Subspace Direct Sums

Definition

Suppose $U_1, \dots , U_m$ are subspaces of a vector space $V$. The sum $M = U_1 + \cdots + U_m$ is called a direct sum if each element $x \in M$ can only be written in one way as a sum

\[ x = u_1 + \cdots + u_m \]

for $u_i \in U_i$, $i = 1, \dots , m$. We call the sum direct. Conversely, we call $U_1, \dots , U_m$ a decomposition of $M$.

Notation

If $M$ is a direct sum of $U_1, \dots , U_m$, we use the notation $\oplus$. We write

\[ M = U_1 \oplus \cdots \oplus U_m \]