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Simple, construct a basis of R2 with an inner product

  1. Mar 26, 2008 #1
    1. The problem statement, all variables and given/known data
    Construct an orthogonal basis of R2 equipped with the non-standard inner product defined for all X, Y belonging to R2, by

    <X,Y> = X^T AY
    with
    A =
    2 1
    1 3

    3. The attempt at a solution
    So it seems pretty trivial, but I can't seem to get the answer. So my approach is
    1) First get any basis that fits with this inner product
    2) Use Gram-Scmidt to orthogonalize that basis

    So I know I can start with the canonical basis of R2: (1,0) (0,1)
    My problem is, how can I satisfy the inner product if I start from the canonical basis?

    I have also attempted to solve the equations of two arbitrary vectors v1, v2
    where v1=(x1, y2) and v2=(x2,y2) and given the matrix A we know:
    x1x2 + y1y2 = 1
    x1^2 + y1^2 = 2
    x2^2 + y2^2 = 3
    But if I plug in the canoical basis for v1, the solution fails. It only works when both points are non-zero, i.e. v1 = (1, 1)


    Any help would be appreciated.

    EDIT:
    Actually now that I am looking at my solution, I obtained (via my method of substituting (1,1) for x1, y1 in my equations) the basis that works with this inner product as:
    ( 1 - [1+sqrt(5)]/2 , [1+sqrt(5)]/2 ) and (1, 1)

    They seem to be linearly independent, did I do this right? (I know I still have to orthogonalize them)
     
    Last edited: Mar 26, 2008
  2. jcsd
  3. Mar 26, 2008 #2
    Anyone?
     
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