Can a vector space have multiple spanning sets? Maybe this is obvious, but can a vector space have multiple spanning sets or is there only a single spanning set for every vector space?
Thanks

If $S$ spans $V$, so does $\{\alpha v:v\in S\}$ for every non-zero scalar $\alpha$. Unless $V$ is over the two-element field, this automatically gives you more than one spanning set. More trivially, both $V$ and $V\setminus\{\vec 0\}$ span $V$.
Less trivially, if $\dim V\ge 2$, let $u$ and $v$ be linearly independent vectors in $V$. Extend $\{u,v\}$ to a basis $B$ for $V$ containing $u$ and $v$. Then $B\cup\{u+v\}$ is a spanning set different from $B$.
