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.
Kernel and image of a linear map
Definition
For a linear map , its kernel and image are respectively
That both sets are linear subspaces, and that a trivial kernel characterises injectivity, is proved in The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial ↗.
Depends on
Used by
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism Definition
- Rank and nullity of a linear map with finite-dimensional domain Definition
- The tangent space to a regular level set Definition
- The quotient by the kernel is isometric to the range with its induced quotient norm Example
- A linear functional annihilating the kernel of a surjection is a unique transpose multiple Lemma
- Extending a basis of the kernel to a basis of the domain gives a basis of the image Lemma
- Kernel and rank sequences of powers stabilise once equality occurs Lemma
- Orthogonal complement of an eigenspace is invariant Lemma
- Positive square root of a compact positive operator Lemma
- Singular values equal approximation numbers Lemma
- A linear map T of ℝⁿ sends Lebesgue measurable sets to Lebesgue measurable sets, with λₙ(T[E])=|det T| λₙ(E) when T is invertible and T[E] Lebesgue null when it is not Theorem
- Spectral theorem for compact self adjoint operators Theorem
- The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial Theorem
- Universal property of the quotient vector space Theorem
Dependency tree · two levels
3 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
- Axler, Linear Algebra Done Right, Chapter 3 (standard reference, not scraped)