Linear form/Introduction/Section

Definition

Let ${}K$ be a field and let ${}V$ be a ${}K$-vector space. A linear mapping

$V\longrightarrow K$

is called a Linear form on ${}V$.

Example

A linear form on ${}K^{n}$ is of the form

$K^{n}\longrightarrow K,\left(x_{1},\,\ldots ,\,x_{n}\right)\longmapsto \sum _{i=1}^{n}a_{i}x_{i},$

for a tuple ${}\left(a_{1},\,\ldots ,\,a_{n}\right)$. The projections

$p_{j}\colon K^{n}\longrightarrow K,\left(x_{1},\,\ldots ,\,x_{n}\right)\longmapsto x_{j},$

are the easiest linear forms.

The zero mapping to ${}K$ is also a linear form, called the zero form.

Example

Many important examples of linear forms on some vector spaces of infinite dimension arise in analysis. For a real interval ${}[a,b]$, the set of functions ${}\operatorname {Map} \,{\left([a,b],\mathbb {R} \right)}$, or the set of continuous functions ${}C([a,b],\mathbb {R} )$, or the set of continuously differentiable functions ${}C^{1}([a,b],\mathbb {R} )$ form real vector spaces. For a point ${}P\in [a,b]$, the evaluation ${}f\mapsto f(P)$ is a linear form (because addition and scalar multiplication is defined pointwisely on these spaces). Also, the evaluation of the derivative at ${}P$,

$C^{1}([a,b],\mathbb {R} )\longrightarrow \mathbb {R} ,f\longmapsto f'(P),$

is a linear form. For ${}C([a,b],\mathbb {R} )$, the integral, that is, the mapping

$C([a,b],\mathbb {R} )\longrightarrow \mathbb {R} ,f\longmapsto \int _{a}^{b}f(t)dt,$

is a linear form. This rests on the linearity of the integral.

The kernel of the zero form is the total space; for any other linear form ${}f\in \operatorname {Hom} _{K}{\left(V,K\right)}$ with ${}f\neq 0$, the dimension is ${}\dim _{K}{\left(V\right)}-1$. This follows from the dimension formula. With the exception of the zero form, a linear form is always surjective.