I wanna know about a theorem of linear algebra

Question

Is there some theorem that says that if the number of vectors of a base $S$ is less than the dimension of a vector space $V$ than it cannot span the vector space?

Answer 1

I suuppose that you're after this:

Theorem: If $V$ is a vector space, if $S\subset V$ and if $\#S<\dim V$, then $\operatorname{span}(S)\neq V$.

Yes, this is a standard Linear Algebra theorem.

Proof: I will assume that $\dim V<+\infty$. If $S\subset V$ and $\operatorname{span}(S)=V$, then, if $S'$ is a maximal linearly independent subset of $S$ such that $\operatorname(S')=V$ (it must exist, since $S$ is finite), $S'$ is a basis of $V$. Therefore$$\#S\geqslant\#S'=\dim V.$$