Change of Basis, Coordinate Vectors, Mapping, Transition Matrix - Linear Algebra

What is the meaning and purpose of change of basis and why it is useful using an example, defining it as a linear transformation using matrix notation. Coordinate vectors (including unit vectors) relative to a basis are defined, and also the transition matrix formula.

0:00 Coordinate vectors, mapping to R^n
2:13 Change of basis
9:24 Transition matrix
16:32 Calculating the transition matrix between bases for R^n