Let $\mathrm{W}$ be the vector subspace of $\mathrm{V}$ that is spanned by the base $\mathrm{B}$.
If I can show that $\dim{V} = \dim{W}$, does this imply that $\mathrm{B}$ spans $\mathrm{V}$, and so, that $\mathrm{B}$ is a base of $\mathrm{V}$?
My intuition says so. I don't need a formal proof, just to know if this is true.
Yes, this is true. If W is a subspace of V and they have the same dimension then they are the same vector space.