Petersson trace formula: Difference between revisions

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The '''Steinitz exchange lemma''' is a basic theorem in [[linear algebra]] used, for example, to show that any two [[Basis (linear algebra)|bases]] for a finite-[[Dimension (vector space)|dimensional]] [[vector space]] have the same number of elements. The result is named after the German mathematician [[Ernst Steinitz]]. The result is often called the '''Steinitz–Mac&nbsp;Lane exchange lemma''', also recognizing the generalization<ref>
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{{citation|last=Mac&nbsp;Lane|first=Saunders|authorlink=Saunders Mac Lane|year=1936|title=Some interpretations of abstract linear dependence in terms of projective geometry|journal=American Journal of Mathematics|volume=58|pages=236–240|doi=10.2307/2371070| jstor=2371070 | issue=1|publisher=The Johns Hopkins University Press}}.</ref>
by [[Saunders Mac Lane|Saunders Mac&nbsp;Lane]]
of Steinitz's lemma to [[matroid]]s.<ref>
{{citation|editor-last=Kung|editor-first=Joseph P. S.|title=A Source Book in Matroid Theory|publisher=Birkhäuser|mr=0890330|isbn=0-8176-3173-9|location=Boston|year=1986}}.
</ref>


== Statement ==
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If {''v''<sub>1</sub>, ..., ''v''<sub>''m''</sub>} is a set of ''m'' [[Linear independence|linearly independent]] vectors in a vector space ''V'', and {''w''<sub>1</sub>, ..., ''w''<sub>''n''</sub>} [[Linear span|span]] ''V'' then ''m''&nbsp;≤&nbsp;''n'' and, possibly after reordering the ''w''<sub>''i''</sub>, the set {''v''<sub>1</sub>, ..., ''v''<sub>''m''</sub>, ''w''<sub>''m''&nbsp;+&nbsp;1</sub>, ..., ''w''<sub>''n''</sub>} spans ''V''.
 
==Proof==
 
We are going to show that for any integer <math>k</math> satisfying <math>0\leq k\leq m</math>, the following assertion is valid. Choosing <math>k=m</math> gives the result.
 
(A) The set <math>\{ v_1,\ldots, v_k,w_{k+1},\ldots,w_n\}</math> spans <math>V</math> (where the <math>w_j</math> have possibly been reordered, and the reordering depends on <math>k</math>).
 
We will prove (A) by induction over <math>k</math>: Being clear for <math>k=0</math>, the only thing that needs to be done is the inductive step.
 
Assume that (A) holds for some <math>k</math> satisfying <math>0\leq k<m</math>. Since <math>v_{k+1}\in V</math>, and <math>\{ v_1,\ldots, v_k,w_{k+1},\ldots,w_n\}</math> spans <math>V</math> (by the induction hypothesis), there exist <math>\mu_1,\ldots,\mu_n</math> such that
:<math>v_{k+1}=\sum_{j=1}^k \mu_j v_j+\sum_{j=k+1}^n \mu_j w_j.</math>
At least one of <math>\{\mu_{k+1},\ldots,\mu_n\}</math> must be non-zero, otherwise this equality would contradict the linear independence of <math>\{ v_1,\ldots,v_m \}</math>; note that this additionally implies that <math>k<n</math>. By reordering the <math>w_{k+1},\ldots,w_n</math>, we may assume that <math>\mu_{k+1}</math> is not zero. Therefore, we have
: <math>w_{k+1}= \frac{1}{\mu_{k+1}}\left(v_{k+1} - \sum_{j=1}^k \mu_j v_j - \sum_{j=k+2}^n \mu_j w_j\right)</math>
In other words, <math>w_{k+1}</math> is in the span of <math>\{ v_1,\ldots, v_{k+1},w_{k+2},\ldots,w_n\}</math> and so the latter must be the whole of <math>V</math>. We have thus shown that (A) holds for <math>k+1</math>, completing the inductive step.
 
==Applications==
The Steinitz exchange lemma is a basic result in [[computational mathematics]], especially in [[numerical linear algebra|linear algebra]] and in [[Matroid#Greedy_algorithm|combinatorial algorithms]].<ref>Page v in Stiefel:
{{cite book|last=Stiefel|first=Eduard L.|authorlink=Eduard Stiefel|title=An introduction to numerical mathematics|edition=Translated by Werner C. Rheinboldt & Cornelie J. Rheinboldt from the second German|publisher=Academic Press|location=New York|year=1963|pages=x+286|mr=181077}}
</ref>
 
== References ==
 
<references/>
 
* Julio R. Bastida, ''Field extensions and Galois Theory'', [[Addison–Wesley|Addison–Wesley Publishing Company]] (1984).
 
[[Category:Linear algebra]]
[[Category:Lemmas]]
[[Category:Matroid theory]]

Latest revision as of 21:41, 25 October 2014

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