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Wikipedia / 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.

Stating the theorem

Linear transformations

is the rank (the dimension of its image is the nullity (the dimension of its kernel This theorem can be refined via the splitting lemma to be a statement about an isomorphism Taking dimensions, the rank–nullity theorem follows.

Matrices

Linear maps can be represented with matrices matrix is the underlying field. is , the number of columns of matrix

Proofs

Here we provide two proofs. The first operates in the general case, using linear maps. The second proof with rank linearly independent.

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.

First proof

.

is a subspace be such a basis.

By the Steinitz exchange lemma; it remains to be shown that it is also linearly independent to conclude that it is a basis.

.

.

This concludes our proof.

Second proof

linearly independent). We will show that:

whose columns form a basis.

Without loss of generality where

  • linearly independent column vectors, and

  • .

(see rank factorization

identity matrix

.

are linearly independent.

We next prove that any must be a linear combination.

.

constitute a basis for the null space. Therefore, the nullity. This concludes our proof.

A third fundamental subspace

is the dimension of the null space, which represents the kernel is considered alongside its image and kernel: the cokernel is the quotient space) together with the rank–nullity theorem is sometimes called the fundamental theorem of linear algebra.

Reformulations and generalizations

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

is a short exact sequence

is an exact sequence of finite-dimensional vector spaces, then The rank–nullity theorem for finite-dimensional vector spaces may also be formulated in terms of the index

in detail. 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.

Citations

References

  • .