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bigoplus

Represents a large direct sum operator commonly used in abstract algebra and linear algebra to denote the sum of vector spaces or modules.

Overview

A fundamental notation in advanced mathematics that indicates the direct sum of multiple mathematical structures, particularly useful when working with algebraic structures.

  • Essential in representation theory and module theory
  • Frequently appears in decomposition theorems and direct sum decompositions
  • Used to construct new algebraic structures from existing ones
  • Common in graduate-level mathematics and theoretical physics when dealing with tensor products and direct sums of vector spaces

Examples

Direct sum of vector spaces V₁ through Vₙ.

V=i=1nViV = \bigoplus_{i=1}^n V_i
V = \bigoplus_{i=1}^n V_i

Direct sum decomposition of a module M.

M=M1M2M3M = M_1 \bigoplus M_2 \bigoplus M_3
M = M_1 \bigoplus M_2 \bigoplus M_3

Direct sum of matrix spaces over different fields.

Rm×n=k=1rspan{Ek}\mathbb{R}^{m \times n} = \bigoplus_{k=1}^r \text{span}\{E_k\}
\mathbb{R}^{m \times n} = \bigoplus_{k=1}^r \text{span}\{E_k\}