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Orthogonal Complement


The orthogonal complement of a subspace V of the vector space R^n is the set of vectors which are orthogonal to all elements of V. For example, the orthogonal complement of the space generated by two non proportional vectors u, v of the real space R^3 is the subspace formed by all normal vectors to the plane spanned by u and v.

In general, any subspace V of an inner product space E has an orthogonal complement V^_|_ and

 E=V direct sum V^_|_.

This property extends to any subspace V of a space E equipped with a symmetric or differential k-form or a Hermitian form which is nonsingular on V.


See also

Fredholm's Theorem, Orthogonal Decomposition, Orthogonal Sum

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Orthogonal Complement." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/OrthogonalComplement.html

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