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Basis of a subspace

  1. Oct 7, 2010 #1
    1. The problem statement, all variables and given/known data

    I need to find a basis for the following:

    S = {f are polynomials of degree less than or equal to 4| f(0) = f(1) = 0}

    2. The attempt at a solution

    A general polymial is of the form:

    p(x) = ax^4 + bx^3 + cx^2 + dx + e

    Now for p(0) = p(1) = 0 I must have:

    e = 0 and a + b + c + d + e = 0

    Which basically becomes:

    a + b + c + d = 0.

    Here is where I get stuck. I'd usually solve for a variable and then substitute it back into the polynomial and then group like coefficients to get a basis, but it just isn't working!

    Any help?

    NEVERMIND!!!!!!!!!!!!!!! I've figured out what I was doing wrong lol Apparently I don't know how to factor LOL
     
    Last edited: Oct 7, 2010
  2. jcsd
  3. Oct 7, 2010 #2

    Mark44

    Staff: Mentor

    And e is still 0.
    Notwithstanding that you have figured this out, you can find the solution set for the system of equations
    a + b + c + d = 0
    e = 0

    by simply solving for a.

    a = -b - c - d
    b = b
    c = ......c
    d = ............d
    e = 0

    Then any "vector" of coefficients looks like <a, b, c, d, e> = b<-1, 1, 0, 0, 0> + c<-1, 0, 1, 0, 0> + d<-1, 0, 0, 1, 0>, where b, c, and d are any real scalars.

    Putting this back in terms of polynomials, a basis is {-x4 + x3, -x4 + x2, -x4 + x}.

    Another basis, with all leading coefficients positive is {x4 - x3, x4 - x2, x4 - x}.
     
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