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The direct sum is also commutative up to isomorphism, i.e. for any algebraic structures and of the same kind. The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding direct product. This is false, however, for some algebraic objects, like nonabelian groups.
The subspace V × {0} of V ⊕ W is isomorphic to V and is often identified with V; similarly for {0} × W and W. (See internal direct sum below.) With this identification, every element of V ⊕ W can be written in one and only one way as the sum of an element of V and an element of W. The dimension of V ⊕ W is equal to the sum of the ...
The group operation in the external direct sum is pointwise multiplication, as in the usual direct product. This subset does indeed form a group, and for a finite set of groups {H i} the external direct sum is equal to the direct product. If G = ΣH i, then G is isomorphic to Σ E {H i}. Thus, in a sense, the direct sum is an "internal ...
The binary operation, called vector addition or simply addition assigns to any two vectors v and w in V a third vector in V which is commonly written as v + w, and called the sum of these two vectors. The binary function, called scalar multiplication, assigns to any scalar a in F and any vector v in V another vector in V, which is denoted av ...
This applies also when E and F are linear subspaces or submodules of the 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 is an abelian group, vector space, or module (respectively) equipped with two monomorphisms: and : such that is the internal direct sum of ...
A given direct sum decomposition of into complementary subspaces still specifies a projection, and vice versa. If X {\displaystyle X} is the direct sum X = U ⊕ V {\displaystyle X=U\oplus V} , then the operator defined by P ( u + v ) = u {\displaystyle P(u+v)=u} is still a projection with range U {\displaystyle U} and kernel V {\displaystyle V} .
If V is the direct sum of its weight spaces V = ⨁ λ ∈ h ∗ V λ {\displaystyle V=\bigoplus _{\lambda \in {\mathfrak {h}}^{*}}V_{\lambda }} then V is called a weight module ; this corresponds to there being a common eigenbasis (a basis of simultaneous eigenvectors) for all the represented elements of the algebra, i.e., to there being ...
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 ...