Definitions from the Web
Term: Direct Sum
Description: In mathematics, the direct sum is a binary operation defined on mathematical structures, such as vector spaces, groups, or modules. It combines two structures to create a new one by preserving the independent nature of each structure.
Senses:
- Algebraic Sense: In algebra, the direct sum of two vector spaces or modules is the set of all possible sums of pairs of vectors or elements, respectively, taken from each individual space or module.
- Representation Theory Sense: In representation theory, the direct sum of two representations is a new representation that combines the original representations by applying them independently.
Usages:
- Algebra Example: The direct sum of vector spaces V and W, denoted as V ⊕ W, is the set of vectors obtained by taking one vector from V and one vector from W, with vector addition and scalar multiplication defined component-wise.
- Representation Theory Example: When two irreducible representations are combined via direct sum, their individual properties are preserved, and this new representation can provide insights into the original representations.
Sample Sentences:
- The direct sum of the vector spaces ℝ³ and ℂ² is a new vector space with the dimension 5.
- The direct sum of two cyclic groups G and H produces a new group G ⊕ H with the elements (g, h), where g ∈ G and h ∈ H.
- The direct sum of two irreducible representations of a symmetry group allows us to study the symmetry properties of the combined representation.
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