# Subspace

• Nov 1st 2008, 10:26 AM
slevvio
Subspace
Show that L and {0} are the only subspaces of $L=\{t\bold{v}:t \in F\} \subseteq V$ where V is a vector space over a field F and $\bold{v} \in V$.

I don't know how to show they are the only subspaces of L and any help would be appreciated, thank you.
• Nov 1st 2008, 10:41 AM
Laurent
Quote:

Originally Posted by slevvio
Show that L and {0} are the only subspaces of $L=\{t\bold{v}:t \in F\} \subseteq V$ where V is a vector space over a field F and $\bold{v} \in V$.

I don't know how to show they are the only subspaces of L and any help would be appreciated, thank you.

If $M$ is a subspace of $L$, either it is equal to $\{0\}$ or it contains a non-zero vector $\bold{w}$. But if $\bold{0}\neq \bold{w}\in M\subset L$, there is a non-zero $t\in F$ such that $\bold{w}=t\bold{v}$, hence $\bold{v}=\frac{1}{t}\bold{w}\in M$, and as a consequence every scalar multiple of $\bold{v}$ is in $M$: $L\subset M$ and finally $M=L$.
• Nov 2nd 2008, 03:34 AM
slevvio
Thanks I see this now