Subspace Sums
Why
Definition
Suppose $U_1, \dots , U_m$ are subsets of $V$
The sum of $U_1, \dots ,
U_m$ is the set
\[
\Set{u_1 + \cdots + u_m}{u_1 \in U_1, \dots , u_m \in U_m}
\]
For subspaces
The sum of two subspaces is a subspace. Moreover, it is the smallest subspace containing both subspaces.
Suppose $U_1, \dots , U_m$ are subspaces of a
vector space $V$.
The $U_1 + \cdots + U_m$ is the smallest
subspace containing $U_1, \dots , U_m$