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Rank-nullity theorem

 

 
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Rank-nullity theorem
 
 
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In mathematicsMathematics

Mathematics is the discipline that deals with concepts such as quantity, structure, space and change....
, the rank-nullity theorem of linear algebraLinear algebra

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces , linear transformations, and...
, in its simplest form, relates the rank and the nullityNullity

Nullity may refer to:* Nullity , a legal declaration that no marriage had ever come into being...
 of a matrix together with the number of columns of the matrix. Specifically, if A is an m-by-n matrix over the fieldField (mathematics)

In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and ...
 F, then
rank A + nullity A = n.


This applies to linear maps as well. Let V and W be vector spaceVector space

In mathematics, a vector space is a collection of objects that, informally speaking, may be scaled and added....
s over the field F and let T : V → W be a linear map. Then the rank of T is the dimension of the image of T, the nullity the dimension of the kernelKernel (algebra)

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measure...
 of T, and we have
dim (im T) + dim (ker T) = dim V

thus, equivalently,
rank T + nullity T = dim V.

This is in fact more general than the matrix statement above, because here V and W may even be infinite-dimensional.

To prove the theorem, one starts with a basisFacts About Basis (linear algebra)

In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector s...
 of the kernel of T, and extends it to a basis of all of V. It is then not too difficult to show that T applied to the "new" basis vectors yields a basis of the image of T.

Reformulations and generalizations

This theorem is a statement of the first isomorphism theorem of algebra to the case of vector spaces.

In more modern language, the theorem can also be phrased as follows: if
0 → U → V → R → 0

is a short exact sequence of vector spaces, then
dim(U) + dim(R) = dim(V)

Here R plays the role of im T and U is ker T.

In the finite-dimensional case, this formulation is susceptible to a generalization: if
0 → V1 → V2 → ... → Vr → 0

is an exact sequenceExact sequence

In mathematics, especially in homological algebra and other applications of abelian category theory, as well as in group theory, a...
 of finite-dimensional vector spaces, then

The rank-nullity theorem for finite-dimensional vector spaces may also be formulated in terms of the index of a linear map. The index of a linear map T : V → W, where V and W are finite-dimensional, is defined by
index T = dim(ker T) - dim(cokerCokernel

In mathematics, the cokernel of a morphism f : X → Y is an object Q and a morphism q : Y →...
 T).

Intuitively, dim(ker T) is the number of independent solutions x of the equation Tx = 0, and dim(coker T) is the number of independent restrictions that have to be put on y to make Tx = y solvable. The rank-nullity theorem for finite-dimensional vector spaces is equivalent to the statement
index T = dim(V) - dim(W).

We see that we can easily read off the index of the linear map T from the involved spaces, without any need to analyze T 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.