Are All Bases Sets of Orthogonal Vectors?

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SUMMARY

Not all bases are sets of orthogonal vectors. A clear example is the basis for R² consisting of the vectors {\vec{i}, \vec{i}+ \vec{j}} which are nonorthogonal under the standard inner product. While orthonormal bases simplify component calculations, the concept of "orthogonal" and "orthonormal" is contingent upon the inner product defined on the vector space. It is possible to define an inner product such that any given basis can be transformed into an orthonormal basis.

PREREQUISITES
  • Understanding of vector spaces and bases
  • Familiarity with inner product definitions
  • Knowledge of orthogonal and orthonormal vectors
  • Basic concepts of R² geometry
NEXT STEPS
  • Explore the properties of inner products in vector spaces
  • Study the process of converting a basis to an orthonormal basis
  • Learn about Gram-Schmidt orthogonalization
  • Investigate applications of orthogonal bases in linear transformations
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Mathematicians, physics students, and anyone studying linear algebra or vector spaces will benefit from this discussion.

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Would all bases be sets of orthogonal (but not necessarily orthonormal) vectors?
 
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No. Try to think of a basis for R^2 consisting of nonorthogonal vectors -- there are plenty.
 
For example, {[itex]\vec{i}[/itex], [itex]\vec{i}+ \vec{j}[/itex] } is a basis for R2 and they are not orthogonal (with the "usual" inner product). It happens to be easier to to find components in an orthonormal basis.

In any case, "orthogonal" as well as "orthonormal" depend upon an innerproduct defined on the vector space. Given any basis it is always possible to define an innerproduct in which that basis is orthonormal.
 

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