Can Each Vector Be Made Orthonormal Directly in Gram-Schmidt Process?

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SUMMARY

The discussion focuses on the Gram-Schmidt process and the orthonormalization of vectors. It confirms that users can choose to normalize each newly calculated vector immediately after orthogonalization without affecting the overall process. The order of orthogonalization and normalization is flexible, allowing for normalization to occur at any stage. This flexibility ensures that the integrity of the vectors is maintained throughout the process.

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  • Understanding of the Gram-Schmidt process
  • Knowledge of vector spaces and linear algebra
  • Familiarity with orthogonal and orthonormal vectors
  • Basic skills in mathematical proofs and problem-solving
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators teaching vector space concepts and numerical methods.

gbartel
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I have a question regarding the Gram-Schmidt process. When solving for the new subspace, is it okay to make a newly calculated vector orthonormal before calculating the rest of the vectors? I've always first solved for the vectors then made all the calculated vectors orthonormal, but I'd like to make each vector orthonormal directly after each one is calculated. I figured this might corrupt the vectors beyond the first one.

Thanks in advance!
 
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Yes, that is okay. You can first orthogonalize all and then normalize all, or immediately normalize after you have orthogonalized, or normalize some in between the orthogonalization, whatever you want, the order does not matter.
 
Awesome thanks!
 

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