# Subspace

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

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 $\displaystyle L=\{t\bold{v}:t \in F\} \subseteq V$ where V is a vector space over a field F and $\displaystyle \bold{v} \in V$.

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

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