Khan Academy: Using Orthogonal Change of Basis Matrix

Video defining an orthogonal matrix as a square matrix C whose columns form an orthonormal set. Shows that C will be invertible and that multiplying C transpose by C will result in the identity matrix. The result that C transpose equals...

A video lesson explaining how to change from coordinates with respect to a given basis to traditional Cartesian coordinates and vice versa by using a change of basis matrix. Includes concrete examples in R3. [17:55]

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]

Using an invertible change of basis matrix to go between different coordinate systems.

Video continuing the example from the previous video where a transformation matrix with respect to an alternate basis was found given a transformation matrix with respect to the standard basis, an alternate basis and a change of basis...

Video reviewing the relationship between transformation matrices with respect to different bases. Addresses manipulating the equation to write it in terms of the transformation matrix with respect to the standard basis. The relationship...

Video discussing transformation matrices and how a vector is able to be mapped to the same resulting vector by using a transformation matrix with respect to the standard basis as well as one with respect to a different basis. Reviews how...

Video showing how to change coordinate systems to make finding the transformation matrix with respect to the standard basis easier. First, a specific linear transformation is defined and the regular method for constructing the...

Video first introduces the identity matrix and shows that when any vector is multiplied by the identity matrix the result is the original vector. Also shows that the columns of the identity matrix are called the standard basis for R^n....

Video first introduces the identity matrix and shows that when any vector is multiplied by the identity matrix the result is the original vector. Also shows that the columns of the identity matrix are called the standard basis for R^n....