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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 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.
where is the exterior product and is the direct sum. The individual elements of this algebra are then called Grassmann numbers . It is standard to omit the wedge symbol ∧ {\displaystyle \wedge } when writing a Grassmann number once the definition is established.
If (V,φ) and (W,ψ) are representations of (say) a group G, then the direct sum of V and W is a representation, in a canonical way, via the equation (,) = (,). The direct sum of two representations carries no more information about the group G than the two representations do individually. If a representation is the direct sum of two proper ...
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.
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 symmetric algebra S(V) can be built as the quotient of the tensor algebra T(V) by the two-sided ideal generated by the elements of the form x ⊗ y − y ⊗ x. All these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring.
In mathematics, the term "graded" has a number of meanings, mostly related: . In abstract algebra, it refers to a family of concepts: . An algebraic structure is said to be -graded for an index set if it has a gradation or grading, i.e. a decomposition into a direct sum = of structures; the elements of are said to be "homogeneous of degree i ".
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