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Math Help - linear algebra question: Subspaces

  1. #1
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    Oct 2008
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    linear algebra question: Subspaces

    Need a little help figuring whether the following subsets are actually subspaces:

    1. the plane of vectors (b1, b2, b3) with b1=b2
    2. the plane of vectors with b1 = 1
    3. all linear combinations of v=(1,4,0) and w=(2,2,2)
    4. all vectors that satisfy b1+b2+b3 = 0
    5. all vectors with b1<= b2 <= b3

    i'm having difficulties figuring which one satisfy the subspace requirements and how exactly the prove it... please help!!!
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  2. #2
    Senior Member
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    A set is a subspace if and only if all of the following conditions are satisfied:

    It is contained in or equal to a vector space.
    It is non-empty
    It is closed under multiplication by a scalar
    It is closed under addition.


    The first 2 of these are generally very easy to prove. For the 3rd use the fact that it is non-empty and assume that u is in the set and prove that this implies ku is in the set for any scalar k. For the third, assume that u and v are in the set and show that this implies u+v is in the set.
    Last edited by badgerigar; October 7th 2008 at 10:52 PM. Reason: grammar
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