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