Dependence relation

<h2 id="examples">Examples</h2>
<ul><li>Let 
  
    
      
        V
      
    
    {\displaystyle V}
  
 be a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
  
    
      
        F
        .
      
    
    {\displaystyle F.}
  
 The relation 
  
    
      
        ◃
      
    
    {\displaystyle \triangleleft }
  
, defined by  
  
    
      
        υ
        ◃
        S
      
    
    {\displaystyle \upsilon \triangleleft S}
  
 if 
  
    
      
        υ
      
    
    {\displaystyle \upsilon }
  
 is in the <a href="/facts/Linear_subspace/2wtKTgHL">subspace</a> spanned by 
  
    
      
        S
      
    
    {\displaystyle S}
  
, is a dependence relation. This is <a href="/facts/Logical_equivalence/ysCWmsxA">equivalent</a> to the definition of <a href="/facts/Linear_independence/8JrHfIIa">linear dependence</a>.</li>
<li>Let 
  
    
      
        K
      
    
    {\displaystyle K}
  
 be a <a href="/facts/Field_extension/L7yIDtyK">field extension</a> of 
  
    
      
        F
        .
      
    
    {\displaystyle F.}
  
 Define 
  
    
      
        ◃
      
    
    {\displaystyle \triangleleft }
  
 by  
  
    
      
        α
        ◃
        S
      
    
    {\displaystyle \alpha \triangleleft S}
  
 if 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  
 is <a href="/facts/Algebraic_element/Bl87J5mB">algebraic</a> over 
  
    
      
        F
        (
        S
        )
        .
      
    
    {\displaystyle F(S).}
  
 Then 
  
    
      
        ◃
      
    
    {\displaystyle \triangleleft }
  
 is a dependence relation. This is equivalent to the definition of <a href="/facts/Algebraic_dependence/H8wV21eD">algebraic dependence</a>.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Matroid/5bLcGlRF">matroid</a></li></ul>
<p><i>This article incorporates material from Dependence relation on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i>
</p>

Dependence relation open-in-new

Dependence relation