Linear envelope

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A vector and its linear envelope .

In the linear algebra is the linear hull (including the instep tensioning [from English, of [Linear] span ], chuck , product or final called) a subset of a vector space over a body the set of all linear combinations of vectors and scalars of . The linear envelope forms a sub-vector space , which is also the smallest sub-vector space that it contains.


Constructive definition

If is a vector space over a field and a subset of the vector space, then is

the linear envelope of . The linear hull is the set of all finite linear combinations of the .

In the case of a finite subset , this definition is simplified to


The linear envelope of the empty set is the zero vector space , that is


because the empty sum of vectors gives the zero vector by definition .

Other definitions

The following definitions are equivalent to the constructive definition:

  • The linear hull of a subset of a vector space is the smallest subspace that contains the set .
  • The linear hull of a subset of a vector space is the intersection of all subspaces of which contain.


As symbols for the linear span of is or , , , or uses. If finite, for example , double brackets are avoided by using the notations , or .


Be two sets subsets of -Vektorraumes: . Then:

  1. ,
  2. ,
  3. .

These three properties characterize the linear hull as a hull operator .

The following also applies:

  • The linear hull of a subset of a vector space is a subspace of .
  • For every subspace of a vector space holds .
  • A set of vectors is a generating system of their linear envelope. If, in particular, a set of vectors is a generating system of a subspace, then this is its linear envelope.
  • The sum of two subspaces is the linear envelope of the union, so .
  • In the set of subspaces of a vector space (including the total space) the operation “form the linear envelope of the union” can be introduced as a two-digit combination. The dual link for this is the formation of intersections. With these links it then forms an association .
  • If there are subspaces of a vector space, then the following applies to the dimensions of the linear envelope and the dimensional formula :


  • The linear envelope of a single vector is a straight line through the origin.
  • The two vectors and are elements of the real vector space . Its linear envelope is the - plane.
  • Let be the vector space of the formal power series to the field and the set of monomials . Then the linear envelope of the subspace of the polynomials is :


  • Gerd Fischer : Linear Algebra. An introduction for first-year students (basic math course). 17th edition, Vieweg + Teubner-Verlag, Wiesbaden 2010. ISBN 9783834809964 , 384 pages.

Individual evidence

  1. a b Dietlinde Lau: Algebra and Discrete Mathematics 1. Springer, ISBN 978-3-540-72364-6 , page 162
  2. ^ Siegfried Bosch : Linear Algebra. Springer, 2001, ISBN 3-540-41853-9 , pp. 29-30