Quadratic space

A quadratic space $(𝑉, 𝑞)$ over $𝕂$ is a vector space $𝑉$ over $𝕂$ equipped with a quadratic form $𝑞$ , or equivalently*¹ a symmetric bilinear form #m/def/geoalg

𝑣 \cdot 𝑤 = 𝑏 𝑞 (𝑣, 𝑤) 2 = 𝑞 (𝑣 + 𝑤) - 𝑞 (𝑣) - 𝑞 (𝑤) 2

The value of $𝑞 (𝑣)$ is called the quadrance² of $𝑣 \in 𝑉$ .

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.
Iff every vector is isotropic then $𝑉$ is totally isotropic.
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.
The set $r a d 𝑉$ of all degenerate vectors in $𝑉$ is called the radical.
An isometry $𝑓 : (𝑉, 𝑞) \to (𝑉', 𝑞')$ is a linear map such that $𝑞' (𝑓 𝑣) = 𝑞 (𝑣)$ for all $𝑣 \in 𝑉$ .
A bijective isometry is called an orthogonal transformation, and these form the Orthogonal group of a quadratic space