Basic selection set

The basic selection theorem is an elementary theorem of linear algebra , one of the branches of mathematics . It is related to Steinitz's exchange lemma , the basic complement theorem and the bound lemma .

Formulation of the sentence

The sentence is as follows:

In every vector space over any body , a base can always be selected from a finite generating system . ${\displaystyle V}$ ${\displaystyle K}$

In particular, every finitely generated vector space has a finite basis.

Evidence sketch

For a subset of the vector space denote its linear envelope . ${\displaystyle E\subset V}$ ${\displaystyle \operatorname {span} (E)}$

Now let be a finite generating system of . If it is already linearly independent , then one is done, because with that there is a linearly independent generating system, i.e. a basis of . ${\displaystyle E}$ ${\displaystyle V}$ ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle V}$

On the other hand, even linearly dependent , so it can be an element selected with . Then , is itself also a finite generating system of , the number of which , however, is reduced by. ${\displaystyle E}$ ${\displaystyle e\in E}$ ${\displaystyle e\in \operatorname {span} (E\setminus \{e\})}$ ${\displaystyle \operatorname {span} (E)=\operatorname {span} (E\setminus \{e\})}$ ${\displaystyle E\setminus \{e\}}$ ${\displaystyle V}$ ${\displaystyle E}$ ${\displaystyle 1}$

If you are now linearly independent, i.e. a basis, you are done. Otherwise repeat the procedure. ${\displaystyle E\setminus \{e\}}$

In this way, after a finite number of steps, one arrives at a finite subset that represents a basis of . ${\displaystyle E^{*}\subseteq E}$ ${\displaystyle V}$

Related theorem

Related to the base selection set is the following fundamental theorem of linear algebra :

Every vector space has a basis.

The proof of this related theorem requires in the case that the underlying vector space is not finitely generated, the use of one of the maximality principles of set theory such as that of Zorn's lemma . The theorem therefore only arises if the axiom of choice is accepted . In contrast to this, the basic selection theorem always applies because it already presupposes the finiteness of a generating system.

generalization

As the proof sketch shows, it can be shown in the same way that a basis can always be selected from a finite generating system of a matroid .

Web link

• Extract from a script from the University of Regensburg (PDF)

literature

• Gerd Fischer : Linear Algebra . 17th updated edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-0996-4 .

Individual evidence

1. ↑ ^{a b} Gerd Fischer : Linear Algebra . 17th updated edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-0996-4 , pp. 88 .