# 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