Can 3 Vectors ever span $\mathbb{R}^2$?

Question

If I had: $(0,1)$, $(1,2)$, and $(1,3)$, couldn’t I always set the constant that multiplies the third vector equal to 0, and then the other two vectors could successfully span all of $\mathbb{R}^2$ since they aren’t scalar multiples of each other?

That makes sense to me when I visualize that in my head, but is that wrong? Could that work?

Answer 1

Of course, because it only takes two vectors to span $\Bbb R^2$.