Linear form

from Wikipedia, the free encyclopedia

The linear form is a term from the mathematical branch of linear algebra . It is used to describe a linear mapping from a vector space into the underlying body .

In the context of functional analysis , i.e. in the case of a topological - or - vector space, the linear forms considered are mostly continuous linear functionals .


Let it be a body and a vector space. A mapping is called linear form if the following applies to all vectors and scalars :

  1. (Additivity);
  2. (Homogeneity).

The set of all linear forms over a given vector space forms its dual space and thus itself again naturally a vector space.


General properties for linear forms are for example:

  • Like any linear map, they are completely determined by their values ​​for an arbitrary basis of .
  • They are either trivial (identical everywhere ) or surjective .
  • If two of them have the same kernel , the only difference is that they are multiplied by a scalar.

The following also applies specifically to linear functionals:

  • They are continuous exactly when their core is closed .
  • Their absolute amount is always a semi-norm on .
  • Linear functionals are exactly the mappings , where a vector and denote the standard scalar product.

Linear form as a tensor

A linear form is a first order covariant tensor ; they are therefore sometimes also called 1-form . 1-forms form the basis for the introduction of differential forms .

Related terms

Is specific and to change the second condition in starting, whereby the complex conjugate of designated, to obtain a semi-linear form .

A mapping that is linear or semilinear in more than one argument is a sesquilinear form , a bilinear form , or generally a multilinear form .