Sum of subspaces

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be linear subspaces of $V$. The sum of $U$ and $W$ is $$U + W = \{u + w : u \in U, \ w \in W\}.$$ For subspaces $U_1, \dots, U_k$, the sum is $U_1 + \cdots + U_k = \{u_1 + \cdots + u_k : u_i \in U_i\}$.

Remarks

The sum $U + W$ is a subspace, and it is the smallest subspace of $V$ that contains $U \cup W$. The sum is the span of $U \cup W$. When $U \cap W = \{0\}$, the sum is a direct sum. For finite-dimensional subspaces, Grassmann's formula gives $\dim(U + W) = \dim U + \dim W - \dim(U \cap W)$.

Explanations

L1 human

The sum of two subspaces collects every vector that we can build from one piece in each subspace. In space, the sum of two different lines through the origin is the plane that contains both lines. It is the smallest subspace that holds both subspaces.