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.
Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms
Definition
Let be a vector space over . A bilinear form on is a function that is linear in each variable separately.
The form is
- symmetric when for all ;
- skew-symmetric when for all ;
- alternating when for every .
These conditions are kept distinct because their relations depend on the characteristic of .
Depends on
Used by
- The standard bilinear form ⟨ x,y⟩=∑_i<nxᵢyᵢ on Fⁿ Definition
- The trace form (x,y)↦ Tr_K/F(xy) of a finite extension Definition
- The form on ℝ² with matrix beginpmatrix0&10&0 endpmatrix is neither symmetric nor alternating Example
- FALSE: ⟨ x,y⟩=∑ᵢxᵢyᵢ makes F₂ⁿ an inner product space False statement
- Determinant criterion: if the matrix (⟨ vᵢ,wⱼ⟩)_i,j is invertible then both v₁,…,vₘ and w₁,…,wₘ are linearly independent Lemma
- Diagonal criterion: if ⟨ vᵢ,wᵢ⟩≠0 and ⟨ vᵢ,wⱼ⟩=0 for i≠ j, then v₁,…,vₘ are linearly independent Lemma
- Triangular criterion: if ⟨ vᵢ,wᵢ⟩≠0 and ⟨ vᵢ,wⱼ⟩=0 for j<i, then v₁,…,vₘ are linearly independent Lemma
- Alternating forms are skew-symmetric; the converse holds when charF≠2, while in characteristic 2 alternating forms are symmetric Theorem
- Bilinear forms on V correspond linearly and bijectively to linear maps V→ V^* Theorem
- Every alternating form on a finite-dimensional space has a basis of symplectic pairs followed by a basis of its radical; in particular its rank is even Theorem
- If charF≠2, quadratic forms and symmetric bilinear forms correspond by q(v)=B(v,v) and B(u,v)=1/2 b_q(u,v) 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
- K. Conrad, Bilinear Forms (standard reference, not scraped)
- H. Pinkham, Linear Algebra, Chapter 7 (standard reference, not scraped)