Khan Academy: Linear Algebra: Invertible Change of Basis Matrix

Video first reviews what it means for a function to be invertible. Shows that if a function is invertible then there is a unique solution to f(x) = y. Also shows that if there is a unique solution to the equation f(x) = y for all y then...

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

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...

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...

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]

Video first explains and gives examples of identity matrices. Then explains what an inverse matrix is and shows how to calculate the inverse of a 2x2 matrix. [14:14]

Video example of finding the inverse of a 3 by 3 matrix. Includes using expansion by minors to determine the determinant and finding the matrix of cofactors and adjugate or adjoint matrix. This video also appears in the strand Algebra:...

Video shows how to determine the inverse of a 3 by 3 matrix using the Gauss-Jordan elimination method. This video also appears in the strand Algebra: Matrices. [13:35]