Definition:Linear Form (Linear Algebra)

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

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Definition

Let $\struct {R, +, \times}$ be a commutative ring.

Let $\struct {R, +_R, \circ}_R$ denote the $R$-module $R$.

Let $\struct {G, +_G, \circ}_R$ be a module over $R$.

Let $\phi: \struct {G, +_G, \circ}_R \to \struct {R, +_R, \circ}_R$ be a linear transformation from $G$ to the $R$-module $R$.

$\phi$ is called a linear form on $G$.

Also known as

A linear form is also known as a linear functional.

Also see

• Definition:Algebraic Dual
• Definition:Bilinear Form

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

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations