is hom(v, w) is a vector Space?if explain it with example.
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Answer:
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: