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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 ${0}^{⊥} = V$ .

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 29 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Sheldon Axler, Linear Algebra Done Right, 4th ed., §6C (standard reference, not scraped)
Sergei Treil, Linear Algebra Done Wrong, Ch. 5, §5.2 (standard reference, not scraped)

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