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 = \bigoplus_{i=1}^n V_i

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

Direct sum decomposition of a module M.

M = 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.

\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\}