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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, ... ,v_n } are a set of basis vectors for V

            {w₁, w₂, ... ,w_m } 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 {F_ij: i=1,n; j=1,m} where

            F_ij maps v_i into w_j 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 x'

       y 0

       z 0

Function F₁₂:

       x y'

       y 0

       z 0

Function F₂₁:

       x 0

       y x'

       z 0

Function F₂₂:

       x 0

       y y'

       z 0

Function F₃₁:

       x 0

       y 0

       z x'

Function F₃₂:

       x 0

       y 0

       z 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

             {E_ij: i=1,m; j=1,n}

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

Question. Are F_ij and E_ij 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 {E_ij: 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 {F_ij: i=1,n; j=1,m} also represent a basis for Hom(V,W). Is it the same basis? Do F_ij in any cases correspond directly to E_ij? Each of the functions F_ij 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 F_ij 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)

            ...................

            e_n = (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)

            ...................

            g_m = (0, 0, 0,..., 1)

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

                     w = E_ij e_i

Here E_ji selects the i-th element of elementary vector 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 F_ij . In this case F_ij corresponds to the matrix E_ji.