Rank–nullity theorem

The rank theorem or dimension set is a set from the mathematical subfield of linear algebra. He points to a relationship between the dimensions of the set of definitions, the core and the image of a linear map between two vector spaces.

Set

Is a linear map from a vector space into a vector space, then, for the dimensions of the set of definitions, the core and the image of the figure, the equation

If one uses the terms defect for the dimension of the core and rank for the dimension of the image of the figure, that is the rank theorem:

Evidence

Proof of the homomorphism

The theorem follows immediately from the homomorphism

Since the factor space is isomorphic to a complementary space of in is true

Now

Is follows from the equivalence of isomorphism and equality of dimension

Proof by base addition

If a set is a base of which is complemented by a set to a basis of ( is then a basis of a complementary space of ), then

A base of image. If we now consider the restriction of the linear hull

Then is injective and

Thus is an isomorphism between and the image of. Therefore applies

The homomorphism theorem follows also - through the transition from the complementary space to the factor space.

Reversal

The rate applies to vector spaces of arbitrary (even infinite ) dimension. In the finite dimensional case, the dimension of the image space can be from the dimension of the core as

Calculate. According to vice versa then we also have

In the infinite-dimensional case, the dimension of the image space can be set by means of the rank not from the dimension of the core ( or vice versa) calculated when the core has the same dimension as the entire room. Otherwise, the dimension of the image space is equal to the dimension of.

Generalization

A far-reaching generalization of the rank theorem is the statement that the alternating sum of the dimensions of the individual components of a complex chain is equal to the alternating sum of the dimensions of its homology groups. See the Euler characteristic of a chain complex.