1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Basis and Adjoint.

  1. Feb 25, 2007 #1
    1. The problem statement, all variables and given/known data

    Apply the Gram-Schmidt orthogonalization procedure to the canonical basis [itex]1, x, x^2, x^3, x^4[/itex] in order to find an orthonormal basis for the space P4([0, 1]) with respect to the inner product <p(x), q(x)> =int(0,1) p(x)q(x) dx

    AND USE THIS BASIS TO FIND THE ADJOINT OF THE LINEAR MAP! [itex]A(ax^4 + bx^3 +cx^2 + dx + e) = cx^2[/itex] (=cx^2 ... no cx^3... latex acts funny?)




    2. Relevant equations



    3. The attempt at a solution

    Finding the orthogonal basis using the Gram-Schmit algorithm is slimply plugging numbers into a formula, so that is straight forward.

    My Basis is:

    [itex]e1 =1[/itex]

    [itex]e2 = x-1/2[/itex]

    [itex]e3 = x^2+1/6-x[/itex]

    [itex]e4 = x^3-1/20+3/5*x-3/2*x^2[/itex]

    [itex]e5 = x^4+1/70-2/7*x+9/7*x^2-2*x^3[/itex]


    Could somebody please guide me as to how I would

    USE THIS BASIS TO FIND THE ADJOINT OF THE LINEAR MAP [itex]A(ax^4 + bx^3 + cx^2 + dx + e) = cx^2[/itex] (=cx^2 ... no bx^2... latex acts funny?)

    I honestly have no idea where to start?

    thought a good place to start would be to find the map of e_i with respect to A,

    A(e1) = 0
    A(e2) = 0
    A(e3) = x^2
    A(e4) = 3/2*x^2
    A(e5) = 9/7*x^2
     
    Last edited: Feb 25, 2007
  2. jcsd
  3. Feb 25, 2007 #2

    AKG

    User Avatar
    Science Advisor
    Homework Helper

    The adjoint A* of a map A is the linear transformation satisfying <Ax,y> = <x,A*y> for all x, y in your vector space. Since you have supposedly correctly found a basis which is orthnormal w.r.t. the given inner product, find A* should be straightforward.

    Compute <A(ei), ej> for 1 < i, j < 5. This will give you <ei, A*(ej)> for each i, j. Since the basis is orthonormal, you get:

    A*(ej) = <A(e1),ej>e1 + ... + <A(e5),ej>e5

    You should be able to go from here.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Basis and Adjoint.
  1. Adjoint of an adjoint? (Replies: 16)

  2. A basis (Replies: 3)

  3. Adjoint of an operator (Replies: 1)

  4. Adjoint question (Replies: 4)

Loading...