- 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
- 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

- Home
- Math
- Linear Algebra
- Linear Algebra: Parametric Representations of Lines