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 $V$.

Explanations

L1 human

A linear form takes a vector and returns one number, in a linear way. On the plane, the map $(x, y) \mapsto 2x - 3y$ is a linear form. Its kernel is the line $2x = 3y$, unless the form is zero.