Prove Weighted Unit Circle Ellipse: Inner Product on lR^2

In summary, the unit circle for an inner product on lR^2 is defined as the set of all vectors of unit length, where the norm squared is equal to (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). This can also be compared to the dot product in l_2 and l_infty to show that everything in between is an ellipse. Another method is to consider an orthonormal basis in the inner product.
  • #1
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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?
 
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  • #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.
 
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  • #3
Sounds awkward to me! Have you considered just looking at an orthonormal basis in that inner product?
 

What is the Weighted Unit Circle Ellipse?

The Weighted Unit Circle Ellipse is a mathematical concept that describes an ellipse with a center at the origin and a major axis of length 2. It is also known as the unit circle ellipse or the unit disk.

What is the Inner Product on lR^2?

The Inner Product on lR^2 is a mathematical operation that takes two vectors in the plane and produces a scalar value. It is also known as the dot product or scalar product, and it is defined as the product of the magnitudes of the two vectors, multiplied by the cosine of the angle between them.

How is the Weighted Unit Circle Ellipse related to the Inner Product on lR^2?

The Weighted Unit Circle Ellipse can be described using the Inner Product on lR^2. Specifically, the squared length of a vector in the plane is equal to the inner product of the vector with itself. This relationship can be used to prove the properties of the Weighted Unit Circle Ellipse.

What is the proof of the Weighted Unit Circle Ellipse using the Inner Product on lR^2?

The proof of the Weighted Unit Circle Ellipse using the Inner Product on lR^2 involves showing that the squared length of a vector on the ellipse is equal to the inner product of the vector with itself. This is done by defining the ellipse using its equation, substituting the coordinates of a point on the ellipse into the equation, and using the properties of the Inner Product on lR^2 to simplify the equation.

Why is understanding the Weighted Unit Circle Ellipse and Inner Product on lR^2 important?

Understanding the Weighted Unit Circle Ellipse and Inner Product on lR^2 is important because it is a fundamental concept in mathematics and has many applications in fields such as physics, engineering, and computer science. It also helps to develop critical thinking and problem-solving skills, which are essential for success in many areas of life.

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