Direct Sum (Noun)
Meaning
A union of two disjoint sets in which every element is the sum of an element from each of the disjoint sets.
Classification
Nouns denoting groupings of people or objects.
Examples
- The direct sum of two vector spaces V and W is a new vector space formed by taking the Cartesian product of V and W, and defining addition and scalar multiplication component-wise.
- The direct sum of two groups G and H is an abelian group when both G and H are abelian.
- In algebra, the tensor product and the direct sum are used to combine two algebraic structures.
- Finite direct sums are used to define Hilbert spaces in functional analysis.
- The direct sum is particularly useful for categorizing the finite simple groups in group theory.