DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 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 projection $P_{W} v$ is the $W$ -component in $V = W \oplus W^{⊥}$

Definition

Let $W$ be a subspace of a finite-dimensional inner product space $V$ . The orthogonal-decomposition theorem (For a subspace $W$ of a finite-dimensional inner product space, $V = W \oplus W^{⊥}$ ) gives

$V = W \oplus W^{⊥}$

associates to every $v \in V$ unique vectors $w \in W$ and $z \in W^{⊥}$ with $v = w + z$ . The orthogonal projection onto $W$ is the function

$P_{W} : V ⟶ W, P_{W} v = w .$

Equivalently, $P_{W} v$ is the unique vector of $W$ such that $v - P_{W} v \in W^{⊥}$ .

Depends on

For a subspace $W$ of a finite-dimensional inner product space, $V=W\oplus W^\perp$

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 11 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., definition 6.55 (standard reference, not scraped)
Sergei Treil, Linear Algebra Done Wrong, Definition 5.3.1 (standard reference, not scraped)

The orthogonal projection PWvPW​v is the WW-component in V=W⊕W⊥V=W⊕W⊥