Find the ker (T) for T(x,y) = (3x, x-y, y)

find the ker (T) for T(x,y) = (3x, x-y, y)

The kernel is the set {(0,0)}. Remember that the kernel is defined to the set of all vectors that map to the zero vector under T. That is, we need numbers x and y that satisfy the following system:

3x = 0
x-y = 0
y = 0

It is easy to see that (0,0) is the only ordered pair satisfying this system.