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 annihilator of and the preannihilator of
Definition
Let be a linear subspace. Its annihilator in the algebraic dual is
For a linear subspace , its preannihilator in is
Both are linear subspaces because all their defining equations are homogeneous and linear. The superscript position records which side of the evaluation pairing is being annihilated.
Depends on
Used by
- In infinite dimension, distinct subspaces of V^* can have the same preannihilator Counterexample
- The annihilator of the coordinate plane z=0 in ℝ³ is the line spanned by the third coordinate functional Example
- Assuming choice, ^∘(U^∘)=U; in finite dimension, dim U^∘=dim V-dim U Theorem
- Assuming choice, ker T^*=(imT)^∘ and imT^*=(ker T)^∘; in finite dimensions rankT^*=rankT Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 17 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
- H. Pinkham, Linear Algebra, §6.4 (standard reference, not scraped)
- K. Conrad, Infinite-Dimensional Dual Spaces (standard reference, not scraped)