Inner Products in R2 other than euclidian

Thread starter corey2014
Start date Nov 13, 2011

In summary, an inner product in R2 is a mathematical operation that takes two vectors and returns a scalar value. While the most commonly used inner product in R2 is the Euclidean inner product, there are other options such as the Hermitian inner product and the Minkowski inner product. The Hermitian inner product is a generalization of the Euclidean inner product for complex vectors and is useful in complex analysis and quantum mechanics. On the other hand, the Minkowski inner product is important in special relativity and is calculated by subtracting the square of the time component from the sum of the squares of the space components. The choice of inner product can also affect the geometry of a vector space, as different inner products result

Nov 13, 2011

corey2014

asdf

Last edited: Nov 14, 2011

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Nov 13, 2011

D H

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Hint: An inner product for a real vector space has to be symmetric, linear, and positive definition.

Nov 13, 2011

corey2014

Yes, i know the definition, however i don't know how to show that it has to be linear, or that the dot product has to be of that form...

1. What is an inner product in R2 other than the Euclidean inner product?

An inner product is a mathematical operation that takes two vectors and returns a scalar value. In R2, the most commonly used inner product is the Euclidean inner product, which is defined as the dot product of the two vectors. However, there are other inner products that can be used in R2, such as the Hermitian inner product and the Minkowski inner product.

2. How is the Hermitian inner product different from the Euclidean inner product?

The Hermitian inner product is a generalization of the Euclidean inner product, which is defined for complex vectors. It takes into account the complex conjugate of one of the vectors, making it more suitable for applications in complex analysis and quantum mechanics.

3. What is the significance of the Minkowski inner product in R2?

The Minkowski inner product is a special type of inner product that is used in the study of special relativity. It is defined using the Minkowski metric, which takes into account the concept of space-time intervals. This inner product is important in understanding the geometrical properties of space-time in special relativity.

4. How do you calculate the Minkowski inner product in R2?

The Minkowski inner product is calculated by taking the dot product of two vectors in R2 and then subtracting the square of the time component from the sum of the squares of the space components. This results in a scalar value that represents the space-time interval between the two vectors.

5. Can the choice of inner product affect the geometry of a vector space?

Yes, the choice of inner product can affect the geometry of a vector space. Different inner products can result in different notions of distance and angle between vectors, which can change the geometrical properties of the vector space. For example, the Minkowski inner product results in a different concept of distance than the Euclidean inner product, leading to different geometric properties in the vector space.