Linear envelope

A vector and its linear envelope . ${\displaystyle a}$ ${\displaystyle \langle a\rangle }$

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. ${\displaystyle A}$ ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle A}$

definition

Constructive definition

If is a vector space over a field and a subset of the vector space, then is ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle A\subset V}$

${\displaystyle \langle A\rangle =\left\{\left.\textstyle \sum \limits _{i=1}^{n}\lambda _{i}a_{i}\right|\lambda _{i}\in K,a_{i}\in A,n\in \mathbb {N} \right\}}$

the linear envelope of . The linear hull is the set of all finite linear combinations of the . ${\displaystyle A}$ ${\displaystyle a_{i}}$

In the case of a finite subset , this definition is simplified to ${\displaystyle A}$

${\displaystyle \langle \{a_{1},a_{2},\dotsc ,a_{n}\}\rangle =\{\lambda _{1}a_{1}+\lambda _{2}a_{2}+\dotsb +\lambda _{n}a_{n}\mid \lambda _{1},\lambda _{2},\dotsc ,\lambda _{n}\in K\}}$.

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

${\displaystyle \langle \emptyset \rangle =\{0\}}$,

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 . ${\displaystyle A}$ ${\displaystyle V}$ ${\displaystyle A}$
• The linear hull of a subset of a vector space is the intersection of all subspaces of which contain. ${\displaystyle A}$ ${\displaystyle V}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle A}$

notation

As symbols for the linear span of is or , , , or uses. If finite, for example , double brackets are avoided by using the notations , or . ${\displaystyle A}$ ${\displaystyle \operatorname {span} (A)}$ ${\displaystyle \operatorname {Span} (A)}$ ${\displaystyle \langle A\rangle }$ ${\displaystyle L(A)}$ ${\displaystyle \operatorname {lin} A}$ ${\displaystyle {\mathcal {L}}(A)}$ ${\displaystyle A}$ ${\displaystyle A=\{a_{1},\dotsc ,a_{n}\}}$ ${\displaystyle \langle a_{1},\dotsc ,a_{n}\rangle }$ ${\displaystyle L\{a_{1},\dotsc ,a_{n}\}}$ ${\displaystyle {\mathcal {L}}\{a_{1},\dotsc ,a_{n}\}}$

properties

Be two sets subsets of -Vektorraumes: . Then: ${\displaystyle K}$ ${\displaystyle A,B\subseteq V}$

1. ${\displaystyle A\subseteq \langle A\rangle }$,
2. ${\displaystyle A\subseteq B\Rightarrow \langle A\rangle \subseteq \langle B\rangle }$,
3. ${\displaystyle \langle A\rangle =\langle \langle A\rangle \rangle }$.

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 . ${\displaystyle V}$ ${\displaystyle V}$

• For every subspace of a vector space holds . ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle \langle U\rangle =U}$

• 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 . ${\displaystyle U_{1}+U_{2}=\{u_{1}+u_{2}\mid u_{1}\in U_{1},u_{2}\in U_{2}\}}$ ${\displaystyle U_{1},U_{2}}$ ${\displaystyle U_{1}+U_{2}=\langle U_{1}\cup U_{2}\rangle }$

• 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 . ${\displaystyle T}$ ${\displaystyle T}$

• If there are subspaces of a vector space, then the following applies to the dimensions of the linear envelope and the dimensional formula : ${\displaystyle U,V}$

${\displaystyle \dim(U+V)+\dim(U\cap V)=\dim U+\dim V}$.

Examples

• The linear envelope of a single vector is a straight line through the origin. ${\displaystyle \langle a\rangle }$ ${\displaystyle a\in \mathbb {R} ^{2}\setminus \{(0,0)\}}$

• The two vectors and are elements of the real vector space . Its linear envelope is the - plane. ${\displaystyle (3,0,0)}$ ${\displaystyle (0,2,0)}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \langle (3,0,0),(0,2,0)\rangle }$ ${\displaystyle x}$ ${\displaystyle y}$

• 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 : ${\displaystyle K[[X]]=\left\{\left.\textstyle \sum \limits _{k=0}^{\infty }\lambda _{k}x^{k}\right|\lambda _{k}\in K\right\}}$ ${\displaystyle K}$ ${\displaystyle A=\{x^{k}\mid k\in \mathbb {N} \}}$ ${\displaystyle A}$

  ${\displaystyle \langle A\rangle =\left\{\left.\textstyle \sum \limits _{i=0}^{n}\lambda _{i}x^{i}\right|\lambda _{i}\in K,n\in \mathbb {N} \right\}=K[X]}$.

literature

• 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