Bilinear form

A bilinear form $𝐵 : 𝑉 \times 𝑉 \to 𝕂$ is a multilinear map from a vector space $𝑉$ to its underlying field $𝕂$ , linalg i.e. linear in both arguments separately

𝐵 (𝛼 𝑢 + 𝛽 𝑣, 𝑤) = 𝛼 𝐵 (𝑢, 𝑤) + 𝛽 𝐵 (𝑣, 𝑤) 𝐵 (𝑢, 𝛼 𝑣 + 𝛽 𝑤) = 𝛼 𝐵 (𝑢, 𝑣) + 𝛽 𝐵 (𝑢, 𝑤)

Further terminology

A bilinear form is called symmetric iff $𝐵 (𝑥, 𝑦) = 𝐵 (𝑦, 𝑥)$ .
A bilinear form is called nondegenerate iff $𝑉 \to 𝑉 * : 𝑣 \mapsto 𝐵 (𝑣, -)$ is an isomorphism, equivalently $𝐵 (𝑣, 𝑤) = 0$ for all $𝑤 \in 𝑉$ implies $𝑣 = 0$ .

Properties

Correspondence between quadratic forms and symmetric bilinear forms away from 2

Table of Contents

Bilinear form

Further terminology

Properties

See also

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