Orthonormal Basis - Definition & Examples

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Thread starter The black vegetable
Start date May 11, 2019
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Basis General relativity Orthonormal basis

In summary, an orthonormal basis is a set of vectors in a vector space that are both orthogonal and normalized. It is important because it simplifies computations and has useful properties. To find an orthonormal basis, we use the Gram-Schmidt process on a set of linearly independent vectors. A vector space can have multiple orthonormal bases, but they will all have the same dimension. Some examples of orthonormal bases include standard basis vectors, eigenvectors, and Fourier basis vectors. They are commonly used in various applications of linear algebra.

May 11, 2019

The black vegetable

TL;DR Summary: Trying to find your Orthonormal basis

Is this correct? If not any hints on how to find

Many thanks

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May 11, 2019

Orodruin

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It seems to me you are missing some sign in your definition of M. Otherwise notationally dubious but in essence fine.

Also, wouldn’t something like ”the black adder” sound better?

May 11, 2019

Tardigrade

thanks Orodruin very helpful, But I think I have a better idea, I think I'm missing a sign in my definition of M :)

What is an orthonormal basis?

An orthonormal basis is a set of vectors in a vector space that are all mutually perpendicular (orthogonal) to each other and have a length of 1 (normalized). This means that the dot product of any two vectors in the basis is 0 and the magnitude of each vector is 1.

Why is an orthonormal basis important?

An orthonormal basis is important because it provides a convenient and efficient way to represent vectors in a vector space. It allows for easy calculations involving dot products, projections, and transformations. It is also a key concept in linear algebra and is used in many applications such as signal processing, computer graphics, and quantum mechanics.

How do you find an orthonormal basis?

To find an orthonormal basis, you first need to find a set of linearly independent vectors in the vector space. Then, you can use the Gram-Schmidt process to orthogonalize the vectors and normalize them to have a length of 1. This will result in an orthonormal basis for the vector space.

What are some examples of orthonormal bases?

Some examples of orthonormal bases include the standard basis in Euclidean space, where the basis vectors are the unit vectors along the x, y, and z axes. Another example is the Fourier basis, which is used in signal processing and consists of complex exponential functions. In quantum mechanics, the eigenstates of a system are often used as an orthonormal basis.

Can an orthonormal basis exist in any vector space?

An orthonormal basis can exist in any finite-dimensional vector space. However, in an infinite-dimensional vector space, an orthonormal basis may not exist for all vector spaces. For example, in the space of continuous functions, an orthonormal basis does not exist. In these cases, alternative bases such as a Schauder basis may be used.