Orthogonal Complement¶

Definition¶

Let $V$ be a real vector space with an inner product, written $\boldsymbol{x}\cdot\boldsymbol{y}$ (in $\mathbb{R}^n$ : the dot product), and let $W\subseteq V$ $V$ . The orthogonal complement of $W$ is

W^\perp=\{\boldsymbol{v}\in V\colon{}\boldsymbol{v}\cdot\boldsymbol{w}=0\quad \text{for all }\boldsymbol{w}\in W\},

the set of vectors orthogonal to every vector of $W$ . It is read " $W$ perp". Usually $W$ is a subspace.

Interpretation¶

$W^\perp$ collects all directions perpendicular to $W$ . In $\mathbb{R}^3$ the complement of a plane through the origin is its normal line, and the complement of a line through the origin is the plane perpendicular to it.

A line W through the origin of R^3 spanned by (1,1,1), and the plane W perp through the origin, x1 + x2 + x3 = 0, spanned by (-1,1,0) and (-1,0,1) and meeting W at a right angle.

The figure draws the first example below: $W=\operatorname{span}\{(1,1,1)^{\top}\}$ is a line, and $W^\perp$ is the plane $x_1+x_2+x_3=0$ with basis $(-1,1,0)^{\top},(-1,0,1)^{\top}$ . Every vector of the plane is orthogonal to $(1,1,1)^{\top}$ , the two meet only in $\boldsymbol{0}$ , and the dimensions add up to $1+2=3$ $3$ .

Switch between a line and a plane $W$ in $\mathbb R^3$ : $W^\perp$ is the plane or line through $\boldsymbol{0}$ perpendicular to it, and $\dim W+\dim W^\perp=3$ $3$ . Move $\boldsymbol{b}$ to see it split as $\boldsymbol{p}+\boldsymbol{e}$ with $\boldsymbol{p}\in W$ and $\boldsymbol{e}\in W^\perp$ $W^{⊥}$ .

"Complement" does not mean set complement: $V\setminus W$ is never a subspace, as it misses $\boldsymbol{0}$ . Nor is $W^\perp$ just some subspace $U$ with $V=W\oplus U$ $U$ : such complements are many, but the orthogonal one is unique. It depends on the inner product; with a different inner product on $\mathbb{R}^n$ the orthogonal complement can change.

To compute it, it suffices to test a spanning set: if $W=\operatorname{span}\{\boldsymbol{w}_1,\ldots,\boldsymbol{w}_k\}$ $span {w_{1}, \dots, w_{k}}$ , then $\boldsymbol{v}\in W^\perp$ if and only if $\boldsymbol{v}\cdot\boldsymbol{w}_i=0$ for $i=1,\ldots,k$ $1, \dots, k$ . In $\mathbb{R}^n$ this is the homogeneous system $M^{\top}\boldsymbol{v}=\boldsymbol{0}$ with $M=\begin{pmatrix}\boldsymbol{w}_1&\cdots&\boldsymbol{w}_k\end{pmatrix}$ $(w_{1} \dots w_{k})$ , so $W^\perp$ is the null space (kernel) of $M^{\top}$ .

Key Results¶

$W^\perp$ is a subspace for every subset $W$ , and $W^\perp=(\operatorname{span}W)^\perp$ $(span W)^{⊥}$ .
$W\cap W^\perp=\{\boldsymbol{0}\}$ for a subspace $W$ , since $\boldsymbol{w}\cdot\boldsymbol{w}=0$ forces $\boldsymbol{w}=\boldsymbol{0}$ $0$ .
If $V$ is finite-dimensional and $W$ a subspace: $V=W\oplus W^\perp$ (a direct sum), $\dim W+\dim W^\perp=\dim V$ $dim V$ , and $(W^\perp)^\perp=W$ $W$ .
Inclusions reverse: $W_1\subseteq W_2$ implies $W_2^\perp\subseteq W_1^\perp$ $W_{1}^{⊥}$ . Also $(W_1+W_2)^\perp=W_1^\perp\cap W_2^\perp$ $W_{2}^{⊥}$ .
$\{\boldsymbol{0}\}^\perp=V$ and $V^\perp=\{\boldsymbol{0}\}$ ${0}$ .
For a real matrix $A$ , $\operatorname{Null}(A)=\operatorname{Row}(A)^\perp$ and $\operatorname{Null}(A^{\top})=\operatorname{Col}(A)^\perp$ (Fundamental Theorem of Linear Algebra).

Examples¶

A line in $\mathbb{R}^3$ . For $W=\operatorname{span}\{(1,1,1)^{\top}\}$ $span {(1, 1, 1)^{⊤}}$ , $W^\perp=\{\boldsymbol{x}\colon x_1+x_2+x_3=0\}$ $0}$ , with basis $(-1,1,0)^{\top},(-1,0,1)^{\top}$ . Dimensions: $1+2=3$ $3$ .

A plane in $\mathbb{R}^3$ . For $W=\{\boldsymbol{x}\colon x_1-2x_2+x_3=0\}$ $0}$ , $W^\perp=\operatorname{span}\{(1,-2,1)^{\top}\}$ $span {(1, - 2, 1)^{⊤}}$ : the coefficient vector is the normal vector of the plane.

A plane in $\mathbb{R}^4$ . For $W=\operatorname{span}\{(1,0,1,0)^{\top},(0,1,0,1)^{\top}\}$ $span {(1, 0, 1, 0)^{⊤}, (0, 1, 0, 1)^{⊤}}$ , the conditions $x_1+x_3=0$ and $x_2+x_4=0$ give

W^\perp=\operatorname{span}\left\{\begin{pmatrix}1\\0\\-1\\0\end{pmatrix},\begin{pmatrix}0\\1\\0\\-1\end{pmatrix}\right\},

and $2+2=4$ $4$ . Note that $(1,0,0,0)^{\top}$ is not in $W$ , yet it is not in $W^\perp$ either: most vectors lie in neither.

Prerequisites¶

Sources¶

Math V: Advanced Linear Algebra · Week 5

Aliases: perp, orthocomplement, perpendicular complement Keywords: W^⊥, W perp, (W^⊥)^⊥ = W, dim W + dim W^⊥ = n, R^n = W ⊕ W^⊥, normal line Symbols: element of, intersection, real numbers, subset

Last updated: 22 October 2026 19:04