In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W {\displaystyle W} of a vector space V {\displaystyle V} equipped with a bilinear form B {\displaystyle B} is the set W ⊥ {\displaystyle W^{\perp }} of all vectors in V {\displaystyle V} that are orthogonal to every vector in W {\displaystyle W} . Informally, it is called the perp, short for perpendicular complement. It is a subspace of V {\displaystyle V} .
Example Let V = ( R 5 , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle V=(\mathbb {R} ^{5},\langle \cdot ,\cdot \rangle )} be the vector space equipped with the usual dot product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } (thus making it an inner product space), and let W = { u ∈ V : A x = u , x ∈ R 2 } , {\displaystyle W=\{\mathbf {u} \in V:\mathbf {A} x=\mathbf {u} ,\ x\in \mathbb {R} ^{2}\},} with
A = ( 1 0 0 1 2 6 3 9 5 3 ) . {\displaystyle \mathbf {A} ={\begin{pmatrix}1&0\\0&1\\2&6\\3&9\\5&3\\\end{pmatrix}}.}
then its orthogonal complement W ⊥ = { v ∈ V : ⟨ u , v ⟩ = 0 ∀ u ∈ W } {\displaystyle W^{\perp }=\{\mathbf {v} \in V:\langle \mathbf {u} ,\mathbf {v} \rangle =0\ \ \forall \ \mathbf {u} \in W\}} can also be defined as W ⊥ = { v ∈ V : A ~ y = v , y ∈ R 3 } , {\displaystyle W^{\perp }=\{\mathbf {v} \in V:\mathbf {\tilde {A}} y=\mathbf {v} ,\ y\in \mathbb {R} ^{3}\},} being
A ~ = ( − 2 − 3 − 5 − 6 − 9 − 3 1 0 0 0 1 0 0 0 1 ) . {\displaystyle \mathbf {\tilde {A}} ={\begin{pmatrix}-2&-3&-5\\-6&-9&-3\\1&0&0\\0&1&0\\0&0&1\end{pmatrix}}.}
The fact that every column vector in A {\displaystyle \mathbf {A} } is orthogonal to every column vector in A ~ {\displaystyle \mathbf {\tilde {A}} } can be checked by direct computation. The fact that the spans of these vectors are orthogonal then follows by bilinearity of the dot product. Finally, the fact that these spaces are orthogonal complements follows from the dimension relationships given below.
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