Let V c1 cos theta c2 sin theta c1 c2 R and W R2 Let T V

Let V = {c_1 cos theta + c_2 sin theta | c_1, c_2 R} and W = R^2. Let T: V rightarrow W be the linear transformation defined by T(c_1 cos theta + c_2 sin theta) = [c_1 c_2]

Solution

A homomorphism is a structure-preserving map between two algebraic structures .An isomorphism is a homomorphism  (i.e. a mathematical mapping) that admits an inverse. Two mathematical objects are isomorphic if an isomorphism exists between them.

By definition, T is a homomorphism as it preserves structures. Further, since T( c₁ cos + c₂ sin) = (c₁ , c₂ )^T , therefore, T^-1 (c₁ , c₂ )^T = c₁ cos + c₂ sin . Now, (c₁ , c₂ )^T R²    and c₁ cos + c₂ sin V , where c₁ , c₂ are arbitrary elements of R. This means that T admits an inverse so that T is an isomorphism. This implies that V W