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Math Help - Subspace

  1. #1
    Senior Member slevvio's Avatar
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    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.
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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 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.
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  3. #3
    Senior Member slevvio's Avatar
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    Thanks I see this now
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