1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: 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


    User Avatar
    Science Advisor

    Sounds awkward to me! Have you considered just looking at an orthonormal basis in that inner product?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook