Multilinear map

Bilinear form

A bilinear form 𝐵 :𝑉 ×𝑉 →K is a bilinear map from a module 𝑉 to its ground ring K, #m/def/linalg i.e. linear in both arguments separately

Further terminology

• A bilinear form is called symmetric iff 𝐵(𝑥,𝑦) =𝐵(𝑦,𝑥).
• A bilinear form is called nondegenerate iff 𝑉 →𝑉∗ :𝑣 ↦𝐵(𝑣, −) is an isomorphism, equivalently 𝐵(𝑣,𝑤) =0 for all 𝑤 ∈𝑉 implies 𝑣 =0. See self-dual vector space.

Properties

• Correspondence between quadratic forms and symmetric bilinear forms away from 2

See also

• Quadratic space

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