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Given a vector $\mathbf{v} \in \mathbb{R}^n$, the set of vectors in $\mathbb{R}^n$ orthogonal to $\mathbf{v}$, namely $$\{\mathbf{u} \in \mathbb{R}^n: \mathbf{u} \cdot \mathbf{v}=0\},$$ forms a subspace. In fact, it is the null space of the $1 \times n$ matrix $$\left( \begin{matrix} v_1 & v_2 & \cdots & v_n \\ \end{matrix} \right)$$ if $\mathbf{v}=(v_1,v_2,\ldots,v_n)$.

Question: Is there a specific name or notation for this vector space?

Rebecca J. Stones
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    It is called a hyperplane. – John Douma Aug 27 '13 at 22:53
  • The notation is $\langle \mathbf{v}\rangle^\perp$ and if $F$ is a subspace or even a set then it's orthogonal is denoted by $F^\perp$. –  Aug 27 '13 at 22:56
  • @user69810 The tangent space of a point at the hyperplane, and the plane is defined by using $\mathbf{v}$ as normal, to be precise. – Shuhao Cao Aug 27 '13 at 23:00
  • @ShuhaoCao Don't you need to start with some kind of surface to get a tangent space? In this case we are given the vector but we are not told that it is the normal to a surface. – John Douma Aug 27 '13 at 23:08
  • @user69810: The set ${(1,y):y\in\mathbb R}$ is a hyperplane in $\mathbb R^2$ but is not the space of vectors orthogonal to any vector. –  Aug 27 '13 at 23:13
  • It's not a vector space, since it doesn't have the zero vector. Wikipedia distinguishes these into "vector hyperplane" (those that are vector spaces) and "affine hyperplane" (translations of vector hyperplanes). – Rebecca J. Stones Aug 27 '13 at 23:20

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We usually denote it "W perp", a W with an upside down T as a superscript. We call it the orthogonal complement.

$$W^\perp$$

Andrea Mori
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Demetri Pananos
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