The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different...

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest...

In mathematics, a group G is called the direct sum of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra...

such as the direct sum and the Kronecker sum. Two matrices must have an equal number of rows and columns to be added. In which case, the sum of two matrices...

group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol ⋉. There are two closely...

product of topological spaces is another instance. There is also the direct sum – in some areas this is used interchangeably, while in others it is a...

mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is...

a combination of algebraic objects Direct sum of groups Direct sum of modules Direct sum of permutations Direct sum of topological groups Einstein summation...

mathematics, a topological group G {\displaystyle G} is called the topological direct sum of two subgroups H 1 {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}}...

Πi∈I Ri coincides with the direct sum of the additive groups of the Ri. In this case, some authors call R the "direct sum of the rings Ri" and write ⊕i∈I...

Whitney sum (named for Hassler Whitney) or direct sum bundle of E and F is a vector bundle E ⊕ F over X whose fiber over x is the direct sum Ex ⊕ Fx of...

canonically isomorphic to the direct sum of Vi. In this case, H is called the internal direct sum of the Vi. A direct sum (internal or external) is also...

Look up direct in Wiktionary, the free dictionary. Direct may refer to: Directed set, in order theory Direct limit of (pre), sheaves Direct sum of modules...

dimensions). Note that any element in the direct sum of two vector spaces of matrices could be represented as a direct sum of two matrices. A block diagonal matrix...

similar to the corresponding statements for groups. The direct product of vector spaces and the direct sum of vector spaces are two ways of combining an indexed...

sets, particularly on infinite products, as in the product topology or direct sum. This use of the prefix "co" to describe a property possessed by a set's...

grading or gradation, which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers. For "pure" vector...

vector space or module V. 2.  Direct sum: if E and F are two abelian groups, vector spaces, or modules, then their direct sum, denoted E ⊕ F {\displaystyle...

each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups...

the direct sum of representations please refer to the section on direct sums of representations. A representation is called isotypic if it is a direct sum...

categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum...

the algebraic direct sum by requiring certain maps be continuous; the result retains many nice properties from the operation of direct sum in finite-dimensional...

a direct sum of unitaries acting on 1-dimensional complex spaces (eigenspaces), but an antiunitary operator may only be decomposed into a direct sum of...

In combinatorics, the skew sum and direct sum of permutations are two operations to combine shorter permutations into longer ones. Given a permutation...

notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted...

direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum...

abelian group with basis B {\displaystyle B} may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member...

it is the direct sum of simple (irreducible) submodules. For a module M, the following are equivalent: M is semisimple; i.e., a direct sum of irreducible...

to a direct sum of copies of the ring R. These are the modules that behave very much like vector spaces. Projective Projective modules are direct summands...

{a}}_{i}} as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals...