R^n/Orthogonal complement/Example

Let ${}V=\mathbb {R} ^{n}$, endowed with the standard inner product. For the one-dimensional linear subspace ${}\mathbb {R} e_{i}$, generated by the standard vector ${}e_{i}$, the orthogonal complement consists of all vectors ${}{\begin{pmatrix}x_{1}\\\vdots \\x_{i-1}\\0\\x_{i+1}\\\vdots \\x_{n}\end{pmatrix}}$, where the ${}i$-th entry is ${}0$. For a one-dimensional linear subspace ${}\mathbb {R} v$, generated by a vector

${}v={\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}\neq 0\,,$

the orthogonal complement can be found by determining the solution space of the linear equation

${}a_{1}x_{1}+\cdots +a_{n}x_{n}=0\,.$

The orthogonal space

${}U=(\mathbb {R} v)^{\perp }={\left\{{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\mid a_{1}x_{1}+\cdots +a_{n}x_{n}=0\right\}}\,$

has dimension ${}n-1$; this is a so-called hyperplane. The vector ${}v$ is called a normal vector for the hyperplane ${}U$.

For a linear subspace ${}U\subseteq \mathbb {R} ^{n}$ that is given by a basis (or a generating system) ${}v_{i}={\begin{pmatrix}a_{i1}\\\vdots \\a_{in}\end{pmatrix}}$, ${}i=1,\ldots ,k$, the orthogonal complement is the solution space of the system of linear equations

${}A{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}=0\,,$

where ${}A={\left(a_{ij}\right)}$ is the matrix formed by the vectors ${}v_{i}$ as rows.