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Math Help - F, vector space of all functions Reals to Reals. Which of these subsets are subspaces

  1. #1
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    F, vector space of all functions Reals to Reals. Which of these subsets are subspaces

    F is the vector space of all functions from reals to reals. Which of the following subsets of F are subspaces?
    a) the set of all polynomial functions of degree greater than 3
    b) the set of all polynomial functions of degree less than 3
    c) the set of all functions satisfying f(2) = 0
    d) the set of all functions satisfying f(2) > 0

    I'm not sure i remember the process correctly..
    1. prove the 0(x) function exists
    2. prove the function holds under vector addition
    3. prove the function holds under scalar multiplication

    Like I said I'm not sure if this is the process of how to prove if they are subspaces or not. If anyone could give me some clues or do one of them as an example it would be greatly appreciated!
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Yes, those are the three things you need to do.

    Now, the first set, (a), is not a vector space; it is missing the zero function.

    For (b), can you add two polynomials of degree less than 3 to get a polynomial of degree 3 or more? Does multiplication by a non-zero scalar change the degree of a polynomial?

    (c), again, is a vector space. You just need to do the checks and think about why they work.

    (d) is not. I will leave you to work out why...
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