Direct Sum

definitionvector-spaces

Definition

The sum $V_1+\cdots +V_m$ is called a direct sum, and denoted by $V_1\oplus \cdots \oplus V_m$, if every element of the sum can be written uniquely as $w=v_1+\cdots +v_m$.

Properties

• Suppose $V_1,\dots ,V_m$ are subspaces of $V$. Then, $V_1+\cdots +V_m=V_1\oplus \cdots V_m$ if and only if the only way to write $0$ as a sum of $v_1+\cdots +v_m$ is if all the $v_k$‘s are $0$.
• Suppose $U$ and $W$ are subspace of $V$, then $U+W=U\oplus W$ if and only if $U\cap W=\{0\}$.

Examples