Alphabeta MathLinear Transformations, Rank-Nullity and Quotient Spaces
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How 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.
✓ 4 results · all verified · 0 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 4 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Linear map between vector spaces over the same field
Lemma
A linear map preserves zero, negatives, and subtraction
Definition
Kernel and image of a linear map
Theorem
The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial
Lemma
Extending a basis of the kernel to a basis of the domain gives a basis of the image
Definition
Rank and nullity of a linear map with finite-dimensional domain
Theorem