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Math Help - Subspaces and Bases

  1. #1
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    Subspaces and Bases

    This problem seems like a trick question, I am not sure how to approach it.

    Is the following a subspace? If so, find the bases and dimension.
    P ∈ R[x] ; P^2 = 0

    I know that it is a subspace if:
    The zero vector is included in the set
    Closed under addition
    Closed under scalar multiplication

    From first glance it seems like a subspace but how do I prove this and would it have a 0 basis?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by natcapat View Post
    QAlop{:"}]=-091`qThis problem seems like a trick question and I am not quite sure how to approach it. The simplest questions really throw me off.

    Is the following a subspace?
    }"{p/;olikmujnbgveaq
    Sorry - what is the question...?

    (you might want to use LaTeX, but use [tex ] and [/tex ] instead of [math ] and [/math ]).
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  3. #3
    Behold, the power of SARDINES!
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    Quote Originally Posted by natcapat View Post
    This problem seems like a trick question, I am not sure how to approach it.

    Is the following a subspace? If so, find the bases and dimension.
    P ∈ R[x] ; P^2 = 0

    I know that it is a subspace if:
    The zero vector is included in the set
    Closed under addition
    Closed under scalar multiplication

    From first glance it seems like a subspace but how do I prove this and would it have a 0 basis?
    If by

    R[x] \text{ You mean } \mathbb{R}[x]

    Then since the real numbers are a field the polynomial ring must be an integral domain.
    If R is just a Ring the we will need some information about the units of R.
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by TheEmptySet View Post
    If by

    R[x] \text{ You mean } \mathbb{R}[x]

    Then since the real numbers are a field the polynomial ring must be an integral domain.
    If R is just a Ring the we will need some information about the units of R.
    ..ignore this post...I was wrong...
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  5. #5
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    That is my problem I wasn't given any more information than just that. I understand that it is all real number in the field of polynomials but I do not know what do after. The problem given was exactly this:

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  6. #6
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by natcapat View Post
    That is my problem I wasn't given any more information than just that. I understand that it is all real number in the field of polynomials but I do not know what do after. The problem given was exactly this:

    Well, what polynomials square to give 0? Hint: there aren't very many of them...! (TheEmptySet tells you why, above.)
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  7. #7
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    Note that when we say a polynomial (or any function) is 0, we mean it is 0 for all x.
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  8. #8
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    hint #2: such a polynomial must be constant (why?).
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