Verify that T R3 rightarrow R given Tx y z x y 0 2x z is

Verify that T: R^3 rightarrow R given T(x y z) = (x + y 0 2x - z) is a linear transformation. (a). T((x_1 y_1 z_1) + (x_2 y_2 z_2))

Solution

T( x y z) = (x + y, 0 , 2x - z)

Prperties of linear transformation:

vector addition : T( x1 y1 z1) = (x1 + y1, 0 , 2x1 - z1)

T( x2 y2 z2) = (x2 + y2, 0 , 2x2 - z2)

T(x1 y1 z1 ) + T(x2 y2 z2) = ( x1+ x2 +y1+y2, 0 , 2(x1+x2) - (z1 +z2) )

= T( x1+x2 , y1+y2 , z1+z2)

b) scalar multiplication : T(cx cy cz) = ( cx + cy , 0 , 2xc -cz)

= c( x +y , 0 , 2x - z)

T(cX) = cT(X)

It follows the prpoerties of linear transformation.Hence it is a linear transformation