Determining if it is a linear transformation?

Question

I'm having a hard time determining if $Q\colon\mathbb{R}^2\to\mathbb{R}^3$ defined by $Q(X,Y)=(3Y,0,X+Y)$ is a linear transformation. I tried using the method I learned: $Q(U+V)=Q(U)+Q(V)$.

Answer 1

If a transformation Q is linear; then it satisfies two conditions: 1) Q(U+V)=Q(U)+Q(V)
2) Q(aU)=aQ(U), where a is some scalar

Q(x,y)=(3y,0,x+y) , Q(z,t)=(3t,0,z+t)
Q(x,y)+Q(z,t)=(3y,0,x+y)+(3t,0,z+t)=(3(y+t),0,x+y+z+t)=Q(x+z,y+t)
Q(ax,ay)=(3ay,0,ax+ay)=a(3y,0,x+y)=aQ(x,y)

So indeed, Q is a linear transformation

Answer 2

Another way to see $Q$ is indeed a linear transformation is to look at its effect on a general straight line through the origin, which has parametric equation $(at, bt, ct)$ for some constants $a,b,c$.

$Q(at,bt,ct) = (3bt, 0, (a+b)t) = (a't, b't, c't)$

where $a'=3b$, $b'=0$, $c'=a+b$.

In other words, $Q$ maps each straight line through the origin to another straight line through the origin - which is an alternative definition of a linear transformation.