Complex conjugate of a vector space

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the complex conjugate of a <a href="/facts/Complex_numbers/2w7mqI7e">complex</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
  
    
      
        V
        
      
    
    {\displaystyle V\,}
  
 is a complex vector space 
  
    
      
        
          
            V
            ¯
          
        
      
    
    {\displaystyle {\overline {V}}}
  
 that has the same elements and additive group structure as 
  
    
      
        V
        ,
      
    
    {\displaystyle V,}
  
 but whose <a href="/facts/Scalar_multiplication/jroFDfbM">scalar multiplication</a> involves <a href="/facts/Complex_conjugate/7TEaoQDe">conjugation</a> of the scalars. In other words, the scalar multiplication of 
  
    
      
        
          
            V
            ¯
          
        
      
    
    {\displaystyle {\overline {V}}}
  
 satisfies

α
        
        ∗
        
        v
        =
        
          
          
            
              α
              ¯
            
          
          ⋅
          
          v
          
        
      
    
    {\displaystyle \alpha \,*\,v={\,{\overline {\alpha }}\cdot \,v\,}}

where 
  
    
      
        ∗
      
    
    {\displaystyle *}
  
 is the scalar multiplication of 
  
    
      
        
          
            V
            ¯
          
        
      
    
    {\displaystyle {\overline {V}}}
  
 and 
  
    
      
        ⋅
      
    
    {\displaystyle \cdot }
  
 is the scalar multiplication of 
  
    
      
        V
        .
      
    
    {\displaystyle V.}

The letter 
  
    
      
        v
      
    
    {\displaystyle v}
  
 stands for a vector in 
  
    
      
        V
        ,
      
    
    {\displaystyle V,}
  
 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  
 is a complex number, and 
  
    
      
        
          
            α
            ¯
          
        
      
    
    {\displaystyle {\overline {\alpha }}}
  
 denotes the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> of 
  
    
      
        α
        .
      
    
    {\displaystyle \alpha .}

</p><p>More concretely, the complex conjugate vector space is the same underlying real vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate <a href="/facts/Linear_complex_structure/gzGklcAX">linear complex structure</a> 
  
    
      
        J
      
    
    {\displaystyle J}
  
 (different multiplication by 
  
    
      
        i
      
    
    {\displaystyle i}
  
).
</p>

Complex conjugate of a vector space open-in-new

Complex conjugate of a vector space