Rank–nullity theorem

From Wikipedia, the free encyclopedia

In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to 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. ^[1]

This applies to linear maps as well. Let V and W be vector spaces over the field F and let T : V → W be a linear map. 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

or, 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.

One can refine this statement (via the splitting lemma or the below proof) to be a statement about an isomorphism of spaces, not just dimensions: in addition to:

$\dim \operatorname{im}\,T + \dim \ker T = \dim V$

one in fact also has:

$\operatorname{im}\,T \oplus \ker T \simeq V.$

More generally, one can consider the image, kernel, coimage, and cokernel, which are related by the fundamental theorem of linear algebra.

[edit] Proof

Suppose $\{\mathbf{u}_1, \ldots, \mathbf{u}_m\}$ forms a basis of ker T. We can extend this to form a basis of V: $\{\mathbf{u}_1, \ldots, \mathbf{u}_m, \mathbf{w}_1, \ldots, \mathbf{w}_n\}$ .

Since the dimension of ker T is m and the dimension of V is m+n, it suffices to show that the dimension of image T is n.

Let us conjecture that $\{T\mathbf{w}_1, \ldots, T\mathbf{w}_n \}$ forms a basis of image T.

Let v be an arbitrary vector in V. There exist unique scalars such that:

$\mathbf{v}=a_1 \mathbf{u}_1 + \cdots + a_m \mathbf{u}_m + b_1 \mathbf{w}_1 +\cdots + b_n \mathbf{w}_n$

$\Rightarrow T\mathbf{v} = a_1 T\mathbf{u}_1 + \cdots + a_m T\mathbf{u}_m + b_1 T\mathbf{w}_1 +\cdots + b_n T\mathbf{w}_n$

$\Rightarrow T\mathbf{v} = b_1 T\mathbf{w}_1 + \cdots + b_n T\mathbf{w}_n \; \; \because T\mathbf{u}_i = 0$

Thus, $\{T\mathbf{w}_1, \ldots, T\mathbf{w}_n \}$ spans image T.

We only now need to show that this list is not redundant; that is, that $\{T\mathbf{w}_1, \ldots, T\mathbf{w}_n \}$ are linearly independent. We can do this by showing that a linear combination of these vectors is zero iff. the coefficient on each vector is zero. Let:

$c_1 T\mathbf{w}_1 + \cdots + c_n T\mathbf{w}_n = 0 \Leftrightarrow T\{c_1 \mathbf{w}_1 + \cdots + c_n \mathbf{w}_n\}=0$

$\therefore c_1 \mathbf{w}_1 + \cdots + c_n \mathbf{w}_n \in \operatorname{ker} \; T$

Then, since $\mathbf{u}_i$ span ker T, there exists a set of scalars $d i$ such that:

$c_1 \mathbf{w}_1 + \cdots + c_n \mathbf{w}_n = d_1 \mathbf{u}_1 + \cdots + d_m \mathbf{u}_m$

But, since $\{\mathbf{u}_1, \ldots, \mathbf{u}_m, \mathbf{w}_1, \ldots, \mathbf{w}_n\}$ form a basis of V, all $c_i, \; d_i$ must be zero.

Therefore, $\{T\mathbf{w}_1, \ldots, T\mathbf{w}_n \}$ is linearly independent and indeed a basis of image T. This proves that the dimension of image T is n, as desired.

In more abstract terms, the map $T\colon V \to \operatorname{image}\,T$ splits.

[edit] Reformulations and generalizations

This theorem is a statement of the first isomorphism theorem of algebra to the case of vector spaces; it generalizes to the splitting lemma.

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 → ... → V_r → 0

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

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

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.

[edit] See also

[edit] Notes

^ Meyer (2000), page 199.

[edit] References

Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871-454-8, http://www.matrixanalysis.com/ .

[0] Meyer (2000), page 199.

[1]

Rank–nullity theorem

From Wikipedia, the free encyclopedia

Contents

[edit] Proof

[edit] Reformulations and generalizations

[edit] See also

[edit] Notes

[edit] References

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