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.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf | theorem in linear algebra ⓘ |
| alsoKnownAs |
dimension theorem
ⓘ
rank-nullity theorem NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.