5.8 Change of basis
Given an n-dimensional vector space V and a linear transformation V → V, we can choose different bases for V. Let’s call them 
and 
. With respect to X, the linear transformation has a matrix AX. With respect to Y it has a different matrix AY, but they both implement the same linear transformation. If v is a vector in V, it has one set of coordinates corresponding to X and another set for Y.
How do we change from one set of coordinates for v to the other? What is the relationship between AX and AY?
This topic can be confusing because it is easy to get yourself working with the wrong matrix or its inverse. If you start with the basic idea of what you are trying to represent, the rest takes care of itself.
Let’s look at an example that demonstrates how choice of basis can make things easier.
Suppose you had city blocks laid out in a nice rectilinear pattern so that we can use the basis vectors x1 = (1, 0) and x2 = (0,2) to...
