matt grime

Changing basis changes T. I assume that when b1=(1,1) this is b1 relative to the standard basis e1 and e2 (well, 'standard' doesn't matter - it's just another basis).

There are several ways to do this. You could just try to work out what Te1 and Te2 are from the action of T on b1 and b2 (e.g. note that e2=b2-b1, so you know what Te2 is now, but don't forget to express Te2 in terms of e1 and e2.).