The orthogonal complement of a given subspace is all the vectors in the encompassing linear space which are orthogonal to every vector in . Typically the orthogonal complement of a subspace is written as and is usually read as "W perp".

Formally given a subspace and it's encompassing vector space we define the orthogonal complement as:

This is the set of all vectors in that are orthogonal to all of the vectors in .

Computing

Given a subspace of generated from the span of the vectors we can determine the orthogonal complement of as follows:

alternatively given a matrix

and conversely