Linear maps on finite dimensional vector spaces

Let $V$ and $W$ are finite dimensional vector spaces such that dim $V= n$ and dim $W=n$

Definition 1 (Linear Operator)  

Example 1  

Theorem 1.1   Let $S,T\in L(V,W)$ such that $S(v_i)=T(v_i)$ for every element of the basis then $S(x)= T(x)$ for every $x\in V$.

Theorem 1.2   Let $V$ be a finite dimensional vector space. Let

$$B=\{v_1,v_2,\dots,v_n\}$$

be a basis of $V$ and $\{w_1, w_2, \dots, w_n\}$ be arbitrary elements of $W$. Then there exists a unique $T\in L(V,W)$ such that $T(v_i)=w_i$ for $i=1,2,\dots, n$.

Exercise 1   Let $v\in V$ and $w\in W$. Find the necessary and sufficient conditions for the existence of a linear map $T\in L(V,W)$ such that $T(v)=w$.