Let L R2 rightarrow R2 be defined by Show that L is not a li

Let L: R^2 rightarrow R^2 be defined by Show that L is not a linear transformation by finding vectors x, and,y such that L(x + y) notequalto L(x) + L(y): Prove your answer by calculating (for your choice of x, y):

Solution

Let x = ( 1 , 2)^T and y = ( 2, 0)^T . Then, x + y = (3, 2)^T.

Also, L ( x + y) = L (3, 2 )^T = ( 3*2, -3)^T = ( 6 , -3)^T and L( x ) + L( y ) = L( 1, 2)^T + L ( 2, 0)^T = ( 1*2, -1)^T + ( 2*0, -2)^T = (2, -1)^T + ( 0, -2)^T = ( 2 , -3)^T (3 , 2)^T. Thus, L ( x + y) L( x ) + L( y ). Hence L is not a linear transformation, as it is not closed under addition.