for the monomial basis is . so whatever matrix represents f in this basis, we have:
and
now the column (a,c) is f(1,0) = f(1+0x) = f(1). so let's find this first. remember, matrix multiplication is linear. this means that 9f(1,4) + 4f(-2,-9)
= f(9,36) + f(-8,-36) = f(1,0).
but f(1,4) = (1,-2), and f(-2,-9) = (2,4). so 9f(1,4) + 4f(-2,-9) = 9(1,-2) + 4(2,4) = (9,-18) + (8,16) = (17,-2). so a = 17, c = -2.
now let's find the second column (b,d), which is f(0,1) = f(0+1x) = f(x).
-2f(1,4) - f(-2,-9) = f(-2,-8) + f(2,9) = f(0,1). so f(0,1) = -2(1,-2) - (2,4) = (-2,4) + (-2,-4) = (-4,0). so b = -4, d = 0.
finally, let's check:
and
so that is indeed the correct matrix.