---- > [!definition] Definition. ([[orthogonal direct sum of modules]]) > Let $R$ be a (say, [[commutative ring|commutative]] [[ring]]) and let $M_{1}$ and $M_{2}$ be $R$-[[module|modules]] equipped with respective [[bilinear map|bilinear forms]] $B_{1},B_{2}$. Their [[direct sum of modules|direct sum]] $M_{1} \oplus M_{2}$, endowed with the [[bilinear map|bilinear form]] $\begin{align} B: (M_{1} \oplus M_{2}) \times (M_{1} \oplus M_{2}) &\to R \\ \big( (u_{1}, u_{2}) , (v_{1}, v_{2}) \big) & \mapsto B_{1}(u_{1}, v_{1}) + B_{2}(u_{2}, v_{2}) \end{align}$ is called the **orthogonal direct sum** of $M_{1}$ and $M_{2}$. (Note that if $B_{1},B_{2}$ are in fact [[inner product|inner products]] $\langle -,- \rangle_{1}, \langle -,- \rangle_{2}$, then denoting $B=\langle -,- \rangle$ one has $\langle (u_{1},u_{2}) , (v_{1},v_{2}) \rangle \ = \langle u_{1}, v_{1} \rangle + \langle u_{2}, v_{2} \rangle = 0 \iff u_{1} \perp v_{1} \text{ and } u_{2} \perp v_{2} ,$ justifying the name.) ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```