Khan Academy: Linear Algebra: 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...

Video shows that the distributive property is true for multiplying matrices. [9:52]

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 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 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 creating a basis out of n linearly independent eigenvectors. Reviews transformations and change of basis matrices. Shows that when the eigenbasis is used as an alternate coordinate system, the transformation is a diagonal matrix...

Video starts with reviewing matrix notation. Defines multiplying a matrix by a vector and shows examples. Shows how the matrix-vector products can be interpreted as the dot product of column vectors with a matrix or as a weighted linear...

Video demonstrating that the area of a parallelogram created by graphing the column vectors of a 2x2 matrix is the equivalent to the absolute value of the determinant of the matrix. Pythagorean theorem, in conjunction with calculating a...