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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 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

{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.