Are All Bases Sets of Orthogonal Vectors?

pr0me7heu2 · Mar 10, 2008

Would all bases be sets of orthogonal (but not necessarily orthonormal) vectors?

morphism · Mar 10, 2008

No. Try to think of a basis for R^2 consisting of nonorthogonal vectors -- there are plenty.

HallsofIvy · Mar 10, 2008

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.

Are All Bases Sets of Orthogonal Vectors?

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