Multilinear map

Bilinear form

A bilinear form 𝐵 :𝑉 ×𝑉 →𝕂 is a multilinear map from a vector space 𝑉 to its underlying field 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.

Properties

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

See also

• Quadratic space

#state/tidy | #lang/en | #SemBr