Khan Academy: Change of Basis: Alternate Basis Transformation Matrix Example

Showing that the transformation matrix with respect to basis B actually works. Brief point on why someone would want to operate in a different basis to begin with.

Understanding alternate coordinate systems.

A video lesson explaining how to convert between traditional Cartesian coordinates and coordinates with respect to a given basis by using an invertible change of basis matrix. Includes two concrete examples in R2. [13:34]

Video reviews from previous videos that a function is invertible if and only if it is onto and one-to-one. Shows how to determine if a linear transformation is onto by showing that the columns of the transformation matrix span the...

Video shows that the transformation matrix of the composition of linear functions is the product of the transformation matrices of each linear transformation in the composition. [16:30]

Video explains what a basis for a vector subspace is and its criteria: the vectors must span the subspace and be linearly independent. Also gives an informal definition of basis as the minimum set of vectors that spans the subspace....

Video reviewing the process of finding eigenvalues and then using that process to find the eigenvalues for a specific 3x3 matrix. Part of the process requires finding the determinant of a matrix and solving a cubic equation. [14:08]

Video first shows an intuitive sense of what composing linear transformations does and the notation used. Shows that the composition of linear transformations is also a linear transformation. [12:21]

Video defines the column space as a linear combination of the column vectors and relates it to span. Shows that the column space is a valid subspace. Shows several ways to interpret a column space. This video also appears in the strand...

A video lesson that goes through four example problems dealing with conic sections. Topics covered include: finding the vertex of a parabola from its equation; finding the asymptotes of a hyperbola and then sketching its graph;...