rank–nullity theorem
E621086
The rank–nullity theorem is a fundamental result in linear algebra that relates the dimension of a vector space to the sum of the dimensions of the kernel and image of a linear transformation.
All labels observed (2)
| Label | Occurrences |
|---|---|
| rank-nullity theorem | 1 |
| rank–nullity theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6832962 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: rank–nullity theorem Context triple: [linear algebra, hasKeyTheorem, rank–nullity theorem]
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A.
Sylvester’s law of inertia
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
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B.
linear algebra
Linear algebra is a branch of mathematics that studies vectors, vector spaces, linear transformations, and systems of linear equations, forming a foundation for many areas of science and engineering.
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C.
Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
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D.
Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
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E.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: rank–nullity theorem Target entity description: The rank–nullity theorem is a fundamental result in linear algebra that relates the dimension of a vector space to the sum of the dimensions of the kernel and image of a linear transformation.
-
A.
Sylvester’s law of inertia
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
-
B.
linear algebra
Linear algebra is a branch of mathematics that studies vectors, vector spaces, linear transformations, and systems of linear equations, forming a foundation for many areas of science and engineering.
-
C.
Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
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D.
Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
-
E.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf | theorem in linear algebra ⓘ |
| alsoKnownAs |
dimension theorem
ⓘ
rank-nullity theorem ⓘ
linked to:
rank–nullity theorem
|
| appliesTo |
homomorphisms of modules over a field (vector spaces)
ⓘ
linear transformation ⓘ matrix ⓘ |
| assumes | finite dimension of the domain vector space ⓘ |
| criterion |
A linear map T: V → W from a finite-dimensional space is injective if and only if rank(T) = dim(V).
ⓘ
A linear map T: V → W is injective if and only if nullity(T) = 0. ⓘ For T: V → W with dim(V) = dim(W), T is injective if and only if it is surjective. ⓘ |
| expresses | dim(V) = dim(ker T) + dim(im T) ⓘ |
| field | linear algebra ⓘ |
| generalizationOf | rank theorem for matrices ⓘ |
| holdsFor | finite-dimensional vector spaces ⓘ |
| holdsOver | any field ⓘ |
| implies |
The image of a linear transformation from a finite-dimensional space is finite-dimensional.
ⓘ
The kernel of a linear transformation from a finite-dimensional space is finite-dimensional. ⓘ |
| importance | fundamental structural result about linear maps on finite-dimensional spaces ⓘ |
| involvesConcept |
basis
ⓘ
dimension of image ⓘ dimension of kernel ⓘ dimension theorem for vector spaces ⓘ image of a linear transformation ⓘ kernel of a linear transformation ⓘ linear independence ⓘ span ⓘ subspace ⓘ vector space ⓘ |
| isTaughtIn | undergraduate linear algebra courses ⓘ |
| relatedTo |
column space
ⓘ
first isomorphism theorem for vector spaces ⓘ fundamental theorem of linear algebra ⓘ
linked to:
Singular value decomposition
row space ⓘ solution space of homogeneous linear equations ⓘ |
| relatesConcept |
dimension of a vector space
ⓘ
nullity of a linear transformation ⓘ rank of a linear transformation ⓘ |
| statement |
For a linear transformation T: V → W between finite-dimensional vector spaces, dim(V) = rank(T) + nullity(T).
ⓘ
For an m×n matrix A over a field, n = rank(A) + nullity(A). ⓘ |
| typicalProofUses |
basis extension
ⓘ
direct sum decomposition ⓘ isomorphism between quotient space and image ⓘ |
| usedFor |
analyzing solution spaces of linear systems
ⓘ
characterizing injective linear maps ⓘ characterizing surjective linear maps ⓘ computing dimension of kernel from rank ⓘ computing rank from nullity ⓘ proving isomorphism theorems in linear algebra ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: rank–nullity theorem Description of subject: The rank–nullity theorem is a fundamental result in linear algebra that relates the dimension of a vector space to the sum of the dimensions of the kernel and image of a linear transformation.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: rank–nullity theorem