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Math Help - Subspace proof

  1. #1
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    Subspace proof

    Here attached is the question:

    How do I prove it as an if and only statement.

    We have learnt the definition of subspace being when the set is empty, closed under vector addition and closed under multiplication.

    I can prove for when c=0 that it is a subspace. How do I prove for all other values, that it is not a subspace?
    Attached Thumbnails Attached Thumbnails Subspace proof-subspace.png  
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    (0,0) must be in X(why?)! Hence a*0+b*0+c!=0 - absurd!
    Last edited by Also sprach Zarathustra; November 25th 2010 at 10:52 AM.
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  3. #3
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    Quote Originally Posted by Also sprach Zarathustra View Post
    (0,0) must be in X(why?)! Hence a*0+b*0=c!=0 - absurd!
    What?

    I know that when c = 0, X is a subspace of R2 but I don't know how to prove that this is the case IF and ONLY IF c = 0

    I can prove it for c=1, c=2 etc but not a general proof to show that for all non zero values for c, X is not a subspace...
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  4. #4
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    Oh I see what you mean, sorry. So do you have to start by proving that 0,0 is in X which shows that C has to equal 0?
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  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
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    1. X subspace ==> (0,0) must be in X! Hence a*0+b*0+c=0 or c=0

    2. c=0 ==> ax_1 + bx_2=0 ==> NOW prove that X is closed under vector addition and closed under multiplication.
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  6. #6
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    Quote Originally Posted by Also sprach Zarathustra View Post
    1. X subspace ==> (0,0) must be in X! Hence a*0+b*0+c=0 or c=0

    2. c=0 ==> ax_1 + bx_2=0 ==> NOW prove that X is closed under vector addition and closed under multiplication.
    Thanks, I can do that now.

    The only question I have is, how does this prove that c can't be any other number in order for it to be a subspace of X?

    Haven't we just taken an example to find a possible value for c, then prove c=0 is a subspace of X?
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  7. #7
    MHF Contributor Also sprach Zarathustra's Avatar
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    c is constant!

    see again post #5 (1)
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  8. #8
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    Quote Originally Posted by Also sprach Zarathustra View Post
    c is constant!

    see again post #5 (1)
    Thanks alot.

    Sorry for being a bit 'slow.' Only introduced to subspaces yesterday so not fully understanding it all yet..
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