Definition

A sum of subspaces $U_1+...+U_m$ is called a Direct Sum of subspaces if whenever $u\in U_1+...+U_m$ there is exactly one choice of $u_i\in U_i, i=1,2,....,m$ such that $u=u_1+...u_m$. When $U_1+...+U_m$ is a Direct Sum, use the notation: $$U_1\oplus ... \oplus U_m$$ ## Intuition You will notice that this is similar is description to linear independence and you can choose to think of it as a notion of it but for subspaces rather than elements.