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Math Help - Vector Subspaces

  1. #1
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    Vector Subspaces

    Hoping to get some help with questions that Im stuck on for exam revision
    Show that:
    a) U = (x,y,z) x+y=z is a subspace of R3
    b) U= (x,y,z), y is rational and is not a subspace of R3
    c) U= A|A2=A,AϵMmn is not a subspace of Mmn
    d)U={F|f(x+pi)=f(x), FϵC(-infinity, + infinity) is a subspace of C(-inifity, +inifinity)
    e) u= p(x)|p(1)=0, p(x)ϵPn is a subspace of Pn
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  2. #2
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    Re: Vector Subspaces

    Quote Originally Posted by rjd871 View Post
    Hoping to get some help with questions that Im stuck on for exam revision
    Show that:
    a) U = (x,y,z) x+y=z is a subspace of R3
    b) U= (x,y,z), y is rational and is not a subspace of R3
    c) U= A|A2=A,AϵMmn is not a subspace of Mmn
    d)U={F|f(x+pi)=f(x), FϵC(-infinity, + infinity) is a subspace of C(-inifity, +inifinity)
    e) u= p(x)|p(1)=0, p(x)ϵPn is a subspace of Pn
    Cheers
    Have you tried to help yourself? If you have then post what you have done.
    If you have not, then why do you think we can help if you do not tell us what you don't understand?
    How does one prove a set is a subspace?
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  3. #3
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    Re: Vector Subspaces

    Some of those can't see because. . .well the type is all screwy. You might want to post a photo of the exam questions. And these all really follow from the definition, of which there are three conditions:

    1. Subspace contains the 0-vector.
    2. Subspace is closed under vector addition.
    3. Subspace is closed under scalar multiplication.

    With those three conditions to check, all of these are fairly boilerplate. Also, I'm fairly sure (b) isn't written correctly.
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  4. #4
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    Re: Vector Subspaces

    the reason the U of (b) is not a subspace is because we can consider (0,1,0) which is clearly in U (0 is an element of R, and 1 is rational) and then take e (the natural logarithm base) times it (since the scalar e is in the field R) to obtain e(0,1,0) = (0,e,0), which is not in U (since e is irrational), thus U is not closed under scalar multiplication.
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