Determine if the transformations are linear operators?

a) T(x, y) = (2x, y)


b.) T(x, y) = (-y, x)


c) T(x, y) = (x+1, y)





How would you determine if these are linear operators? I know that these conditions must satisfy: T(u+v) = T(u) + T(v),


T(ku) = kT(u). I am supposed to check both properties individually.


But I do not understand how to apply these properties.

Determine if the transformations are linear operators?
u1 =(x1,y1)


u2 = (x2,y2)


T(u1+u2) = T(u1) +T(u2) ?


u1+u2 = (x1+x2, y1+y2)


T(u1+u2) = T(x1+x2, y1+y2)


by definition: T(x1+x2, y1+y2) =


(2(x1+x2), y1+y2) = (2x1+2x2, y1+y2)


T(u1) =T(x1,y1) = (2x1, y1)


T(u2) = T(x2, y2) = (2x2, y2)


T(u1) +T(u2) = (2x1, y1) + (2x2, y2) =


(2x1+2x2, y1 +y2) .





Do the same to T(ku) to show if it is


kT(u)
Reply:pick two numerical values for x and two for y, substitute the values in the transforms (conditions) and then substitute the valued conditions in the two checks.