Orthogonal Set

A subset {v_1,...,v_k} of a vector space V, with the inner product <,>, is called orthogonal if <v_i,v_j>=0 when i!=j. That is, the vectors are mutually perpendicular.

Note that there is no restriction on the lengths of the vectors. If the vectors in an orthogonal set all have length one, then they are orthonormal.

The notion of orthogonal makes sense for an abstract vector space over any field as long as there is a symmetric quadratic form. The usual orthogonal sets and groups in Euclidean space can be generalized, with applications to special relativity, differential geometry, and abstract algebra.

See also

Clifford Algebra, Homogeneous Space, Hyperbolic Geometry, Lie Group, Lorentzian Inner Product, Orthogonal Group, Orthogonal Transformation, Orthonormal Basis, Symmetric Quadratic Form

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Orthogonal Set." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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