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Weighted unit circle

  1. Mar 6, 2007 #1
    Prove that the unit circle, for an inner product on lR^2 is defined as the set of all vectors of unit length ||v|| = 1, of the non-standard inner product [tex]v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2[/tex] is an ellipse.

    I know that norm squared will be [tex](v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2) (v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2)[/tex], but I don't really want to multiply that all out to show that it looks like an ellipse. Is there a better way, maybe manipulating the inner product somehow?
    Last edited: Mar 6, 2007
  2. jcsd
  3. Mar 6, 2007 #2
    I figured out another way. Take the dot product (l_2) in |R^2 and compare it with the l_infty inner product.

    [tex] B_2 = set(v \in lR^2 | v_1^2 + v_2^2 = 1)[/tex]
    [tex] B_\infty = set(v \in lR^2 | max(|v_1|,|v_2|) = 1)[/tex]

    Everything in between will be an ellipse.
    Last edited: Mar 6, 2007
  4. Mar 7, 2007 #3


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    Sounds awkward to me! Have you considered just looking at an orthonormal basis in that inner product?
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