Row and column vectors

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a column vector with ⁠
 
 
 
 m
 
 
 {\displaystyle m}
 
⁠ elements is an 
 
 
 
 m
 ×
 1
 
 
 {\displaystyle m\times 1}
 
 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> consisting of a single column of ⁠
 
 
 
 m
 
 
 {\displaystyle m}
 
⁠ entries, for example,

x
        
        =
        
          
            [
            
              
                
                  
                    x
                    
                      1
                    
                  
                
              
              
                
                  
                    x
                    
                      2
                    
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    x
                    
                      m
                    
                  
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle {\boldsymbol {x}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}.}

Similarly, a row vector is a 
 
 
 
 1
 ×
 n
 
 
 {\displaystyle 1\times n}
 
 matrix for some ⁠
 
 
 
 n
 
 
 {\displaystyle n}
 
⁠, consisting of a single row of ⁠
 
 
 
 n
 
 
 {\displaystyle n}
 
⁠ entries,

a
        
        =
        
          
            [
            
              
                
                  
                    a
                    
                      1
                    
                  
                
                
                  
                    a
                    
                      2
                    
                  
                
                
                  …
                
                
                  
                    a
                    
                      n
                    
                  
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle {\boldsymbol {a}}={\begin{bmatrix}a_{1}&a_{2}&\dots &a_{n}\end{bmatrix}}.}

(Throughout this article, boldface is used for both row and column vectors.)
The <a href="/facts/Transpose/8wmsagGS">transpose</a> (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector:

[
              
                
                  
                    
                      x
                      
                        1
                      
                    
                    
                    
                      x
                      
                        2
                      
                    
                    
                    …
                    
                    
                      x
                      
                        m
                      
                    
                  
                
              
              ]
            
          
          
            
              T
            
          
        
        =
        
          
            [
            
              
                
                  
                    x
                    
                      1
                    
                  
                
              
              
                
                  
                    x
                    
                      2
                    
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    x
                    
                      m
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle {\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}^{\rm {T}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}}

and

[
              
                
                  
                    
                      x
                      
                        1
                      
                    
                  
                
                
                  
                    
                      x
                      
                        2
                      
                    
                  
                
                
                  
                    ⋮
                  
                
                
                  
                    
                      x
                      
                        m
                      
                    
                  
                
              
              ]
            
          
          
            
              T
            
          
        
        =
        
          
            [
            
              
                
                  
                    x
                    
                      1
                    
                  
                  
                  
                    x
                    
                      2
                    
                  
                  
                  …
                  
                  
                    x
                    
                      m
                    
                  
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle {\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}^{\rm {T}}={\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}.}

The set of all row vectors with n entries in a given <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> (such as the <a href="/facts/Real_numbers/R02gw5Pb">real numbers</a>) forms an n-dimensional <a href="/facts/Vector_space/M1pxMLx2">vector space</a>; similarly, the set of all column vectors with m entries forms an m-dimensional vector space.
The space of row vectors with n entries can be regarded as the <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> of the space of column vectors with n entries, since any linear functional on the space of column vectors can be represented as the left-multiplication of a unique row vector.

Row and column vectors open-in-new

Row and column vectors