[<< wikibooks] Linear Algebra/Change of Basis
Representations, whether of vectors or of maps, vary with the bases.
For instance, with respect to the two bases 
  
    
      
        
          
            
              E
            
          
          
            2
          
        
      
    
    {\displaystyle {\mathcal {E}}_{2}}
   and 

  
    
      
        B
        =
        ⟨
        
          
            (
            
              
                
                  1
                
              
              
                
                  1
                
              
            
            )
          
        
        ,
        
          
            (
            
              
                
                  1
                
              
              
                
                  −
                  1
                
              
            
            )
          
        
        ⟩
      
    
    {\displaystyle B=\langle {\begin{pmatrix}1\\1\end{pmatrix}},{\begin{pmatrix}1\\-1\end{pmatrix}}\rangle }
  for 
  
    
      
        
          
            R
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {R} ^{2}}
  , the vector 

  
    
      
        
          
            
              
                e
                →
              
            
          
          
            1
          
        
      
    
    {\displaystyle {\vec {e}}_{1}}
   has two different representations.

  
    
      
        
          
            
              R
              e
              p
            
          
          
            
              
                
                  E
                
              
              
                2
              
            
          
        
        (
        
          
            
              
                e
                →
              
            
          
          
            1
          
        
        )
        =
        
          
            (
            
              
                
                  1
                
              
              
                
                  0
                
              
            
            )
          
        
        
        
          
            
              R
              e
              p
            
          
          
            B
          
        
        (
        
          
            
              
                e
                →
              
            
          
          
            1
          
        
        )
        =
        
          
            (
            
              
                
                  1
                  
                    /
                  
                  2
                
              
              
                
                  1
                  
                    /
                  
                  2
                
              
            
            )
          
        
      
    
    {\displaystyle {\rm {Rep}}_{{\mathcal {E}}_{2}}({\vec {e}}_{1})={\begin{pmatrix}1\\0\end{pmatrix}}\qquad {\rm {Rep}}_{B}({\vec {e}}_{1})={\begin{pmatrix}1/2\\1/2\end{pmatrix}}}
  Similarly, with respect to 
  
    
      
        
          
            
              E
            
          
          
            2
          
        
        ,
        
          
            
              E
            
          
          
            2
          
        
      
    
    {\displaystyle {\mathcal {E}}_{2},{\mathcal {E}}_{2}}
   and 
  
    
      
        
          
            
              E
            
          
          
            2
          
        
        ,
        B
      
    
    {\displaystyle {\mathcal {E}}_{2},B}
  , the identity map
has two different representations.

  
    
      
        
          
            
              R
              e
              p
            
          
          
            
              
                
                  E
                
              
              
                2
              
            
            ,
            
              
                
                  E
                
              
              
                2
              
            
          
        
        (
        
          id
        
        )
        =
        
          
            (
            
              
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
              
            
            )
          
        
        
        
          
            
              R
              e
              p
            
          
          
            
              
                
                  E
                
              
              
                2
              
            
            ,
            B
          
        
        (
        
          id
        
        )
        =
        
          
            (
            
              
                
                  1
                  
                    /
                  
                  2
                
                
                  1
                  
                    /
                  
                  2
                
              
              
                
                  1
                  
                    /
                  
                  2
                
                
                  −
                  1
                  
                    /
                  
                  2
                
              
            
            )
          
        
      
    
    {\displaystyle {\rm {Rep}}_{{\mathcal {E}}_{2},{\mathcal {E}}_{2}}({\text{id}})={\begin{pmatrix}1&0\\0&1\end{pmatrix}}\qquad {\rm {Rep}}_{{\mathcal {E}}_{2},B}({\text{id}})={\begin{pmatrix}1/2&1/2\\1/2&-1/2\end{pmatrix}}}
  With our point of view that the objects of our studies are vectors and
maps, in fixing bases 
we are adopting a scheme of tags or names for these objects, that are
convienent for computation.
We will now see how to translate among these names— we will
see exactly how representations vary as the bases vary.