Orthogonal Complement (original) (raw)
TOPICS
Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology
Alphabetical Index New in MathWorld
The orthogonal complement of a subspace 
of the vector space 
is the set of vectors which are orthogonal to all elements of 
. For example, the orthogonal complement of the space generated by two non proportional vectors 
,
of the real space 
is the subspace formed by all normal vectors to the plane spanned by 
and 
.
In general, any subspace 
of an inner product space 
has an orthogonal complement 
and
This property extends to any subspace 
of a space 
equipped with a symmetric or differential 
-form or a Hermitian form which is nonsingular on 
.
See also
Fredholm's Theorem, Orthogonal Decomposition, Orthogonal Sum
This entry contributed by Margherita Barile
Explore with Wolfram|Alpha
More things to try:
Cite this as:
Barile, Margherita. "Orthogonal Complement." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/OrthogonalComplement.html