Gram-Schmidt Orthonormalization: Modifying Vectors & Spanning Space

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SUMMARY

The discussion centers on the Gram-Schmidt Orthonormalization process, specifically addressing the manipulation of vectors through scalar multiplication before orthogonalization and normalization. It is established that multiplying or dividing the original vectors by a scalar does not affect the spanning of the vector space. The scalar can be factored out during normalization, confirming that the resultant vectors will still span the same space as the original vectors.

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  • Understanding of Gram-Schmidt Orthonormalization
  • Familiarity with vector spaces and spanning sets
  • Knowledge of linear transformations and projections
  • Basic concepts of normalization in linear algebra
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators teaching vector space concepts and orthonormalization techniques.

dntmn
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I have a question about Gram Schmidt Orthonormalization. I know you can orthogonalize then normalize.

My question is can you (multipy/divide) the given vectors by a scalar to give new vectors, then orthogonalize(by taking projection), then normalize the result?

would the result still span the space of the orgional vectors?
 
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Of course, just imagine the scalar hanging out front the whole time, then it goes away when you normalize at the end.
 
Thank you!
 
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