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If is a vector subspace of a real or complex vector space then there always exists another vector subspace of , called an algebraic complement of in , such that is the algebraic direct sum of and (which happens if and only if the addition map is a vector space isomorphism). In contrast to algebraic direct sums, the existence of such a ...
This vector space is the coproduct (or direct sum) of countably many copies of the vector space F. Note the role of the finiteness condition here. One could consider arbitrary sequences of elements in F, which also constitute a vector space with the same operations, often denoted by F N - see below. F N is the product of countably many copies of F.
When the scalar field is the real numbers, the vector space is called a real vector space, and when the scalar field is the complex numbers, the vector space is called a complex vector space. [4] These two cases are the most common ones, but vector spaces with scalars in an arbitrary field F are also commonly considered.
In particular, the direct sum of square matrices is a block diagonal matrix. The adjacency matrix of the union of disjoint graphs (or multigraphs) is the direct sum of their adjacency matrices. Any element in the direct sum of two vector spaces of matrices can be represented as a direct sum of two matrices. In general, the direct sum of n ...
The symmetric tensors of degree n form a vector subspace (or module) Sym n (V) ⊂ T n (V). The symmetric tensors are the elements of the direct sum = (), which is a graded vector space (or a graded module). It is not an algebra, as the tensor product of two symmetric tensors is not symmetric in general.
Each of the H i is included as a closed subspace in the direct sum of all of the H i. Moreover, the H i are pairwise orthogonal. Conversely, if there is a system of closed subspaces, V i, i ∈ I, in a Hilbert space H, that are pairwise orthogonal and whose union is dense in H, then H is canonically isomorphic to the direct sum of V i.
In mathematics, a group G is called the direct sum [1] [2] of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information.
This parallels the extension of the scalar product of vector spaces to the direct sum above. The resulting abelian group is called the direct sum of G and H and is usually denoted by a plus symbol inside a circle: It is customary to write the elements of an ordered sum not as ordered pairs (g, h), but as a sum g + h.