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Math Help - confusion about telling wether or not a subset is a subspace

  1. #1
    Member Jskid's Avatar
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    confusion about telling wether or not a subset is a subspace

    Is the following subset of R^3 a subspace R^3? The set of all vectors of the form (a,b,2)

    The answer key has:
    W is not a subspace. To show this let \vec u=(a_1, b_1, c_1) and \vec v = (a_2, b_2, c_2) be in W. Then c_1=c_2=2 Now \vec u + \vec v = (a_1+a_2, b_1+b_2, c_1+c_2)=(a_1,b_1+b_1,4) which is not in W

    Two things confuse me. 1) Why is it b_1+b_1 not b_1+b_2? 2)I don't see how the property " \vec u + \vec v is in V" fails to hold because (a_1+b_1+b_1,4) is in 3 space isn't it?
    Last edited by Jskid; February 17th 2011 at 01:34 PM. Reason: changed + to ,
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  2. #2
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    Quote Originally Posted by Jskid View Post
    Is the following subset of R^3 a subspace R^3? The set of all vectors of the form (a,b,2)
    Look no further. Is <0,0,0> in the set?
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  3. #3
    Member Jskid's Avatar
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    Quote Originally Posted by Plato View Post
    Look no further. Is <0,0,0> in the set?
    No it's not but I thought I didn't have to verify that rule if it's a subspace of a vector space? 0 \dot \vec u = \vec 0
    And that still don't see where b_1+b_1 came from?

    EDIT:All though I do see how that answers (a_1+b_1+b_1,\textbf{4}) not being in W
    Last edited by Jskid; February 17th 2011 at 01:11 PM. Reason: see post
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  4. #4
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    \vec u + \vec v = (a_1+a_2, b_1+b_2, c_1+c_2)=(a_1+b_1+b_1,4)

    should be

    \vec u + \vec v = (a_1+a_2, b_1+b_2, c_1+c_2)=(a_1+a_2, b_1+b_2, 4).

    Just fyi. The first expression loses a dimension (!), loses a term, and changes an index.
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  5. #5
    Member Jskid's Avatar
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    Quote Originally Posted by Ackbeet View Post
    \vec u + \vec v = (a_1+a_2, b_1+b_2, c_1+c_2)=(a_1+b_1+b_1,4)

    should be

    \vec u + \vec v = (a_1+a_2, b_1+b_2, c_1+c_2)=(a_1+a_2, b_1+b_2, 4).

    Just fyi. The first expression loses a dimension (!), loses a term, and changes an index.
    Good catch, that should've been a comma not a plus.
    The answer manual has b_1+b_1 NOT b_1+b_2 That's what I'm trying to ask about?
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  6. #6
    Member Jskid's Avatar
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    I get it now, I think my solution manual has a typo that got me very confused.
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  7. #7
    A Plied Mathematician
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    Yeah, it's a typo.
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