The discussion centers on proving that the orthogonal complement of the span of a set \( S \), denoted as \( \langle S \rangle^{\bot} \), is equal to the orthogonal complement of \( S \), denoted as \( S^{\bot} \). The user successfully demonstrated that \( S^{\bot} \subseteq \langle S \rangle^{\bot} \) and sought clarification on proving the reverse inclusion. The argument presented shows that if \( y \in \langle S \rangle^{\bot} \), then \( y \) is orthogonal to all vectors in \( S \), confirming \( \langle S \rangle^{\bot} \subseteq S^{\bot} \) is valid.
PREREQUISITESStudents and educators in linear algebra, mathematicians focusing on vector spaces, and anyone seeking to deepen their understanding of orthogonal complements and spans in mathematical contexts.