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Website owner: James Miller
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 {v1, v2, v3} be three basis vectors for vector space V and {w1, w2} be two basis vectors for W. Then a basis for Hom(V,W) would be the following six functions:
Function F11:
v1
w1
v2
0
v3
0
Function F12:
v1
w2
v2
0
v3
0
Function F21:
v1
0
v2
w1
v3
0
Function F22:
v1
0
v2
w2
v3
0
Function F31:
v1
0
v2
0
v3
w1
Function F32:
v1
0
v2
0
v3
w2
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.
Example. A basis for the linear space of all 2x3 matrices is the set of six 2x3 matrices:
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