Rank-nullity theorem
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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 : 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.
In the finite-dimensional case, this formulation is susceptible to a generalization: if
- 0 → V1 → V2 → ... → 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 : V → W, 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.