The orthogonal complement of a subspace S within R^n, denoted S⊥, consists of all vectors orthogonal to every vector in S, and can be characterized as the null space of the matrix whose rows are a spanning (or basis) set for S. This is a topic within linear algebra concerning orthogonality and subspace decomposition, linking the concept of orthogonal complements to null-space computation via row reduction and to dimension counting (dim S + dim S⊥ = n). It further establishes that when S and S⊥ together have linearly independent spanning vectors, every vector in the ambient space has a unique decomposition into a component in S and a component in S⊥, a foundational fact underlying orthogonal decomposition in linear algebra.
The orthogonal complement of a subspace S within R^n, denoted S⊥, consists of all vectors orthogonal to every vector in S, and can be characterized as the null space of the matrix whose rows are a sp…