Transformation matrix

<p>
In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformations</a> can be represented by <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a>. If 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is a linear transformation mapping 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
 to 
  
    
      
        
          
            R
          
          
            m
          
        
      
    
    {\displaystyle \mathbb {R} ^{m}}
  
 and 
  
    
      
        
          x
        
      
    
    {\displaystyle \mathbf {x} }
  
 is a <a href="/facts/Column_vector/pPuRr9Xk">column vector</a> with 
  
    
      
        n
      
    
    {\displaystyle n}
  
 entries, then there exists an 
  
    
      
        m
        ×
        n
      
    
    {\displaystyle m\times n}
  
 matrix 
  
    
      
        A
      
    
    {\displaystyle A}
  
, called the transformation matrix of 
  
    
      
        T
      
    
    {\displaystyle T}
  
, such that:

T
        (
        
          x
        
        )
        =
        A
        
          x
        
      
    
    {\displaystyle T(\mathbf {x} )=A\mathbf {x} }

Note that 
  
    
      
        A
      
    
    {\displaystyle A}
  
 has 
  
    
      
        m
      
    
    {\displaystyle m}
  
 rows and 
  
    
      
        n
      
    
    {\displaystyle n}
  
 columns, whereas the transformation 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is from 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
 to 
  
    
      
        
          
            R
          
          
            m
          
        
      
    
    {\displaystyle \mathbb {R} ^{m}}
  
.  There are alternative expressions of transformation matrices involving <a href="/facts/Row_vector/pPuRr9Xk">row vectors</a> that are preferred by some authors.
</p>

Transformation matrix open-in-new

Transformation matrix