Definition

1. Noun:
   • A mathematical construction: In abstract algebra, particularly in module theory and linear algebra, a "direct sum" is a way to combine two or more algebraic structures (like modules, vector spaces, or abelian groups) into a larger structure. The resulting object is built from its components in a specific, controlled manner where each element can be uniquely represented.
   • The specific construction: Formally, for two modules (or similar structures) A and B, their direct sum, denoted A ⊕ B, is the set of all ordered pairs (a, b) where a is in A and b is in B. The operations (like addition) are defined component-wise.

Usage

• The term "direct sum" is used primarily in formal, academic, and technical contexts related to mathematics.
• It functions as a compound noun. The word "direct" modifies the type of "sum," distinguishing it from other types of sums like the direct product or internal sum.
• It is often used with the preposition "of" (e.g., "the direct sum of two subspaces").

Examples

• Noun:
  • The vector space R³ can be expressed as the direct sum of a line and a plane.
  • Every finite-dimensional vector space is isomorphic to a direct sum of copies of its base field.
  • The module M is decomposable if it can be written as a direct sum of two non-zero submodules.

Advanced Usage

• Internal Direct Sum: When referring to submodules or subspaces U and W of a module V, we say V is the internal direct sum of U and W if every element v in V can be written as v = u + w, with u in U and w in W, and U ∩ W = {0}. This is a property of the subobjects within a larger structure.
• External Direct Sum: The construction A ⊕ B described in the main definition is often called the external direct sum. It creates a new object from two separate ones.
• Direct Sum Decomposition: The process or result of expressing a module or vector space as a direct sum of smaller submodules or subspaces.
  • Finding a direct sum decomposition is a central problem in linear algebra.

Variants and Related Words

• Direct Summand (n): One of the components in a direct sum decomposition. If V = U ⊕ W, then both U and W are direct summands of V.
• Direct Product (n): A related but distinct construction. For a finite number of factors, the direct product and the direct sum of modules are the same set, but they differ for infinite families.

Synonyms

• (Internal) Direct Sum: Often synonymous with "direct sum decomposition" in context.
• Whitney Sum (n): In the context of vector bundles in topology, the direct sum operation is often called the Whitney sum.

Related Phrases

• "decompose into a direct sum of": A common phrase meaning to break down a structure into a direct sum of simpler parts.
  • The representation was shown to decompose into a direct sum of irreducible representations.
• "isomorphic to the direct sum of": A phrase stating that two structures have the same form.
  • The homology group is isomorphic to the direct sum of two cyclic groups.

Noun

1. a union of two disjoint sets in which every element is the sum of an element from each of the disjoint sets