---
tag: 0RT96J4D3H
kind: definition
name: Linear form
msc: [15A04]
defines: 07YQ1C78G0
uses: [0RJF668P1T]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_form
    locator: lead section
---

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A *linear form* on $V$ is a [[linear map|0NRYHJPQG2]] $f \colon V \to F$, 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|0XC1RDBY7A]] is a linear form on the space of $n \times n$ matrices.
The linear forms on $V$ form the [[dual space|07JBSZE0V4]] 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.