Weighted unit circle

[Mindscrape]

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 $$v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2$$ is an ellipse.

I know that norm squared will be $$(v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2) (v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2)$$, 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?

 

[Mindscrape]

I figured out another way. Take the dot product (l_2) in |R^2 and compare it with the l_infty inner product.

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

Everything in between will be an ellipse.

 

[HallsofIvy]

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