Theorems from Basis and dimension

If $B=\{u_1,\ldots,u_n\}$ is a basis for $V$, then every vector $v\in V$ has a unique representation

Let $S$ and $L$ be finite subsets of a vector space $V$. If $\operatorname{span}(S)=V$ and $L$ is linearly independent, then

Any two finite bases of a vector space contain the same number of vectors.

Let $V$ be finite-dimensional.

1. If a set $S$ of $n$ vectors spans $V$, then $n\geq\dim(V)$. If $n=\dim(V)$, then $S$ is a basis.
2. If a set $S$ of $m$ vectors is linearly independent in $V$, then $m\leq\dim(V)$. If $m=\dim(V)$, then $S$ is a basis.