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                                  Vector space Hom(V,W)


      

 

Vector Space Hom(V,W). Let V be some abstract n-dimensional vector space over field F and let W be some abstract m-dimensional vector space over field F. Then every linear transformation from V into W can be represented by some mxn matrix over field F. Moreover, every mxn matrix over F represents some linear mapping from V into W. Let A be the set of all mxn matrices over field F. The set A then represents the set of all linear transformations from V into W. This set of all linear mappings from V into W is itself a vector space over F. The vector space consisting of all linear mappings from V into W is denoted by Hom(V,W) and has a dimension of mn i.e. it has mn linearly independent basis vectors.

 

Basis for Hom(V,W). Let


             {v1, v2, ... ,vn } are a set of basis vectors for V

            {w1, w2, ... ,wm } are a set of basis vectors for W.


A set of basis vectors for the vector space Hom(V,W) is given by the set of mn functions {Fij: i=1,n; j=1,m} where

          

            Fij maps vi into wj and all other v's into 0

 

Example. Let V be Euclidean 3-space and W be Euclidean 2-space. Let the set {x,y,z} be three basis vectors for vector space V and {x',y'} be two basis vectors for W. Then a basis for Hom(V,W) would be the following six functions:



Function F11:


       x ole.gif x'

       y ole1.gif 0

       z ole2.gif 0


 

Function F12:


       x ole3.gif y'

       y ole4.gif 0

       z ole5.gif 0



Function F21:


       x ole6.gif 0

       y ole7.gif x'

       z ole8.gif 0



Function F22:


       x ole9.gif 0

       y ole10.gif y'

       z ole11.gif 0



Function F31:


       x ole12.gif 0

       y ole13.gif 0

       z ole14.gif x'




Function F32:


       x ole15.gif 0

       y ole16.gif 0

       z ole17.gif y'



Basis of the vector space A of all mxn matrices over a field F. A basis for the vector space A of all mxn matrices over a field F is given by the set of nm mxn matrices

      

             {Eij: i=1,m; j=1,n}

      

where Eij has a 1 in the i-th row and j-th column, all other entries being zero.



Question. Are Fij and Eij related? If so, how are they related? Hom(V,W) corresponds directly to the set A of all mxn matrices. The set of mn mxn matrices {Eij: i=1,m; j=1,n} represent a basis for A and so they must also represent a basis for Hom(V,W). But the set {Fij: i=1,n; j=1,m} also represent a basis for Hom(V,W). Is it the same basis? Do Fij in any cases correspond directly to Eij? Each of the functions Fij represent a linear mapping from space V into space W and correspond to some member of the set Hom(V,W) of all such mappings. Thus each Fij must correspond to some mxn matrix.



Answer. Let the set of n elementary n-vectors


            e1 = (1, 0, 0,..., 0)

            e2 = (0, 1, 0,..., 0)

            ...................

            en = (0, 0, 0,..., 1)


constitute a basis for V. Let the set of m elementary m-vectors


            g1 = (1, 0, 0,..., 0)

            g2 = (0, 1, 0,..., 0)

            ...................

            gm = (0, 0, 0,..., 1)


constitute a basis for W. Consider the effect of the linear mapping of a basis vector ei into a vector w by the matrix Eij


                     w = Eij ei


Here Eji selects the i-th element of elementary vector ei and maps it into the j-th position in vector w, putting zeros elsewhere. Thus it maps the i-th elementary vector of V into the j-th elementary vector of W. Hence it does the same thing as Fij . In this case Fij corresponds to the matrix Eji.




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