Results 1 to 4 of 4

Thread: Vector space and maximal proper subspace

  1. #1
    Senior Member vincisonfire's Avatar
    Joined
    Oct 2008
    From
    Sainte-Flavie
    Posts
    468
    Thanks
    2
    Awards
    1

    Vector space and maximal proper subspace

    Suppose that V is a vector space over F and $\displaystyle \vec{v} \in V $ is any nonzero vector.
    Show that there is a subspace W of V maximal subject to the condition
    that $\displaystyle \vec{v} \notin W $ ; show that any such W is a maximal proper subspace of V .
    (Do NOT assume that V is finite-dimensional.)
    For the first part I thought that I could say :
    1. Prove that we can exchange one of the vector in basis of V by $\displaystyle \vec{v}$.
    2. Prove that the basis (B) is still one.
    3.Let $\displaystyle W = B \backslash \vec{v} $.
    4. $\displaystyle B \backslash \vec{v} $ is a basis and thus a maximal independant subset that doesn't contain $\displaystyle \vec{v} $.
    5. W is a subspace of V.
    Is it correct?
    For the second part I don't have a good idea of what to do.
    Any hint would be great. Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by vincisonfire View Post
    Suppose that V is a vector space over F and $\displaystyle \vec{v} \in V $ is any nonzero vector.
    Show that there is a subspace W of V maximal subject to the condition
    that $\displaystyle \vec{v} \notin W $ ; show that any such W is a maximal proper subspace of V .
    (Do NOT assume that V is finite-dimensional.)
    the first part is a quick result of Zorn's lemma: let $\displaystyle C$ be the set of all subspaces W of V such that $\displaystyle v \notin W.$ since $\displaystyle v \neq 0,$ the set $\displaystyle C$ contains the 0 subspace and hence it's not empty.

    choose any chain (totally ordered collection) of elements of $\displaystyle (C, \subseteq).$ then the union of those elements is still in $\displaystyle C.$ thus $\displaystyle C$ has a maximal element by Zorn's lemma.

    for the second part, suppose $\displaystyle W$ is a maximal element of $\displaystyle C$ and $\displaystyle W \subsetneq W_1,$ for some subspace $\displaystyle W_1$ of $\displaystyle V.$ the claim is that $\displaystyle W_1=V$: by maximality of $\displaystyle W$ in $\displaystyle C,$ we must have $\displaystyle v \in W_1.$

    thus $\displaystyle W+<v> \subseteq W_1.$ so we only need to show that $\displaystyle W+<v>=V.$ suppose $\displaystyle W+<v> \neq V,$ and let $\displaystyle v_1 \notin W + <v>.$ then $\displaystyle v \notin W + <v_1>$ and thus $\displaystyle W + <v_1> \in C,$ which

    contradicts maximality of $\displaystyle W$ in $\displaystyle C. \ \Box$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member vincisonfire's Avatar
    Joined
    Oct 2008
    From
    Sainte-Flavie
    Posts
    468
    Thanks
    2
    Awards
    1
    The version of Zorn's lemma I know says that $\displaystyle

    C
    $ needs to be closed under union of chains.
    Here it is
    Let U be any set, and B any nonempty collection of subsets of U. Suppose tnat B is closed under unions of chains, then B has a maximal element.

    How is define the union of these chains?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by vincisonfire View Post

    How is define the union of these chains?
    if $\displaystyle \{W_{\alpha}: \ \alpha \in I \}$ is a chain of subspaces (under inclusion), then $\displaystyle \bigcup_{\alpha \in I} W_{\alpha}$ is again a subspace. if $\displaystyle \forall \alpha \in I: \ W_{\alpha} \in C,$ as defined in my solution, then $\displaystyle \bigcup_{\alpha \in I} W_{\alpha} \in C.$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subspace of a vector space
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Dec 15th 2011, 09:57 AM
  2. Subspace of a vector space
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Dec 1st 2011, 01:16 PM
  3. Vector space and its subspace
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Nov 8th 2011, 04:01 AM
  4. Let Z be a proper subspace of an n-dimensional vector space X
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Mar 29th 2011, 11:56 AM
  5. [SOLVED] Subspace of a Vector space: Why not the other way around?
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Mar 27th 2011, 02:17 PM

Search tags for this page

Search Tags


/mathhelpforum @mathhelpforum