Orthogonal Complements and Projections An orthogonal complement to a subspace S is a subspace of the ambient space V that is orthogonal to S, meaning that t...
Orthogonal Complements and Projections
An orthogonal complement to a subspace S is a subspace of the ambient space V that is orthogonal to S, meaning that the only vectors in the orthogonal complement are orthogonal to every vector in S. The orthogonal complement of S is denoted by S orthogonal.
In other words, S orthogonal = V \ S, where V is the ambient space and S is the subspace.
The orthogonal complement of S is a subspace of V that is complementary to S, meaning that S \ S orthogonal.
The projection of a vector x onto a subspace S is a vector that is orthogonal to all vectors in S. The projection of x onto S is denoted by pr_S(x), and it is given by the formula
where v is any vector in S.
The orthogonal complement of S is the set of all vectors in V that are orthogonal to every vector in S.
Orthogonal complements can be used to decompose a vector into its components along different subspaces. For example, if V is the vector space of all 2D vectors and S is the subspace of all vectors with zero determinant, then the orthogonal complement of S is the set of all vectors with positive determinant. This means that we can decompose any vector x in V into a unique sum of vectors in S and the orthogonal complement of S.
Orthogonal complements are also used in many applications in linear algebra, such as least squares, singular value decomposition, and principal component analysis