what you need to do is a two-step process:
step one: find the "change of basis matrix" P. to be explicit, the P we are going to use is the one that changes B-coordinates to standard coordinates.
since [1,0]B = 1(2,0) + 0(1,1) = (2,0), we have:
which makes it clear P is of the form:
similarly, since [0,1]B = (1,1), we have that:
so the overall procedure is this:
B-coordinate input ---> change to standard coordinates --->apply standard T (matrix A) ---> change back to B-coordinates.
to change back to B--coordinates, we need to use P-1, which is:
step 2: find the desired matrix for T, which is:
let's verify that this actually works:
we know that T(x,y) = (x+2y,y).
writing (x,y) in B-coordinates, we get: (x,y) = x(1,0) + y(0,1) = x[1/2,0]B + y[-1/2,1]B = [(x-y)/2,y]B.
applying our [T]B to [(x-y)/2,y]B (to get [T(x,y)]B), we obtain:
[T]B([(x-y)/2,y]B) = [(x-y)/2+y,y]B = [(x+y)/2,y]B.
changing this back to standard coordinates, we have:
[(x+y)/2,y]B = ((x+y)/2)[1,0]B + y[0,1]B = ((x+y)/2)(2,0) + y(1,1) = (x+y,0) + (y,y) = (x+2y,y) = T(x,y).