User-defined orthonormal basis

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SUMMARY

This discussion focuses on creating a user-defined orthonormal basis from a given normalized vector (versor) in a space with dimension N greater than 3. The primary method mentioned is the Gram-Schmidt process, which allows for the generation of N-1 additional linearly independent vectors while preserving the initial vector's integrity. Users expressed satisfaction with the method's effectiveness, although concerns about numerical stability were noted.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly orthonormal bases
  • Familiarity with the Gram-Schmidt process for orthogonalization
  • Knowledge of vector normalization techniques
  • Basic proficiency in mathematical programming or computational tools
NEXT STEPS
  • Research advanced applications of the Gram-Schmidt process in higher dimensions
  • Explore numerical stability techniques in linear algebra computations
  • Learn about alternative methods for generating orthonormal bases, such as QR decomposition
  • Investigate the implications of orthonormal bases in machine learning and data science
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Mathematicians, data scientists, and engineers who require a solid understanding of orthonormal bases and their applications in higher-dimensional spaces.

markuz
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Does anybody know how to create a orthonormal basis, i.e. a matrix containing orthogonal vectors of norm 1, out of a given direction (normalised vector or versor) in a space with dimension N>3?
With "out of a given direction", I mean that the resulting basis would have the first vector equal to the provided versor.
 
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In principle, all you need is to find N - 1 more linearly independent vectors, and then you can apply the Gram-Schmidt process. That preserves the first vector, so if you put your given vector first in the list it will remain unaltered.
 
thanks

Thanks it works! Not sure about the numerical stability but will keep an eye on it.
Thanks again
Cheers!
 

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