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For an arbitrary family of groups indexed by , their direct sum [2] is the subgroup of the direct product that consists of the elements () that have finite support, where by definition, () is said to have finite support if is the identity element of for all but finitely many . [3] The direct sum of an infinite family () of non-trivial groups is ...
The direct sum is a submodule of the direct product of the modules M i (Bourbaki 1989, §II.1.7). The direct product is the set of all functions α from I to the disjoint union of the modules M i with α(i)∈M i, but not necessarily vanishing for all but finitely many i. If the index set I is finite, then the direct sum and the direct product ...
For any non-negative integer n, =, the cartesian product of n copies of R as a left R-module, is free. If R has invariant basis number, then its rank is n. A direct sum of free modules is free, while an infinite cartesian product of free modules is generally not free (cf. the Baer–Specker group).
The direct sum and direct product are not isomorphic for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of category theory : the direct sum is the coproduct , while the direct product is the product.
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix.It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis.
For example, the coproduct in the category of groups, called the free product, is quite complicated. On the other hand, in the category of abelian groups (and equally for vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only finitely many nonzero terms. (It therefore coincides ...
So M breaks up as the direct sum of R-modules, M = e 1 M ⊕ ... ⊕ e n M. Conversely, given an R-module M 0, then M 0 ⊕n is an M n (R)-module. In fact, the category of R-modules and the category of M n (R)-modules are equivalent. The special case is that the module M is just R as a module over itself, then R n is an M n (R)-module.
The 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 E x ⊕ F x of the vector spaces E x and F x. The tensor product bundle E ⊗ F is defined in a similar way, using fiberwise tensor product of vector spaces.