One thing to keep in mind here is that you're just trying to find the coordinates of a point that is given to you in the coordinate system, so the point itself does not change.
A point in the coordinate system has the coordinates (x, y) expressed as the linear combination of the basis vectors:
In the coordinate system, a point (u, v) will have coordinates of the expressed as linear combinations of the basis vectors:
These are the same point (we are just changing the coordinate system), so:
, where and
In order to get by itself, multiply both sides of the equation on the left by to get:
So, the matrix that changes base from to is .