Limit lemma

The bound lemma is a mathematical theorem from linear algebra , with which an upper bound for the number of linearly independent elements in a vector space can be given. With the help of the bound lemma it can be proved, among other things, that a finitely generated vector space has a basis and that every two bases in such a vector space have the same number of elements.

statement

The limit lemma can be formulated as follows:

If a vector space has a generating system consisting of elements, then each vectors are linearly dependent . ${\ displaystyle V}$ ${\ displaystyle n}$ ${\ displaystyle n + 1}$ ${\ displaystyle V}$

proof

If the elements of the generating system and any vectors of the vector space are, then each of these vectors can be used as a linear combination ${\ displaystyle u_ {1}, \ ldots, u_ {n} \ in V}$ ${\ displaystyle v_ {1}, \ ldots, v_ {n + 1} \ in V}$

{\ displaystyle v_ {j} = \ sum _ {i = 1} ^ {n} a_ {ij} u_ {i}}

represent with scalars . A linear combination of the vectors then has the form ${\ displaystyle a_ {ij}}$ ${\ displaystyle v_ {1}, \ ldots, v_ {n + 1}}$

{\ displaystyle \ sum _ {j = 1} ^ {n + 1} c_ {j} v_ {j} = \ sum _ {j = 1} ^ {n + 1} c_ {j} \ sum _ {i = 1} ^ {n} a_ {ij} u_ {i} = \ sum _ {i = 1} ^ {n} \ left (\ sum _ {j = 1} ^ {n + 1} a_ {ij} c_ { j} \ right) u_ {i}}

.

The linear system of equations with now has more unknowns than equations and thus in particular a nontrivial solution (see reduced step form ). It then follows ${\ displaystyle Ac = 0}$ ${\ displaystyle A = (a_ {ij}) _ {i = 1, \ ldots, n, j = 1, \ ldots, n + 1}}$ ${\ displaystyle c = (c_ {j}) _ {j = 1, \ ldots, n + 1}}$

{\ displaystyle \ sum _ {j = 1} ^ {n + 1} c_ {j} v_ {j} = \ sum _ {i = 1} ^ {n} 0 \ cdot u_ {i} = 0}

and thus the linear dependence of the vectors . ${\ displaystyle v_ {1}, \ ldots, v_ {n + 1}}$

use

With the help of the bound lemma a number of further fundamental theorems of linear algebra can be proven. A direct consequence is, for example, that a finitely generated vector space has a basis and that every two bases in such a vector space have the same number of elements (which is called the dimension of the vector space). Furthermore, in a finitely generated vector space, every linearly independent set of vectors can be supplemented to a finite basis ( basic supplement theorem ).

literature

Max Koecher: Linear Algebra and Analytical Geometry , Springer, Berlin, 4th edition, 1997, ISBN 3-540-62903-3

Individual evidence

↑ ^a ^b Max Koecher: Linear Algebra and Analytical Geometry , Springer, Berlin, 4th edition, 1997, §4.4 (page 23)

[Koecher23-1] Max Koecher: Linear Algebra and Analytical Geometry , Springer, Berlin, 4th edition, 1997, §4.4 (page 23)