Search results
Results from the WOW.Com Content Network
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 ...
Let V be a representation of a group G; or more generally, let V be a vector space with a set of linear endomorphisms acting on it. In general, a vector space acted on by a set of linear endomorphisms is said to be simple (or irreducible) if the only invariant subspaces for those operators are zero and the vector space itself; a semisimple representation then is a direct sum of simple ...
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 ...
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 ...
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 ...
Given two I-graded vector spaces V and W, their direct sum has underlying vector space V ⊕ W with gradation ( V ⊕ W ) i = V i ⊕ W i . If I is a semigroup , then the tensor product of two I -graded vector spaces V and W is another I -graded vector space, V ⊗ W {\displaystyle V\otimes W} , with gradation
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} .