Is this transform linear: $T(a,b,c) = (\ a+2,\ \ b-c,\ \ 5c\ )$

Question

This is from chapter on Matrix Linear Transformations in Linear algebra textbook.

Is the the following transform linear:

$$T(a,b,c) = (\ a+2,\ \ b-c,\ \ 5c\ )$$

Book says:

$$T(a=0,b=0,c=0) = (\ 2,\ \ 0,\ \ 0\ ) \ne (0,0,0)$$

Therefore not linear.

My question is: What linearity property is this proving?

I have only two listed in this chapter:

$$T(v+w)=T(v) + T(w)$$

$$T(\alpha v) = \alpha T(v)$$

Maybe they left out a property need to prove this?

Answer 1

Yes, from the first property we have $$T(0)= T(0+0)=T(0)+T(0)\implies T(0)=0$$

So if $T(0)\ne 0$ then this transformation is not linear.