[<< wikibooks] Abstract Algebra/Linear Algebra
The reader is expected to have some familiarity with linear algebra. For example, statements such as

Given vector spaces 
  
    
      
        V
      
    
    {\displaystyle V}
   and 
  
    
      
        W
      
    
    {\displaystyle W}
   with bases 
  
    
      
        B
      
    
    {\displaystyle B}
   and 
  
    
      
        C
      
    
    {\displaystyle C}
   and dimensions 
  
    
      
        n
      
    
    {\displaystyle n}
   and 
  
    
      
        m
      
    
    {\displaystyle m}
  , respectively, a linear map 
  
    
      
        f
        
        :
        
        V
        →
        W
      
    
    {\displaystyle f\,:\,V\to W}
   corresponds to a unique 
  
    
      
        m
        ×
        n
      
    
    {\displaystyle m\times n}
   matrix, dependent on the particular choice of basis.should be familiar. It is impossible to give a summary of the relevant topics of linear algebra in one section, so the reader is advised to take a look at the linear algebra book.
In any case, the core of linear algebra is the study of linear functions, that is, functions with the property 
  
    
      
        f
        (
        α
        x
        +
        β
        y
        )
        =
        α
        f
        (
        x
        )
        +
        β
        f
        (
        y
        )
      
    
    {\displaystyle f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)}
  , where greek letters are scalars and roman letters are vectors.
The core of the theory of finitely generated vector spaces is the following:
Every finite-dimensional vector space 
  
    
      
        V
      
    
    {\displaystyle V}
   is isomorphic to 
  
    
      
        
          
            F
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {F} ^{n}}
   for some field 
  
    
      
        
          F
        
      
    
    {\displaystyle \mathbb {F} }
   and some 
  
    
      
        n
        ∈
        
          N
        
      
    
    {\displaystyle n\in \mathbb {N} }
  , called the dimension of 
  
    
      
        V
      
    
    {\displaystyle V}
  . Specifying such an isomorphism is equivalent to choosing a basis for 
  
    
      
        V
      
    
    {\displaystyle V}
  . Thus, any linear map between vector spaces 
  
    
      
        f
        
        :
        
        V
        →
        W
      
    
    {\displaystyle f\,:\,V\to W}
   with dimensions 
  
    
      
        n
      
    
    {\displaystyle n}
   and 
  
    
      
        m
      
    
    {\displaystyle m}
   and given bases 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
   and 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
   induces a unique linear map 
  
    
      
        [
        f
        
          ]
          
            ϕ
          
          
            ψ
          
        
        
        :
        
        
          
            R
          
          
            n
          
        
        →
        
          
            R
          
          
            m
          
        
      
    
    {\displaystyle [f]_{\phi }^{\psi }\,:\,\mathbb {R} ^{n}\to \mathbb {R} ^{m}}
  . These maps are presicely the 
  
    
      
        m
        ×
        n
      
    
    {\displaystyle m\times n}
   matrices, and the matrix in question is called the matrix representation of 
  
    
      
        f
      
    
    {\displaystyle f}
   relative to the bases 
  
    
      
        ϕ
        ,
        ψ
      
    
    {\displaystyle \phi ,\psi }
  .
Remark: The idea of identifying a basis of a vector space with an isomorphism to 
  
    
      
        
          
            F
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {F} ^{n}}
   may be new to the reader, but the basic principle is the same.