This may be a dumb question, but if I have three linearly independente vectors in $R^3$, will it always span $R^3$
I'm asking this because it's hard to visualize this for every vector. I can imagine particular cases but not the entire case. Is there a proof for it, or is this na observation?
Yes. There are a few ways to see this. The simplest is to note that if it didn't, there would be a fourth vector not in the span and therefore linearly independent of the other three, so you'd have four mutually linearly independent vectors, which is impossible in $\mathbb{R}^3$