T R3 R3 Txyz x1 y1 z1 Is it a linear transformationSolutio

T: R³ -> R³, T(x,y,z) = (x+1, y+1, z+1)

Is it a linear transformation?

Solution

Consider two vectors x1,y1,z1 and x2,y2, z2

T(x1,y1,z1) =(x1+1, y1+1, z1+1)

T(x2,y2,z2) = (x2+1, y2+1, z2+1)

T(x1+x2, y1+y2, z1+z2) = (x1+x2+1,y1+y2+1, z1+z2+1)

But T(x1,y1,z1)+T(x2,y2,z2) = (x1+1, y1+1, z1+1)+(x2+1, y2+1, z2+1)

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

Hence not linear