T<x,y> = <x+y,2xy>
1 Answer

This function is not a linear operator. Linear opeartor must satisfy condition F(a+b) = F(a) + F(b).
If we choose a=(1,0), b=(0,1), then
T(a) = (1,0)
T(b) = (1,0)
T(a+b) = T(1,1) = (2,2)
T(a) + T(b) = (1,0) + (1,0) = (2,0)
We see that T(a+b) is not equal to T(a)+T(b).