Radical of a quadratic space

Let $(𝑉, 𝑞)$ be a quadratic space. The radical $r a d 𝑉$ of $𝑉$ is the set of all degenerate vectors, #m/def/linalg i.e.

r a d 𝑉 = {𝑣 \in 𝑉 : 𝑏 𝑞 (𝑣, 𝑉) = 0}

i.e. the linear kernel of the curried linear map $𝑏 𝑞 : 𝑉 \to 𝖵 𝖾 𝖼 𝗍 𝕂 (𝐾, 𝑉)$ . In particular (away from 2) it is a totally isotropic vector subspace of $𝑉$ .

Properties

Away from 2 all degenerate vectors are isotropic, so any subspace of the radical is a Normal quadratic subspace.

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