Linear Transformation R2->R3 with 'zero' vector

[A440]

Homework Statement

Is T(X,Y)->(X,Y,1) a linear transformation? where X and Y are defined R2 column vectors.

Homework Equations

Attempt to prove T(cX+Y)=cT(X)+T(Y)
Consider T(cx1+y1,cx2+y2)->(cx1+y1,cx2+y2,1)

The Attempt at a Solution

RS=cT(x1,y1)+T(x2,y2)->c(x1,y1,1)+(x2,y2,1)
=(cx1+y1,cx2+y2,1)+(0,0,c)
=(cx1+y1,cx2+y2,1)+c(0,0,1)
The 'zero' vector: T(0,0)->(0,0,1)
therefore T(cx1+y1,cx2+y2)->(cx1+y1,cx2+y2,1)
and T(X,Y)->(X,Y,1) is a linear transformation.

 

[xaos]

zero vectors are only of the form (0,0) and (0,0,0).