Let  V = \mathbb {R} ^{3} , define f, g, h in V* by f(x,y,z) = x-2y ,g(x,y,z)=x+y+z , h(x,y,z)=y-3z.

Prove that {f,g,h} is a bsis for V*, then find a basis for V.

My work so far:

I first show that {f,g,h} is a linearly independent set.

Suppose that a(x-2y)+b(x+y+z)+c(y-3z) = 0, then I solve a=b=c=0, therefore the set is linearly independent.

Now, I want to show that the set generates V*, so really, it needs to span all real number entries in (x,y,z). I can't quite remember how to do that, any hints?

Thanks.