Education resources for parents, students, teachers & administrators
Linear Algebra Course Videos from The Khan Academy
Click on the links below to view specific linear algebra video lessons.
- Introduction to matrices
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- Matrix multiplication (part 1)
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- Matrix multiplication (part 2)
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- Inverse Matrix (part 1)
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- Inverting matrices (part 2)
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- Inverting Matrices (part 3)
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- Matrices to solve a system of equations
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- Matrices to solve a vector combination problem
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- Singular Matrices
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- 3-variable linear equations (part 1)
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- Solving 3 Equations with 3 Unknowns
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- Linear Algebra: Introduction to Vectors
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- Linear Algebra: Vector Examples
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- Linear Algebra: Parametric Representations of Lines
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- Linear Combinations and Span
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- Linear Algebra: Introduction to Linear Independence
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- More on linear independence
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- Span and Linear Independence Example
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- Linear Subspaces
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- Linear Algebra: Basis of a Subspace
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- Vector Dot Product and Vector Length
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- Proving Vector Dot Product Properties
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- Proof of the Cauchy-Schwarz Inequality
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- Linear Algebra: Vector Triangle Inequality
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- Defining the angle between vectors
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- Defining a plane in R3 with a point and normal vector
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- Linear Algebra: Cross Product Introduction
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- Proof: Relationship between cross product and sin of angle
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- Dot and Cross Product Comparison/Intuition
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- Matrices: Reduced Row Echelon Form 1
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- Matrices: Reduced Row Echelon Form 2
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- Matrices: Reduced Row Echelon Form 3
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- Matrix Vector Products
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- Introduction to the Null Space of a Matrix
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- Null Space 2: Calculating the null space of a matrix
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- Null Space 3: Relation to Linear Independence
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- Column Space of a Matrix
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- Null Space and Column Space Basis
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- Visualizing a Column Space as a Plane in R3
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- Proof: Any subspace basis has same number of elements
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- Dimension of the Null Space or Nullity
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- Dimension of the Column Space or Rank
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- Showing relation between basis cols and pivot cols
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- Showing that the candidate basis does span C(A)
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- A more formal understanding of functions
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- Vector Transformations
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- Linear Transformations
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- Matrix Vector Products as Linear Transformations
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- Linear Transformations as Matrix Vector Products
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- Image of a subset under a transformation
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- im(T): Image of a Transformation
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- Preimage of a set
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- Preimage and Kernel Example
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- Sums and Scalar Multiples of Linear Transformations
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- More on Matrix Addition and Scalar Multiplication
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- Linear Transformation Examples: Scaling and Reflections
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- Linear Transformation Examples: Rotations in R2
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- Rotation in R3 around the X-axis
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- Unit Vectors
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- Introduction to Projections
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- Expressing a Projection on to a line as a Matrix Vector prod
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- Compositions of Linear Transformations 1
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- Compositions of Linear Transformations 2
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- Linear Algebra: Matrix Product Examples
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- Matrix Product Associativity
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- Distributive Property of Matrix Products
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- Linear Algebra: Introduction to the inverse of a function
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- Proof: Invertibility implies a unique solution to f(x)=y
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- Surjective (onto) and Injective (one-to-one) functions
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- Relating invertibility to being onto and one-to-one
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- Determining whether a transformation is onto
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- Linear Algebra: Exploring the solution set of Ax=b
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- Linear Algebra: Matrix condition for one-to-one trans
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- Linear Algebra: Simplifying conditions for invertibility
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- Linear Algebra: Showing that Inverses are Linear
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- Linear Algebra: Deriving a method for determining inverses
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- Linear Algebra: Example of Finding Matrix Inverse
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- Linear Algebra: Formula for 2x2 inverse
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- Linear Algebra: 3x3 Determinant
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- Linear Algebra: nxn Determinant
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- Linear Algebra: Determinants along other rows/cols
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- Linear Algebra: Rule of Sarrus of Determinants
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- Linear Algebra: Determinant when row multiplied by scalar
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- Linear Algebra: (correction) scalar muliplication of row
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- Linear Algebra: Determinant when row is added
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- Linear Algebra: Duplicate Row Determinant
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- Linear Algebra: Determinant after row operations
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- Linear Algebra: Upper Triangular Determinant
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- Linear Algebra: Simpler 4x4 determinant
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- Linear Algebra: Determinant and area of a parallelogram
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- Linear Algebra: Determinant as Scaling Factor
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- Linear Algebra: Transpose of a Matrix
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- Linear Algebra: Determinant of Transpose
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- Linear Algebra: Transpose of a Matrix Product
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- Linear Algebra: Transposes of sums and inverses
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- Linear Algebra: Transpose of a Vector
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- Linear Algebra: Rowspace and Left Nullspace
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- Lin Alg: Visualizations of Left Nullspace and Rowspace
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- Linear Algebra: Orthogonal Complements
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- Linear Algebra: Rank(A) = Rank(transpose of A)
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- Linear Algebra: dim(V) + dim(orthogonoal complelent of V)=n
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- Lin Alg: Representing vectors in Rn using subspace members
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- Lin Alg: Orthogonal Complement of the Orthogonal Complement
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- Lin Alg: Orthogonal Complement of the Nullspace
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- Lin Alg: Unique rowspace solution to Ax=b
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- Linear Alg: Rowspace Solution to Ax=b example
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- Linear Alg: Rowspace Solution to Ax=b example
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- Lin Alg: Showing that A-transpose x A is invertible
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- Linear Algebra: Projections onto Subspaces
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- Linear Alg: Visualizing a projection onto a plane
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- Lin Alg: A Projection onto a Subspace is a Linear Transforma
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- Linear Algebra: Subspace Projection Matrix Example
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- Lin Alg: Another Example of a Projection Matrix
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- Linear Alg: Projection is closest vector in subspace
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- Linear Algebra: Least Squares Approximation
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- Linear Algebra: Least Squares Examples
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- Linear Algebra: Another Least Squares Example
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- Linear Algebra: Coordinates with Respect to a Basis
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- Linear Algebra: Change of Basis Matrix
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- Lin Alg: Invertible Change of Basis Matrix
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- Lin Alg: Transformation Matrix with Respect to a Basis
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- Lin Alg: Alternate Basis Tranformation Matrix Example
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- Lin Alg: Alternate Basis Tranformation Matrix Example Part 2
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- Lin Alg: Changing coordinate systems to help find a transformation matrix
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- Linear Algebra: Introduction to Orthonormal Bases
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- Linear Algebra: Coordinates with respect to orthonormal bases
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- Lin Alg: Projections onto subspaces with orthonormal bases
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- Lin Alg: Finding projection onto subspace with orthonormal basis example
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- Lin Alg: Example using orthogonal change-of-basis matrix to find transformation matrix
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- Lin Alg: Orthogonal matrices preserve angles and lengths
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- Linear Algebra: The Gram-Schmidt Process
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- Linear Algebra: Gram-Schmidt Process Example
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- Linear Algebra: Gram-Schmidt example with 3 basis vectors
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- Linear Algebra: Introduction to Eigenvalues and Eigenvectors
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- Linear Algebra: Proof of formula for determining Eigenvalues
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- Linear Algebra: Example solving for the eigenvalues of a 2x2 matrix
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- Linear Algebra: Finding Eigenvectors and Eigenspaces example
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- Linear Algebra: Eigenvalues of a 3x3 matrix
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- Linear Algebra: Eigenvectors and Eigenspaces for a 3x3 matrix
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- Linear Algebra: Showing that an eigenbasis makes for good coordinate systems
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- Vector Triple Product Expansion (very optional)
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- Normal vector from plane equation
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- Point distance to plane
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- Distance Between Planes
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