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.
Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases
Definition
Vectors in an inner product space (Real and complex inner product spaces, with the inner product linear in the first argument) are orthogonal, written , if . Two subspaces are orthogonal if every vector in one is orthogonal to every vector in the other.
A list or set of vectors is orthogonal if every two distinct members are orthogonal. It is orthonormal if it is orthogonal and every member has induced norm (The norm induced by a real or complex inner product) equal to . An orthonormal basis is an ordered basis (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis) that is an orthonormal list.
The empty list is orthonormal and is the orthonormal basis of the zero space. A one-element list is orthogonal; it is orthonormal exactly when its vector has norm .
Depends on
- Real and complex inner product spaces, with the inner product linear in the first argument
- The norm $\lVert v\rVert=\sqrt{\langle v,v\rangle}$ induced by a real or complex inner product
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
Used by
- The orthogonal complement W^⊥={v:⟨ v,w⟩=0 for all w∈ W} Definition
- FALSE: Every orthogonal set is an orthonormal basis False statement
- Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis Theorem
- Every finite orthogonal list of nonzero vectors is linearly independent Theorem
- Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 70 results over 17 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., §§6A–6B (standard reference, not scraped)
- Sergei Treil, Linear Algebra Done Wrong, Ch. 5, §§5.1–5.3 (standard reference, not scraped)