Question

Let F be a field and V a vector space over F with the basis {v1, v2, ..., vn}. (a) Consider the s...

Let F be a field and V a vector space over F with the basis {v1, v2, ..., vn}.

(a) Consider the set S = {T : V -> F | T is a linear transformation}. Define the operations:

(T1 + T2)(v) := T1(v) + T2(v), (aT1)(v) = a(T1(v)) for any v in V, a in F.

Prove tat S with these operations is a vector space over F.

(b) In S, we have elements fi : V -> F defined as fi (a1v1+ ... + anvn) = ai where aj is in F, and 1<= j <= n for any j. Prove that {f1, ...,fn} is a basis of S.

(c) Since S is a vector space over F, you can consider the set of linear transformations from S to F and create a vector space. Call it S'. For v in V, we define the elements kv: S -> F as kv(f) = f(v). Show that the map e: V -> S' defined as e(v) = kv is an isomorphism of vector spaces.

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i e is gme-one Sain Tㅚ a Junar brin sfarv rroohope3- @ rmsder m, sch-S { T :V-, F) is one-one iii) e is onto :- Thinclea il

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