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Math Help - Linear algebra : base, vector space, prove that.

  1. #1
    MHF Contributor arbolis's Avatar
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    Linear algebra : base, vector space, prove that.

    Hi MHF,
    I don't know how to prove what I must prove, but I've done something.
    Let \bold{B} be a base of the vector space V which has a finite dimension. Let S \subseteq V such that \bold{B} \subseteq \text{span } (S). Prove that \text{span } (S)=V.
    My attempt : We have that \dim S \leq \dim V. And also that \dim \bold{B} \leq \dim S, but as \dim \bold{B}=\dim V we have \dim S=\dim V. So the conclusion follows. Am I right?
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by arbolis View Post
    Hi MHF,
    I don't know how to prove what I must prove, but I've done something.
    Let \bold{B} be a base of the vector space V which has a finite dimension. Let S \subseteq V such that \bold{B} \subseteq \text{span } (S). Prove that \text{span } (S)=V.
    My attempt : We have that \dim S \leq \dim V. And also that \dim \bold{B} \leq \dim S, but as \dim \bold{B}=\dim V we have \dim S=\dim V. So the conclusion follows. Am I right?
    hmmm,

    here is some nice way..
    note that every element of V can be written as a linear combination of the elements in B.. but since B \subseteq \mbox{span }{S}, then every element in B can be written as a linear combination of the elements in S... thus every element in V can be written as a linear combination of the elements in S.
    Last edited by kalagota; February 20th 2009 at 06:17 PM. Reason: gave some proof..
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  3. #3
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    We suppose S=\{ \vec{v}_1 \cdots \vec{v}_m \cdots \vec{v}_n  \}, \mathfrak{B}=\{ \vec{v}_1 \cdots \vec{v}_m \}, \mathfrak{B}\subseteq span(S)
    \because V=span(\mathfrak{B})\ and\ S \subseteq V \therefore S=span(\mathfrak{B}), \{ \vec{v}_{m+1} \cdots \vec{v}_n  \}\ are\ linearly\ dependent\ elements\therefore span(S)=span(\mathfrak{B})=V
    Last edited by math2009; February 20th 2009 at 08:58 PM.
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  4. #4
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by math2009 View Post
    We suppose S=\{ \vec{v}_1 \cdots \vec{v}_m \cdots \vec{v}_n  \}, \mathfrak{B}=\{ \vec{v}_1 \cdots \vec{v}_m \}, \mathfrak{B}\subseteq span(S)
    \because V=span(\mathfrak{B})\ and\ S \subseteq V \therefore S=span(\mathfrak{B}), \{ \vec{v}_{m+1} \cdots \vec{v}_n  \}\ are\ linearly\ dependent

    and what do you want to show with this?
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  5. #5
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    I didn't finish writing, please refresh web page.
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  6. #6
    MHF Contributor kalagota's Avatar
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    actually, there is no need to say that the the latter set is a lin. dep set.. as long as you can show that span B = span S, your done..
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