Rank-nullity theorem

From Academic Kids

In mathematics, the rank-nullity theorem of linear algebra, 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 : VW 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 → UVR → 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 → V1V2 → ... → Vr → 0

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

<math>\sum_{i=1}^r (-1)^i\dim(V_i) = 0.<math>

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 : VW, where V and W are finite-dimensional, 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 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.


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