Question

Let
T:U→V
be a linear transformation, where U and V are of the same finite dimension. Then the following but one statements are equivalent
T is a homomorphism
T is an isomorphism
T is 1 - 1
T is onto

Answer 1

32980:

Task.

Let

T:U→V

be a linear transformation, where U and V are of the same finite dimension. Then the following but one statements are equivalent

T is a homomorphism

T is an isomorphism

T is 1 - 1

T is onto

Solution.

Isomorphism is a one-to-one relation onto the map between two sets, which preserves the relations existing between elements in its domain. Let T:U→V be a linear transformation, where U and V are of the same finite dimension. Then T is an isomorphism. Thus, the statement “T is a homomorphism” is not true.

Answer. “T is a homomorphism” is the only non-equivalent statement.