Linear independence and orthogonaliy

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SUMMARY

A finite set S being linearly independent (L.I.) guarantees the existence of a set T of the same size that is pairwise orthogonal, particularly within an inner product space. The Gram-Schmidt process is a reliable method for constructing such a set T from S. Additionally, methods involving eigenspaces and nullspaces can also yield mutually orthogonal sets. It is noted that the basis spanning T may be orthogonal to the basis spanning S.

PREREQUISITES
  • Understanding of linear independence in vector spaces
  • Familiarity with inner product spaces
  • Knowledge of the Gram-Schmidt orthogonalization process
  • Concepts of eigenspaces and nullspaces in linear algebra
NEXT STEPS
  • Study the Gram-Schmidt process in detail for orthogonalization
  • Explore the properties of eigenspaces and their applications
  • Investigate nullspaces and their role in linear transformations
  • Learn about inner product spaces and their significance in linear algebra
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to deepen their understanding of linear independence and orthogonality concepts.

Werg22
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Please refresh my memory; if a finite set S is L.I., then does this imply the existence of a set T of the same size (i.e. |T| = |S|) so that the elements T are pairwise orthogonal?
 
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If it's in an inner product space, you can use Gram-Schmidt to find T.
 
Werg22 said:
Please refresh my memory; if a finite set S is L.I., then does this imply the existence of a set T of the same size (i.e. |T| = |S|) so that the elements T are pairwise orthogonal?

yes. i believe any method you use (eigenspace, nullspace, etc) to find a set that spans T will be mutually orthogonal. i'd probably guess that the basis spanning T would be orthogonal to the basis spanning A as well.
 

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