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Math Help - Subspaces..

  1. #1
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    Subspaces..

    a)
    Let T : \Re^n \rightarrow \Re^m be a linear map. Define  D= \{ x \in \Re^n \mid T x=y \} , for
    some  y \in \Re^m, where  y \neq 0 . Is D a subspace? If not explain.

    b)
    Let H and V be subspaces of  \Re^n . Is  H \cap V a subspace? Is  H \cup V a subspace? Explain your answers.

    I don't think D is a subspace...but I don't know why.
    What does  H \cap V and  H \cup V represent? There is nothing in part a of the problem that talks about H and V...
    Lastly, what exactly are subspaces?

    Step by step process with some explanation would be really nice!

    Thank you for your time!
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Linnus View Post
    a)
    Let T : \Re^n \rightarrow \Re^m be a linear map. Define  D= \{ x \in \Re^n \mid T x=y \} , for
    some  y \in \Re^m, where  y \neq 0 . Is D a subspace? If not explain.
    no, because if y \neq 0, then 0 \notin D.

    b)
    Let H and V be subspaces of  \Re^n . Is  H \cap V a subspace?
    yes, just check the conditions required for a set to be a subspace.

    Is  H \cup V a subspace? Explain your answers.
    it's a subspace iff either H \subseteq V or V \subseteq H. because otherwise choose h \in H - V, \ v \in V - H. then: h + v \notin H \cup V.
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