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Math Help - Property of subspace

  1. #1
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    Property of subspace

    Hey,

    Consider a subset W of a vectorspace V. If I can show that W is a subspace of V would that imply that W itself is a vectorspace?

    Or do I have to go the other way and first show that the subset W fullfills the conditions of a vectorspace and therefore is a subspace.

    Cole someone kindly clarify.

    Thanks.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by surjective View Post
    Hey,

    Consider a subset W of a vectorspace V. If I can show that W is a subspace of V would that imply that W itself is a vectorspace?

    Or do I have to go the other way and first show that the subset W fullfills the conditions of a vectorspace and therefore is a subspace.

    Cole someone kindly clarify.

    Thanks.
    By definition a subspace of a vector space is a subset of a vector space which is a vector space with the same operations as the ambient space. That said, since most of the axioms of a vector space are 'inherited' from the ambient space it suffices to prove closure under linear combinations to prove a subset of a vector space is a subspace.
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  3. #3
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    properties of vectorspace

    Hey,

    Thanks. So if I can show that a non-empty subset is closed under addition and scalar multiplication then the subset would be a subspace and then also a vectorspace. right?

    Thanks
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by surjective View Post
    Hey,

    Thanks. So if I can show that a non-empty subset is closed under addition and scalar multiplication then the subset would be a subspace and then also a vectorspace. right?

    Thanks
    Indeed.
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