Direct sum of representations

The direct sum is an operation from abstract algebra, a branch of mathematics. For example, the direct sum ${\displaystyle \mathbf {R} \oplus \mathbf {R} }$, where ${\displaystyle \mathbf {R} }$ is real coordinate space, is the Cartesian plane, ${\displaystyle \mathbf {R} ^{2}}$. To see how direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups ${\displaystyle A}$ and ${\displaystyle B}$ is another abelian group ${\displaystyle A\oplus B}$ consisting of the ordered pairs ${\displaystyle (a,b)}$ where ${\displaystyle a\in A}$ and ${\displaystyle b\in B}$. To add ordered pairs, we define the sum ${\displaystyle (a,b)+(c,d)}$ to be ${\displaystyle (a+c,b+d)}$; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of any two algebraic structures, such as rings, modules, and vector spaces.

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