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direct sum
This term is a technical expression used exclusively within abstract algebra and linear algebra. It describes a specific way of constructing a larger mathematical object from smaller ones while preserving the independence of the original components. The key characteristic is the uniqueness of representation, meaning any element in the resulting space has exactly one way to be written as a sum of elements from the constituent spaces.
In a pedagogical context, it is often contrasted with the "internal direct sum," where the spaces are already subspaces of a larger containing space, versus the "external direct sum," where the spaces are combined from the outside. Because it is a specialized mathematical term, it does not have a casual or metaphorical usage in general English.
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An algebraic operation that combines two or more structures, such as vector spaces, modules, or abelian groups, into a larger structure where each component is embedded as a subspace and every element of the resulting structure can be uniquely represented as a sum of elements from the original components.
The direct sum of two vector spaces V and W is denoted by V ⊕ W.