Alternate coordinate systems (bases) videos

Orthogonal complements

dim(v) + dim(orthogonal complement of v) = n

Orthogonal complement of the orthogonal complement

Representing vectors in rn using subspace members

Orthogonal complement of the nullspace

Unique rowspace solution to Ax = b

Rowspace solution to Ax = b example

Visualizing a projection onto a plane

Projections onto subspaces

A projection onto a subspace is a linear transformation

Subspace projection matrix example

Projection is closest vector in subspace

Another example of a projection matrix

Least squares approximation

Another least squares example

Least squares examples

Coordinates with respect to a basis

Change of basis matrix

Invertible change of basis matrix

Alternate basis transformation matrix example part 2

Transformation matrix with respect to a basis

Alternate basis transformation matrix example

Changing coordinate systems to help find a transformation matrix

Introduction to orthonormal bases

Coordinates with respect to orthonormal bases

Finding projection onto subspace with orthonormal basis example

Projections onto subspaces with orthonormal bases

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

The Gram-Schmidt process

Orthogonal matrices preserve angles and lengths

Gram-Schmidt example with 3 basis vectors

Gram-Schmidt process example

Proof of formula for determining eigenvalues

Introduction to eigenvalues and eigenvectors

Example solving for the eigenvalues of a 2x2 matrix

Eigenvectors and eigenspaces for a 3x3 matrix

Eigenvalues of a 3x3 matrix

Finding eigenvectors and eigenspaces example

Showing that an eigenbasis makes for good coordinate systems