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Gram-Schmidt Process

  1. May 26, 2012 #1
    1. The problem statement, all variables and given/known data

    The question states that we should use the Gram-Schmidt to find an orthogonal basis for W where W = span {p , q} and p(x) = 1 + x ; q(x) = 1 + 2x^2

    2. Relevant equations

    ...

    3. The attempt at a solution

    Let [itex]\{b1, b2\}[/itex] be such a basis. Using the G-S process:
    [itex]b1 = p = 1 + x[/itex]
    [itex]b2 = q - proj^q _b = q - \frac{<q, b1>} { \|b1\|^2} . b1[/itex]

    Is this the right way? The answer I got was [itex]\{b1, b2\}[/itex] = [itex]\{1 + x, 1/2 - x/2 + 2x^2\}[/itex]

    Note that p and q belong to the set of polynomials of degrees 2 or less (i.e [itex] \in P_2[/itex]) and [itex]< , >[/itex] denotes an inner product of two components.

    Also, presume I need to find further basis -- [itex]b3[/itex] -- is there a generality for [itex]bn[/itex] for example?
     
    Last edited: May 26, 2012
  2. jcsd
  3. May 26, 2012 #2

    Mark44

    Staff: Mentor

    I haven't checked your work closely, but this is the right idea. You can check orthogonality by taking the inner product of these two functions. BTW, you didn't say what the inner product was for your space - an integral of some kind?
    Again, what is the inner product you are using?
    {b1, b2} is the basis for W, which is a proper subspace of P2.

    The dimension of W is 2, so any basis for W will consist of two lin. independent vectors/functions. This means it is not possible to find another basis function, which is what I think you are asking. If you add a vector or function to a basis, the new addition will necessarily be a linear combination of the other elements in the set.

    The dimension of P2 is 3, so a basis for this function space will consist of three lin. independent functions.

    If you are asking whether the Gram-Schmidt process can be extended to find more than two orthogonal basis elements - yes, it can. I'm sure you can find an article on wikipedia about this.
     
  4. May 26, 2012 #3
    Oh right. The inner product is defined on [itex]P_2[/itex] as such: [itex]<p, q> = a_0b_0 + a_1b_1 + a_2b_2[/itex] where [itex]a_ns[/itex] are the constants of [itex]p(x)[/itex] and [itex]b_ns[/itex] are the constants of [itex]q(x)[/itex].
     
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