Change of Basis

If is a vector space with basis and is a basis with basis , and is linear, then there exists a matrix such that . The th column consists of the coordinate representation of the th vector in 's basis expressed with respect to basis .
This is just a more general form of how we've been talking about transformations: before, we weren't including bases, but now we're explicitly stating that the transformation is from one basis to another.

If and are two bases of vector space , then there exists a transition matrix (read: " from to ") such that for all .

If is a vector space with bases and and is linear, then .

So how do we change basis?
We can change basis by multiplying by the transition matrix! If we have a vector in basis (denoted ""), we can find its representation in basis by multiplying by . So, .
That's the same thing as a 2d transformation! We use the original coordinate system, and transform the basis vectors to the other system's coordinates. So, changing basis vectors is matrix-vector multiplication, where the matrices' columns are how we would express the "other" basis vectors!

Change of Basis for Transformations
We can change the basis that a transformation is represented in the same way we would change basis for vectors:

If you read this out loud, it'll make sense: " from base to is equal to the transition matrix from to times from to times the transition matrix from to ."

We'll see an incredibly powerful application of this in the Diagonalization section.