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matqkks
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Why is an orthogonal basis important?
They don't have to be orthogonal. In an arbitrary curved space, it is not generally possible to find basis vectors that are mutually orthogonal.stallionx said:Because basis vectors have got to be orthogonal ( perpendicular ) so that they are Linearly Independent and one of them can not be formed from any combo of others.
WannabeNewton said:They don't have to be orthogonal. In an arbitrary curved space, it is not generally possible to find basis vectors that are mutually orthogonal.
is not this an infraction of " linear independency
An orthogonal basis is a set of vectors that are mutually perpendicular to each other. This means that the angle between any two vectors in the set is 90 degrees. In other words, the vectors are linearly independent and span the entire vector space.
Orthogonal bases have many applications in mathematics and science. One of the most important uses is in linear algebra, where they are used to simplify calculations and solve systems of equations. They are also used in signal processing, image compression, and computer graphics.
Using an orthogonal basis has several benefits, including simplifying calculations, reducing computational complexity, and improving numerical stability. Additionally, orthogonal bases allow for easier visualization and interpretation of data, making it easier to understand and analyze complex systems.
To find an orthogonal basis, one can use the Gram-Schmidt process. This involves taking a set of linearly independent vectors and orthogonalizing them by projecting each vector onto the subspace spanned by the previous vectors. This process results in a set of orthogonal vectors that form an orthogonal basis.
Yes, an orthogonal basis can be non-orthonormal. An orthonormal basis is a special case of an orthogonal basis where all the vectors have unit length. However, in some applications, it may be useful to have an orthogonal basis without the constraint of unit length. In these cases, the basis is still considered orthogonal, but not orthonormal.