Let T(x,y)=(ax+by, cx+dy), where a, b, c, d are constants.
When (x,y)=(1,0)⇒(1,1)=(ax+by, cx+dy). Therefore a=1 and c=1.
When (x,y)=(0,1)⇒(-1,1)=(ax+by, cx+d). Therefore b=-1 and d=1.
Therefore T(x,y)=(x-y, x+y).
T(1,3)=(-2, 4) and T(-9,1)=(-10, -8).
T can be represented in matrix form:
⎡ 1 -1⎤
⎣ 1 1⎦
So
⎡ 1 -1⎤⎡x⎤=⎡x-y ⎤
⎣ 1 1⎦⎣y⎦ ⎣x+y⎦