Rank-nullity theorem Guide, Meaning , Facts, Information and Description
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.
This formulation is susceptible to a generalization: if
- 0 → V1 → V2 → ... → Vr → 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.
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