Linear and multilinear algebra; matrix theory Definition

Linear form

Definition

Let $F$ be a field and let $V$ be a vector space over $F$ . A linear form on $V$ is a linear map $f \colon V \to F$ , where $F$ is a vector space over itself. Thus $f(u + v) = f(u) + f(v)$ and $f(av) = a f(v)$ for all $u, v \in V$ and $a \in F$ .

Remarks

A linear form is also called a linear functional or a covector. On $F^n$ , each linear form is $x \mapsto c_1 x_1 + \cdots + c_n x_n$ for unique scalars $c_1, \dots, c_n$ . The trace is a linear form on the space of $n \times n$ matrices. The linear forms on $V$ form the dual space of $V$ .

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Vector space definition
Linear map definition
Linear form definition

Tag	`0RT96J4D3H`
URI	`m2://0RT96J4D3H`
Kind	definition
Writing	W1 stated
MSC	15A04
Defines	Linear form
Uses	Linear map

Linear form

Definition

Remarks

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