Complementary subspaces

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be subspaces of $V$. The subspaces $U$ and $W$ are complementary when $V = U \oplus W$, that is, when $U + W = V$ and $U \cap W = \{0\}$. Then $W$ is a complement of $U$ in $V$.

Remarks

A complement is not unique: in $\R^2$, each line through $0$ other than the $x$-axis is a complement of the $x$-axis. If $V$ is finite-dimensional and $W$ is a complement of $U$, then $\dim W = \dim V - \dim U$. A complement $W$ of $U$ is isomorphic to the quotient space $V / U$. In an inner product space, the orthogonal complement is one particular complement.

Explanations

L1 human

Two subspaces are complementary when together they fill the space and they share only the zero vector. Each vector then splits in a unique way into a part in each subspace. A line and a plane through the origin in space are complementary when the line does not lie in the plane.