Let b1 = [1 - 3] and b2 = [2 - 5]. the set b = {b1, b2} is a basis for r2. let t : r2 rightarrow r2 is a linear transformation such that t(b1) = 4b1 + 4b2 and t(b2) = 8b1 + 5b2. then the matrix of t relative to the basis b is [t]b = [ ], and the matrix of t relative to the standard basis e for r2 is [t]e = [].

So the first thing we do is to multiply the top equation by 3. the reason i want to do this is i notice there will be a 6y on both equations and can possibly add/subtract them. so we get 6y = -3x + 27. adding that and the second equation we get 6x = 27-15 which is 12. divide by 6 and we get 2. ∴ x = 2. substituting 2 into the first equation we get 2y = -2 + 9 which is 7. divide by 2 and we get 7/2. ∴ y = 7/2, x = 2.