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Hom(V,W). Vector space of all mxn matrices

Vector space of all mxn matrices. Let V be Euclidean n-space consisting of all n-vectors over the field F and W be Euclidean m-space consisting of all m-vectors over the field F. Let A represent the set of all mxn matrices over the field F. For any v in V and a in A the matrix product

w = av

defines a linear mapping which maps a vector v in V into a vector w in W i.e. it defines a mapping from n-dimensional space V into m-dimensional space W. The set A of all mxn matrices over F is also a vector space. It corresponds to the set of all linear operators that map V into W. It is called Hom(V,W). What is the dimension of vector space A? Its dimension is mn.

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 mn 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.

Example. A basis for the linear space of all 2x3 matrices is the set of six 2x3 matrices:

References

Hohn. Elem. Matrix Algebra. p. 186,187

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