Website owner: James Miller
Isomorphisms and homomorphisms effected by linear transformations from one abstract vector space into another
Let V be an n-dimensional and W be an m-dimensional abstract vector space over the field F. Suppose three linearly independent vectors spanning some subspace R of V are imaged into three linearly independent vectors spanning some subspace S of W under a linear transformation T. Then an isomorphism has been established between subspaces R in V and S in W. These two three dimensional subspaces are isomorphic to each other. Now suppose three linearly independent vectors spanning some subspace R in V are imaged into three vectors spanning some two dimensional subspace S of W under linear transformation T. Then a homomorphism has been established between subspaces R in V and S in W. These two subspaces are homomorphic to each other. If n linearly independent vectors in space V were imaged into n linearly independent vectors spanning some subspace S of W then vector space V would be isomorphic to subspace S of W. In general, if p linearly independent vectors spanning some subspace of V are imaged into k linearly independent vectors spanning some subspace of W then an isomorphism is established between the two subspaces if k = p and a homomorphism is established if k < p (the case k > p is impossible).
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