Definition:Bilinear Form (Linear Algebra)

This page is about Bilinear Form in the context of Linear Algebra. For other uses, see Bilinear Form.

Definition

Let $R$ be a ring.

Let $R_R$ denote the $R$-module $R$.

Let $M_R$ be an $R$-module.

A bilinear form on $M_R$ is a bilinear mapping $B : M_R \times M_R \to R_R$.

Also known as

It is usual to gloss over the modular nature of $R_R$ and consider $B$ merely as a mapping from the $R$-module $M$ directly to the ring $R$:

Hence in this manner, a bilinear form on $M$ is defined as a bilinear mapping $B : M \times M \to R$.

Also see

• Definition:Relative Matrix of Bilinear Form
• Definition:Quadratic Form (Linear Algebra)
• Definition:Associated Quadratic Form
• Definition:Bilinear Space
• Definition:Linear Form (Linear Algebra)

• Results about bilinear forms in the context of linear algebra can be found here.

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