Wikipedia / Direct sum
Wikipedia / Direct sum
In mathematics, more specifically in algebra, the direct sum of a collection of abelian groups is an abelian group constructed by combining the given groups in a specific way, described below. If the input abelian groups have additional structure (for example, are vector spaces, modules, or topological abelian groups), then the direct sum also has that structure, typically.
As an example, the direct sum of two abelian groups. To add ordered pairs is real coordinate space, is the Cartesian plane.
are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces). That relies on the fact that the direct sum is associative up to isomorphism of the same kind. The direct sum is also commutative of the same kind.
The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding direct product. That is false, however, for some algebraic objects like nonabelian groups.
In the case where infinitely many objects are combined, the direct sum and direct product are not isomorphic even for abelian groups, vector spaces, or modules. For example, consider the direct sum and the direct product of (countably) infinitely many copies of the integers. An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there is a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both. Often, if a + sign is used, all but finitely many coordinates must be zero, while if some form of multiplication is used, all but finitely many coordinates must be 1.
for all but finitely many i is contained in the direct product, but is strictly smaller when the index set is infinite, because an element of the direct product can have infinitely many nonzero coordinates.
Examples
The xy-plane, a two-dimensional vector space, can be thought of as the direct sum of two one-dimensional vector spaces: the x and y axes. In this direct sum, the x and y, which is the same as vector addition.
. Given an indexed family. Each Ai is called a direct summand of A the phrase "direct product" is used. When the index set is infinite, the direct sum is not the same as the direct product since the direct sum has the extra requirement that all but finitely many coordinates must be zero.
Internal and external direct sums
, the direct sum is said to be external.
.
Types of direct sums
Direct sum of abelian groups
The direct sum of abelian groups is a prototypical example of a direct sum. Given two such groups is the same as their direct product. That is, the underlying set is the Cartesian product This definition generalizes to direct sums of finitely many abelian groups.
their is the subgroup that have finite support is said to have of non-trivial groups is a proper subgroup
Direct sum of modules
The direct sum of modules is a construction that combines several modules into a new module.
The most familiar examples of that construction occur in considering vector spaces, which are modules over a field. The construction may also be extended to Banach spaces and Hilbert spaces.
Direct sum in categories
An additive category is an abstraction of the properties of the category of modules. In such a category, finite products and coproducts agree, and the direct sum is either of them: cf. biproduct.
More generally, in category theory the is often but not always the coproduct in the category of the mathematical objects in question. For example, in the category of abelian groups, the direct sum is a coproduct. That is also true in the category of modules.
Direct sum of group representations
The direct sum of group representations generalizes the direct sum of the underlying modules by adding a group action. Specifically, given a group and two representations (or, more generally, two G-modules
is the natural map obtained by coordinate-wise action as above.
are viewed as modules over the group ring-modules.
Direct sum of rings
The direct product. is not a coproduct in the category of rings, and should not be written as a direct sum. (The coproduct in the category of commutative rings is the tensor product of rings. In the category of rings, the coproduct is given by a construction similar to the free product of groups.)
is an infinite collection of nontrivial rings, the direct sum of the underlying additive groups may be equipped with termwise multiplication, but that produces a rng, a ring without a multiplicative identity.
Direct sum of matrices
Direct sum of topological vector spaces
A topological vector space such as a Banach space, is said to be a is an isomorphism of topological vector spaces (meaning that this linear map is a bijective homeomorphism are said to be That is true if and only if when considered as additive topological groups is the topological direct sum of the topological subgroups is Hausdorff are necessarily closed
called an is the is a vector space isomorphism.
In contrast to algebraic direct sums, the existence of such a complement is no longer guaranteed for topological direct sums.
is said to be a () A vector subspace is called if it is not a complemented subspace. For example, every vector subspace of a Hausdorff TVS that is not a closed subset is necessarily uncomplemented. Every closed vector subspace of a Hilbert space is complemented. But every Banach space that is not a Hilbert space necessarily possess some uncomplemented closed vector subspace.
Homomorphisms
comes equipped with a projection homomorphism for each j in I and a coprojection for each j in I. for every j in I, called the sum of the gj for all j. Thus the direct sum is the coproduct in the appropriate category.