Find Tv by using the standard matrix and the matrix relative

Find T(v) by using the standard matrix and the matrix relative to B and B\'. T: R^2 rightarrow R^3, T(x, y) = (x + y, x, y), v = (9, 7), B = {(1, -1), (0, 1)}, B\' = {(1, 1, 0), (0, 1, 1), (1, 0, 1)} the standard matrix T(v) = (16, 9, 7) the matrix relative to B and B\' (in terms of the standard basis) T(v) =

Solution

we can write (16,9,7) using B\' as
x(1,1,0)+y(0,1,1)+z(1,0,1)=(16,9,7)
(x,x,0)+(0,y,y)+(z,0,z)=(16,9,7)
(x+z,x+y,y+z)=(16,9,7)
so we get three equations
x+z=16...(i)
x+y=9...(ii)
y+z=7...(iii)

solve (i) for x
x+z=16
x=16-z...(iv)
plug into (ii)
x+y=9
16-z+y=9
-z+y=-7...(v)

add (iii) and (v)
2y=0
y=0

plug into (ii)
x+y=9
x+0=9
x=9

plug into (i)
x+z=16
9+z=16
z=7
plug values of x, y and z into x(1,1,0)+y(0,1,1)+z(1,0,1)
Hence answer will be 9(1,1,0)+0(0,1,1)+7(1,0,1)