DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The orthogonal complement $W^{⊥} = {v : ⟨ v, w ⟩ = 0 for all w \in W}$

Definition

For a linear subspace $W$ (Linear subspace of a vector space) of an inner product space $V$ , using the preceding notion of orthogonality (Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases), its orthogonal complement is

$W^{⊥} : = {v \in V : ⟨ v, w ⟩ = 0 for every w \in W} .$

This is a linear subspace: $0 \in W^{⊥}$ , and linearity in the first argument shows that $a u + b v \in W^{⊥}$ whenever $u, v \in W^{⊥}$ and $a, b$ are scalars. One has $V^{⊥} = {0}$ by positive definiteness and ${0}^{⊥} = V$ .

The orthogonal complement $W^{⊥} = {v : ⟨ v, w ⟩ = 0 for all w \in W}$

Definition

Depends on

Dependency tree · 2 levels

Sources

The orthogonal complement W⊥={v:⟨v,w⟩=0 for all w∈W}W⊥={v:⟨v,w⟩=0 for all w∈W}

Definition

Depends on

Dependency tree · 2 levels

Sources

The orthogonal complement $W^{⊥} = {v : ⟨ v, w ⟩ = 0 for all w \in W}$