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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.
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 ...
An abelian category [4] C is called semi-simple if there is a collection of simple objects , i.e., ones with no subobject other than the zero object 0 and itself, such that any object X is the direct sum (i.e., coproduct or, equivalently, product) of finitely many simple objects.
(That is, if W is an invariant subspace, then there is another invariant subspace P such that V is the direct sum of W and P.) If g {\displaystyle {\mathfrak {g}}} is a finite-dimensional semisimple Lie algebra over a field of characteristic zero and V is finite-dimensional, then V is semisimple; this is Weyl's complete reducibility theorem . [ 4 ]
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 ...
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Determining whether a given subspace W is invariant under T is ostensibly a problem of geometric nature. Matrix representation allows one to phrase this problem algebraically. Write V as the direct sum W ⊕ W′; a suitable W′ can always be chosen by extending a basis of W. The associated projection operator P onto W has matrix representation