Alphabeta MathDual Spaces and Bilinear Forms: Examples and Counterexamples
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✓ 12 results · all verified · 2 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 10 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Example
A dual basis and transpose computed for a linear map on F₃²
Example
The annihilator of the coordinate plane z=0 in ℝ³ is the line spanned by the third coordinate functional
Example
For the polynomial space F[x], the canonical map to the algebraic double dual is injective but not surjective
Counterexample
In infinite dimension, distinct subspaces of V^* can have the same preannihilator
Example
The form on ℝ² with matrix beginpmatrix0&10&0 endpmatrix is neither symmetric nor alternating
Example
q(x,y)=2x²+4xy+5y²=2(x+y)²+3y² has inertia (2,0,0)
Example
A rank-two alternating form on ℝ³ has one symplectic pair and a one-dimensional radical
Counterexample
In characteristic 2, a symmetric bilinear form need not have an orthogonal basis
Counterexample
In characteristic 2, distinct quadratic forms can have the same polar form
Counterexample
Positive determinant does not imply positive definiteness
Example
Sylvester's criterion verifies positive definiteness for a concrete symmetric 3×3 matrix
Counterexample