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Thread: Subspace

  1. #1
    Senior Member slevvio's Avatar
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    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.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by slevvio View Post
    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 $\displaystyle M=L$.
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  3. #3
    Senior Member slevvio's Avatar
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    Thanks I see this now
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