Direct sum of subspaces

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U_1, \dots, U_k$ be subspaces of $V$. The space $V$ is the direct sum of $U_1, \dots, U_k$, written $V = U_1 \oplus \cdots \oplus U_k$, when each $v \in V$ can be written as $$v = u_1 + \cdots + u_k, \qquad u_i \in U_i,$$ in exactly one way. Equivalently, $V = U_1 + \cdots + U_k$, and $u_1 + \cdots + u_k = 0$ with $u_i \in U_i$ implies $u_1 = \cdots = u_k = 0$.

Remarks

For two subspaces, $V = U \oplus W$ if and only if $V = U + W$ and $U \cap W = \{0\}$. For three or more subspaces, pairwise zero intersections are not enough: three different lines through $0$ in $\R^2$ are not a direct sum. If $V = U_1 \oplus \cdots \oplus U_k$ and $V$ is finite-dimensional, then $\dim V = \dim U_1 + \cdots + \dim U_k$.

Explanations

L1 human

A space is the direct sum of some subspaces when we can split each vector into pieces, one piece in each subspace. The split must be unique. The plane is the direct sum of the $x$-axis and the $y$-axis. The pieces of a vector are its components along the subspaces.