The question reads:
a) Let v1 = (1, 2,−1), v2 = (0, 1, 1), v = (2, 1,−5). Solve the equation av1 + bv2 = v. Show that your solution is unique.
b) What does the uniqueness of your solution say about v, v1, v2? What does the existence of your solution say about v, v1, v2?
I have solved the equation and found that a=2 and b=-3, which are unique solutions. What I can draw from part b is that this tells me v is a linear combination of v1 and v2, but I suspect the question is asking for more than this. My only other suspicion is that if the equation has unique solutions then v is linearly independent, but I'm not sure if that's necessarily true.
Is there anything else I can conclude from the equation having unique solutions?
$v$ being expressible as a sum of the others means by definition it is not linearly independent from the pair (though it would be L.I from one or the other as that just says not a scalar multiple).
You get that $v$ is in the subspace whose basis is $\{v_1,v_2\}$