The rank–nullity theorem is a theorem in linear algebra, which asserts:
the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f) and the nullity of f (the dimension of the kernel of f). It follows that for linear transformations of vector spaces of equal finite dimension, either injectivity or surjectivity implies bijectivity.
Stating the theorem
Linear transformations Let T : V → W {\displaystyle T:V\to W} be a linear transformation between two vector spaces where T {\displaystyle T} 's domain V {\displaystyle V} is finite dimensional. Then
rank ( T ) + nullity ( T ) = dim V , {\displaystyle \operatorname {rank} (T)~+~\operatorname {nullity} (T)~=~\dim V,}
where rank ( T ) {\textstyle \operatorname {rank} (T)} is the rank of T {\displaystyle T} (the dimension of its image) and nullity ( T ) {\displaystyle \operatorname {nullity} (T)} is the nullity of T {\displaystyle T} (the dimension of its kernel). In other words,
dim ( Im T ) + dim ( Ker T ) = dim ( Domain ( T ) ) . {\displaystyle \dim(\operatorname {Im} T)+\dim(\operatorname {Ker} T)=\dim(\operatorname {Domain} (T)).}
This theorem can be refined via the splitting lemma to be a statement about an isomorphism of spaces, not just dimensions. Explicitly, since T {\displaystyle T} induces an isomorphism from V / Ker ( T ) {\displaystyle V/\operatorname {Ker} (T)} to Im ( T ) , {\displaystyle \operatorname {Im} (T),} the existence of a basis for V {\displaystyle V} that extends any given basis of Ker ( T ) {\displaystyle \operatorname {Ker} (T)} implies, via the splitting lemma, that Im ( T ) ⊕ Ker ( T ) ≅ V . {\displaystyle \operatorname {Im} (T)\oplus \operatorname {Ker} (T)\cong V.} Taking dimensions, the rank–nullity theorem follows.
Matrices Linear maps can be represented with matrices. More precisely, an m × n {\displaystyle m\times n} matrix M represents a linear map f : F n → F m , {\displaystyle f:F^{n}\to F^{m},} where F {\displaystyle F} is the underlying field. So, the dimension of the domain of f {\displaystyle f} is n, the number of columns of M, and the rank–nullity theorem for an m × n {\displaystyle m\times n} matrix M is
rank ( M ) + nullity ( M ) = n . {\displaystyle \operatorname {rank} (M)+\operatorname {nullity} (M)=n.}
Proofs Here we provide two proofs. The first operates in the general case, using linear maps. The second proof looks at the homogeneous system A x = 0 , {\displaystyle \mathbf {Ax} =\mathbf {0} ,} where A {\displaystyle \mathbf {A} } is a m × n {\displaystyle m\times n} with rank r , {\displaystyle r,} and shows explicitly that there exists a set of n − r {\displaystyle n-r} linearly independent solutions that span the null space of A {\displaystyle \mathbf {A} } . While the theorem requires that the domain of the linear map be finite-dimensional, there is no such assumption on the codomain. This means that there are linear maps not given by matrices for which the theorem applies. Despite this, the first proof is not actually more general than the second: since the image of the linear map is finite-dimensional, we can represent the map from its domain to its image by a matrix, prove the theorem for that matrix, then compose with the inclusion of the image into the full codomain.
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