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Basis of P3

  1. Feb 28, 2010 #1
    My mind is shot.

    Let S be a subspace of P3 consisting of all polynomials of the form ax2+bx+2a+3b. Find a basis for S.

    I am not sure where to start.
     
  2. jcsd
  3. Feb 28, 2010 #2
    What are the coefficients for P3 in? Are a, b elements of some field?
    Find a generating set for this subspace. How about {x2, x, ...}? What else should be in there to generate S?
     
  4. Feb 28, 2010 #3
    Try "factoring out a". It can't factor out of everything, but try factoring it out of all the terms you can.

    --------

    If that doesn't make sense, try to get a feel for what the subspace is by choosing values for a and b.
    For instance, which element does a = 2, b = 0 give you?
     
    Last edited: Feb 28, 2010
  5. Feb 28, 2010 #4
    Then set that equal to 0? Also factor out b the same way?
     
  6. Feb 28, 2010 #5
    My hint about factoring is purposefully vague. What did you get when you tried factoring out the a?

    However, yes, if you correctly factor out the "a"s and the "b"s, you should start to see the answer.
     
  7. Feb 28, 2010 #6
    a(x2+2)+b(x+3)
     
  8. Feb 28, 2010 #7
    I can then set up the Wronskian.
     
  9. Feb 28, 2010 #8
    Do you know what to do now, or do you still have questions?
     
  10. Feb 28, 2010 #9
    I am still confused but I evaluated the Wronskian. How can I come up with the basis now?
    W=x^2+6x-2
     
  11. Feb 28, 2010 #10
    We have all the information now. We just need to realize what we have.

    (1)
    Every element of S can be written as
    a(x2+2) + b(x+3)
    This shows that (x2+2) and (x+3) generate S (also, setting a=1,b=0 or vice versa shows that they're in S).

    (2)
    The Wronskian of these two elements is x2+6x-2, which is not identically zero. This shows that these two elements are linearly independent.

    A linearly independent generating set is precisely a basis, so these two polynomials form your basis.
     
  12. Feb 28, 2010 #11
    That is it?
    Thanks.
     
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