Sums and Direct Sums

Sum of Subspaces

Let $U$ and $W$ be subspaces of $V$. We can denote the sum of U and W as the following set : $U+W=\{u+w\in V\mid u\in U,w\in W\}$ In General, if $V_1,....,V_n$ are subspaces of $V$, then the sum is denoted as $V_1+...+V_n=\{v_1+...+v_n\mid v_i\in V_i\}$

If U and W have a specific property, we denote it with soemthing called a “Direct Sum”

Direct Sum

We say $U$ and $W$ is a direct sum of V if and only if for every $v\in V$, $v=u+w$ in only one way (This is basically saying that if $v=u_1+w_1=u_2+w_2⟺u_1=u_2,w_1=w_2$)