Orthogonal complement

Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $W$ be a subspace of $V$. The orthogonal complement of $W$ is $$W^\perp = \{v \in V : \langle v, w \rangle = 0 \text{ for all } w \in W\}.$$

Remarks

$W^\perp$ is a subspace of $V$, and $W \cap W^\perp = 0$. In $\R^3$ with the dot product, the orthogonal complement of a plane through $0$ is its normal line. For a real $m \times n$ matrix $A$, the null space of $A$ is the orthogonal complement of the row space of $A$ in $\R^n$. In finite dimensions, $V = W \oplus W^\perp$, see the orthogonal decomposition.

Explanations

L1 human

The orthogonal complement collects all vectors that are perpendicular to a subspace. The complement of a line in $\R^3$ is a plane, and the complement of a plane is a line. A subspace and its complement together fill the space.