Rank-nullity theorem

Sponsorship the way you would do it

In linear algebra, the rank-nullity theorem, in its simplest form, relates the rank and the nullity of a matrix together with the number of columns of the matrix. Specifically, if A is an m-by-n matrix over the field F, then

rank A + nullity A = n.

This applies to linear transformations as well. Let V and W be vector spaces over the field F and let T : V → W be a linear transformation. Then the rank of T is the dimension of the image of T, the nullity the dimension of the kernel 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 basis 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

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.

This formulation is susceptible to a generalization: if

0 → V₁ → V₂ → ... → V_r → 0

is an exact sequence of vector spaces, then

The rank-nullity theorem may also be formulated in terms of the index of a linear map. The index of T : V → W is defined by

index T = dim(ker T) - dim(coker 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 (at least for the case of 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.