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Linear algebra, subspace of a vector space?

  1. Mar 28, 2013 #1
    1. The problem statement, all variables and given/known data

    1) The set H of all polynomials p(x) = a+x^3, with a in R, is a subspace of the vector space P sub6 of all polynomials of degree at most 6. True or False?

    2) The set H of all polynomials p(x) = a+bx^3, with a,b in R, is a subspace of the vector space P sub6 of all polynomials of degree at most 6. True or False?

    2. Relevant equations

    Eh.. not sure?

    3. The attempt at a solution

    Once more, not too sure. I've been pouring over my Linear Algebra book, but it seems so abstract... I was under the impression that as long as the polynomial didn't have a lower power than the vector space [number] that the polynomial would be in the subspace of the given vector space :\

    Does the coefficient have anything to do with it?

    Some properties (if they help?): A subspace of a vector space V is a subset H of V that has 3 properties:
    a) The zero vector if V is in H.
    b) H closed under vector addition
    3) H closed under scalar multiplication..

    I've already gotten #1 wrong (the answer was false) - I'd like to know why though :(

    Any help would be awesome!
     
  2. jcsd
  3. Mar 28, 2013 #2

    jbunniii

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    Consider property (a). Can the zero vector (polynomial in this case) be written in the form ##p(x) = a + x^3##?

    Also consider property (b). If I have two polynomials of the form ##a + x^3##, and add them together, is the result also of the form ##a + x^3##?

    Finally, consider property (c). If I multiply a polynomial of the form ##a + x^3## by an arbitrary scalar, say ##2##, is the result of the form ##a + x^3##?
     
  4. Mar 28, 2013 #3

    LCKurtz

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    What exactly were you pouring over your book? Coffee? Beer?? And why? :confused:
     
  5. Mar 28, 2013 #4

    Mark44

    Staff: Mentor

    Yeah, not a good idea - the pages will stick together. :tongue:
     
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