# Change of basis

These are some questions I wrote when thinking about how someone might go about trying to understand the idea of changing a basis.  The idea is that you can work through the questions yourself.  I'm afraid there's no fancy online interactive way of checking with this one!  Just work through the various steps for yourself, spending as much time on them as you need.

Let's think about R2, and two bases for R2.  One is the set {(1 0), (0 1)}, which I'll call A, and the other {(1 0), (1 1)}, which I'll call B.  (Hope it's clear what I mean -- I can't easily write vectors vertically here!)

- Happy that these are both bases for R2?  Can you prove it?

A is a basis, so each point can be written in the form a(1 0) + a'(0 1) for suitable a and a'.  OK?  I'll call (a,a') the coordinates of the point with respect to the basis A.  And similarly for B.  Is that clear?

- Take the point with coordinates (2,3) with respect to A.  Can you write down the coordinates of that same point with respect to B?  (Hint: if it has coordinates (2,3) with respect to A, which vector is it?  How do you write that in terms of basis vectors from B?)

- Take the point with coordinates (-1,7) with respect to B.  What are its coordinates with respect to A?

- Take the point with coordinates (z,w) with respect to A.  What are its coordinates with respect to B?  (Try some more numerical examples if you like, before moving to the algebra.)

- Take the point with coordinates (x,y) with respect to B.  What are its coordinates with respect to A?

- Take the point with coordinates (x,y) with respect to B.  I want to find its coordinates with respect to A by multiplying a matrix P by the vector (x y) (written vertically).  What matrix P do I need?  (This is a convenient way of recording the information about how to change from one set of coordinates to the other.)

- Take the point with coordinates (z,w) with respect to A.  How can I find its coordinates with respect to B by somehow using the matrix P?

Now let's think about a linear map T: R2 --> R2.  I'm going to define it by saying that if a point has coordinates (x,y) with respect to A, then T sends it to the point with coordinates (2x+3y, x-2y) with respect to A.  Does that make sense?

- What's the matrix for T with respect to the basis A?

Now you're going to find the matrix for T with respect to the basis B.  OK?  So I want to give you a point by telling you its coordinates with respect to B (say (x,y)), and I want you to tell me where T sends it, by telling me another point with coordinates with respect to B.  (If you like, imagine that I only know about coordinates with respect to B, but I still want to know what T does.)  Make sense?  How are you going to do this?

- I suggest you take (x,y), the coordinates with respect to B that I've given you, and find the coordinates of that point with respect to A (because that's the only way you know how to apply T).

- Now I suggest you apply T, because you can do that if you know the coordinates of the point with respect to A.

- But now your output of that map is a point with coordinates with respect to A, so please transform them back to coordinates with respect to B before telling me!

- Can you write how you'd do the above three steps using the matrices P and T?