Course videos

Introduction to multidimensional vectors

Multidimensional Vectors Examples

Parametric Representations Of Lines

Linear Combinations and Span

Matrix Vector Products

Introduction to the Null Space of a Matrix

Null Space 2: Calculating the null space of a matrix

Null Space 3: Relation to Linear Independence

Column Space of a Matrix

Null Space and Column Space Basis

Visualizing a Column Space as a Plane in R3

Proof: Any subspace basis has same number of elements

Dimension of the Null Space or Nullity

Dimension of the Column Space or Rank

Showing relation between basis cols and pivot cols

Showing that the candidate basis does span C(A)

Linear Subspaces

Basis of a Subspace

Introduction to Linear Independence

More on linear independence

Span and Linear Independence Example

Orthogonal Complements

PROOF : dim(V) + dim(orthogonoal complelent of V)=n

Representing vectors in Rn using subspace members

Orthogonal Complement of the Orthogonal Complement

Orthogonal Complement of the Nullspace

Unique rowspace solution to Ax=b

Rowspace Solution to Ax=b example

Projections onto Subspaces

Visualizing a projection onto a plane

PROOF : A Projection onto a Subspace is a Linear Transformation

Subspace Projection Matrix Example

Another Example of a Projection Matrix

Coordinates with Respect to a Basis

Change of Basis Matrix

Invertible Change of Basis Matrix

Transformation Matrix with Respect to a Basis

Alternate Basis Tranformation Matrix Example

Alternate Basis Tranformation Matrix Example Part 2

Changing coordinate systems to help find a transformation matrix

Introduction to Orthonormal Bases

Coordinates with respect to orthonormal bases

Projections onto subspaces with orthonormal bases

Finding projection onto subspace with orthonormal basis example

Example using orthogonal change-of-basis matrix to find transformation matrix

Orthogonal matrices preserve angles and lengths

The Gram-Schmidt Process

Gram-Schmidt Process Example

Gram-Schmidt example with 3 basis vectors

Introduction to Eigenvalues and Eigenvectors

Proof of formula for determining Eigenvalues

Example solving for the eigenvalues of a 2x2 matrix

Finding Eigenvectors and Eigenspaces example

Eigenvalues of a 3x3 matrix

Eigenvectors and Eigenspaces for a 3x3 matrix

Showing that an eigenbasis makes for good coordinate systems

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