix Product Associativity

65: Distributive Property of Matrix Products

66: Linear Algebra: Introduction to the inverse of a function

67: Proof: Invertibility implies a unique solution to f(x)=y

68: Surjective (onto) and Injective (one-to-one) functions

69: Relating invertibility to being onto and one-to-one

70: Determining whether a transformation is onto

71: Linear Algebra: Exploring the solution set of Ax=b

72: Linear Algebra: Matrix condition for one-to-one trans

73: Linear Algebra: Simplifying conditions for invertibility

74: Linear Algebra: Showing that Inverses are Linear

75: Linear Algebra: Deriving a method for determining inverses

76: Linear Algebra: Example of Finding Matrix Inverse

77: Linear Algebra: Formula for 2x2 inverse

78: Linear Algebra: 3x3 Determinant

79: Linear Algebra: nxn Determinant

80: Linear Algebra: Determinants along other rows/cols

81: Linear Algebra: Rule of Sarrus of Determinants

82: Linear Algebra: Determinant when row multiplied by scalar

83: Linear Algebra: (correction) scalar muliplication of row

84: Linear Algebra: Determinant when row is added

85: Linear Algebra: Duplicate Row Determinant

86: Linear Algebra: Determinant after row operations

87: Linear Algebra: Upper Triangular Determinant

88: Linear Algebra: Simpler 4x4 determinant

89: Linear Algebra: Determinant and area of a parallelogram

90: Linear Algebra: Determinant as Scaling Factor

91: Linear Algebra: Transpose of a Matrix

92: Linear Algebra: Determinant of Transpose

93: Linear Algebra: Transpose of a Matrix Product

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: Linear Alg: Rowspace Solution to Ax=b example

107: Lin Alg: Showing that A-transpose x A is invertible

108: Linear Algebra: Projections onto Subspaces

109: Linear Alg: Visualizing a projection onto a plane

110: Lin Alg: A Projection onto a Subspace is a Linear Transforma

111: Linear Algebra: Subspace Projection Matrix Example

112: Lin Alg: Another Example of a Projection Matrix

113: Linear Alg: Projection is closest vector in subspace

114: Linear Algebra: Least Squares Approximation

115: Linear Algebra: Least Squares Examples

116: Linear Algebra: Another Least Squares Example

117: Linear Algebra: Coordinates with Respect to a Basis

118: Linear Algebra: Change of Basis Matrix

119: Lin Alg: Invertible Change of Basis Matrix

120: Lin Alg: Transformation Matrix with Respect to a Basis

121: Lin Alg: Alternate Basis Tranformation Matrix Example

122: Lin Alg: Alternate Basis Tranformation Matrix Example Part 2

123: Lin Alg: Changing coordinate systems to help find a transformation matrix

124: Linear Algebra: Introduction to Orthonormal Bases

125: Linear Algebra: Coordinates with respect to orthonormal bases

126: Lin Alg: Projections onto subspaces with orthonormal bases

127: Lin Alg: Finding projection onto subspace with orthonormal basis example

128: Lin Alg: Example using orthogonal change-of-basis matrix to find transformation matrix

129: Lin Alg: Orthogonal matrices preserve angles and lengths

130: Linear Algebra: The Gram-Schmidt Process

131: Linear Algebra: Gram-Schmidt Process Example

132: Linear Algebra: Gram-Schmidt example with 3 basis vectors

133: Linear Algebra: Introduction to Eigenvalues and Eigenvectors

134: Linear Algebra: Proof of formula for determining Eigenvalues

135: Linear Algebra: Example solving for the eigenvalues of a 2x2 matrix

136: Linear Algebra: Finding Eigenvectors and Eigenspaces example

137: Linear Algebra: Eigenvalues of a 3x3 matrix

138: Linear Algebra: Eigenvectors and Eigenspaces for a 3x3 matrix

139: Linear Algebra: Showing that an eigenbasis makes for good coordinate systems

140: Vector Triple Product Expansion (very optional)

141: Normal vector from plane equation

142: Point distance to plane

143: Distance Between Planes