Rank–nullity theorem

The rank–nullity theorem is a theorem in linear algebra, which asserts: It follows that for linear transformations of vector spaces of equal finite dimension, either injectivity or surjectivity implies bijectivity.

be a linear transformation between two vector spaces where

This theorem can be refined via the splitting lemma to be a statement about an isomorphism of spaces, not just dimensions.

Linear maps can be represented with matrices.

matrix M represents a linear map

The first[2] operates in the general case, using linear maps.

and shows explicitly that there exists a set of

linearly independent solutions that span the null space of

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.

We may now, by the Steinitz exchange lemma, extend

; it remains to be shown that it is also linearly independent to conclude that it is a basis.

is linearly independent, and more specifically that it is a basis for

{\displaystyle \operatorname {Rank} (T)+\operatorname {Nullity} (T)=\dim \operatorname {Im} T+\dim \operatorname {Ker} T}

whose columns form a basis of the null space of

must be a linear combination of the columns of

constitute a basis for the null space of

is a linear transformation between two finite-dimensional subspaces, with

is considered alongside its image and kernel: the cokernel of

[7][8] This theorem is a statement of the first isomorphism theorem of algebra for the case of vector spaces; it generalizes to the splitting lemma.

In more modern language, the theorem can also be phrased as saying that each short exact sequence of vector spaces splits.

is a short exact sequence of vector spaces, then

In the finite-dimensional case, this formulation is susceptible to a generalization: if

is an exact sequence of finite-dimensional vector spaces, then[9]

The rank–nullity theorem for finite-dimensional vector spaces may also be formulated in terms of the index of a linear map.

is the number of independent restrictions that have to be put on

The rank–nullity theorem for finite-dimensional vector spaces is equivalent to the statement

We see that we can easily read off the index of the linear map

This effect also occurs in a much deeper result: the Atiyah–Singer index theorem states that the index of certain differential operators can be read off the geometry of the involved spaces.