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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Linear algebra is the branch of Mathematics concerned with The column rank of a matrix A is the maximal number of Linearly independent columns of A. Specifically, if A is an m-by-n matrix over the field F, then
- rank A + nullity A = n. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division
This applies to linear maps as well. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Let V and W be vector spaces over the field F and let T : V → W be a linear map. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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. In Mathematics, the dimension of a Vector space V is the cardinality (i In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism
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. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. 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
This theorem is a statement of the first isomorphism theorem of algebra to the case of vector spaces. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural
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 Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group
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
The rank-nullity theorem for finite-dimensional vector spaces may also be formulated in terms of the index of a linear map. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group 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. In the Mathematics of Manifolds and Differential operators the Atiyah–Singer index theorem states that for an elliptic differential operator
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