Show that the transformation T defined by T x1 x2 4x1 2x2

Show that the transformation T defined by T (x_1, x_2) = (4x_1 - 2x_2, 3|x_2|) is not linear.

If T were linear, then:

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

Let x1 = 1 , x_2 = -1

y1 = 1 , y2 = 1

T(x1,x2)= T(1,-1) = (4 *1 - 2*(-1) , 3 |-1| ) = ( 6, 3)

T(y1,y2) = T(1,1) = ( 4*1 - 2*1 , 3|1|) = (2, 3)

T((x1,x2) + (y1,y2)) = T(2,0) = ( 4*2 - 2*0 , 3|0| ) = ( 8, 0 )

Now comparing

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

( 6,3 ) + (2,3) = (8,0)

But these are not equal , so T is not linear.