[[Geometric algebra MOC]]
# Quadratic space

A **quadratic space** $(V, q)$ over $\mathbb{K}$ is a [[vector space]] $V$ over $\mathbb{K}$ equipped with a [[quadratic form]] $q$, or equivalently\*[^cav] a [[Bilinear form#^symmetric]] [[bilinear form]] #m/def/geoalg
$$
\begin{align*}
v \cdot w =  \frac{b_{q}(v,w)}{2} = \frac{q(v+w)-q(v)-q(w)}{2}
\end{align*}
$$
The value of $q(v)$ is called the **quadrance**[^wild] of $v \in V$.

  [^cav]: [[Away from 2]], see [[Correspondence between quadratic forms and symmetric bilinear forms away from 2]]
  [^wild]: This term is due to [[N. Wildberger]], which is not to say that I am a wildbergerian. I just like the word.

## Further terminology

Let $b_{q}$ denote the polar form of $q$.

- A vector $v \in V$ is **isotropic** iff $q(v) = 0$, otherwise it is **anisotropic**; 
  $V$ is isotropic iff it has an isotropic vector. ^isotropic
- Iff every vector is isotropic then $V$ is **totally isotropic**. ^totallyIsotropic
- A vector $v \in V$ is **degenerate** iff $b_{q}(v,u) = 0$ for all $u \in V$, otherwise it is **nondegenerate**; $V$ is degenerate iff it has a degenerate vector and nondegenerate otherwise. ^nondegenerate
- The set $\opn{rad} V$ of all degenerate vectors in $V$ is called the [[Radical of a quadratic space|radical]].
- An **isometry** $f : (V, q) \to (V', q')$ is a [[linear map]] such that $q'(fv) = q(v)$ for all $v \in V$.
- A bijective isometry is called an **orthogonal transformation**, and these form the [[Orthogonal group of a quadratic space]]

## Properties

- [[Canonical tensors over a nondegenerate quadratic space]]

## See also

- [[Category of quadratic spaces]]
- [[Normal quadratic subspace]]

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