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- https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/18%3A_Orthonormal_Bases_and_Complements/18.06%3A_Orthogonal_ComplementsWe call the subspace \(U+V:=span (U\cup V) = \{u+v | u\in U, v\in V \}\) the \(\textit{sum}\) of \(U\) and \(V\). Given two subspaces \(U\) and \(V\) of a space \(W\) such that $$U\cap V=\{0_{W}\}\, ,...We call the subspace \(U+V:=span (U\cup V) = \{u+v | u\in U, v\in V \}\) the \(\textit{sum}\) of \(U\) and \(V\). Given two subspaces \(U\) and \(V\) of a space \(W\) such that $$U\cap V=\{0_{W}\}\, ,$$ the \(\textit{direct sum}\) of \(U\) and \(V\) is defined as: Remark: The set \(U^{\perp}\) (pronounced "\(U\)-perp'') is the set of all vectors in \(W\) orthogonal to \(\textit{every}\) vector in \(U\).
- https://math.libretexts.org/Courses/Mission_College/MAT_04C_Linear_Algebra_(Kravets)/07%3A_Orthogonality/7.02%3A_Orthogonal_ComplementsIt will be important to compute the set of all vectors that are orthogonal to a given set of vectors. It turns out that a vector is orthogonal to a set of vectors if and only if it is orthogonal to th...It will be important to compute the set of all vectors that are orthogonal to a given set of vectors. It turns out that a vector is orthogonal to a set of vectors if and only if it is orthogonal to the span of those vectors, which is a subspace, so we restrict ourselves to the case of subspaces.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/14%3A_Orthonormal_Bases_and_Complements/14.06%3A_Orthogonal_ComplementsWe call the subspace \(U+V:=span (U\cup V) = \{u+v | u\in U, v\in V \}\) the \(\textit{sum}\) of \(U\) and \(V\). Given two subspaces \(U\) and \(V\) of a space \(W\) such that $$U\cap V=\{0_{W}\}\, ,...We call the subspace \(U+V:=span (U\cup V) = \{u+v | u\in U, v\in V \}\) the \(\textit{sum}\) of \(U\) and \(V\). Given two subspaces \(U\) and \(V\) of a space \(W\) such that $$U\cap V=\{0_{W}\}\, ,$$ the \(\textit{direct sum}\) of \(U\) and \(V\) is defined as: Remark: The set \(U^{\perp}\) (pronounced "\(U\)-perp'') is the set of all vectors in \(W\) orthogonal to \(\textit{every}\) vector in \(U\).
