Question

Let V be a vector space over F. Give a constructive proof that V and L(F,V) are isomorphic, i.e., construct an explicit isomorphism between these spaces

Answer 1

To show that V and L(F,V) are isomorphic, we can construct an explicit isomorphism between these spaces by defining a mapping T: V to L(F,V). We then need to prove that T is linear and bijective.

In order to show that V and L(F,V) are isomorphic, we can construct an explicit isomorphism between these spaces.

1. Let's define the mapping T: V → L(F,V) as follows:
   
   For each vector v ∈ V, let T(v) be the linear transformation from F to V defined by T(v)(a) = av for all a ∈ F. This means that T(v) maps a scalar a to the vector av.
2. To prove that T is an isomorphism, we need to show that it is linear and bijective. You can break this down into two parts:
   
   - To show that T is linear, we can verify that it preserves vector addition and scalar multiplication.
   
   - To show that T is bijective, we can establish both injectivity and surjectivity.

By constructing an explicit isomorphism T between V and L(F,V), we have shown that they are isomorphic.