Alphabeta MathHow 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
Definition
For a linear subspace (Linear subspace of a vector space) of an inner product space , using the preceding notion of orthogonality (Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases), its orthogonal complement is
This is a linear subspace: , and linearity in the first argument shows that whenever and are scalars. One has by positive definiteness and .
Depends on
Used by
- If W is T-invariant, then W^⊥ is T^*-invariant Lemma
- The quaternion double cover generates the third homotopy group of SO(3) Lemma
- Every affine hyperplane of ℝⁿ, and hence every proper linear subspace, is Lebesgue null Theorem
- For a subspace W of a finite-dimensional inner product space, V=W⊕ W^⊥ Theorem
- ker T^*=(imT)^⊥ and imT^*=(ker T)^⊥ in finite dimension Theorem
Dependency tree · two levels
9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)