To be a subspace it must confirm three axioms:
Containing the zero vector, closure under addition and closure under scalar multiplication
a) (0,0,0) is contained since (0,2*0,3*0)=(0,0,0)
b) Let (y,2y,3y) be a vector, then (x,2x,3x)+(y+2y+3y)=(x+y,2(x+y),3(x+y)) => closure under addition
c) Let c be a constant, then c*(x,2x,3x)=(cx, 2(cx), 3(cx)) => closure under scalar multiplication
So it's a subspace. I hope this is what you wanted and that will help you!