Linear mapping/Determined on generating set/No mapping/Exercise

Let ${}K$ be a field and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let ${}v_{1},\ldots ,v_{n}$ be a system of generators for ${}V$ and let ${}w_{1},\ldots ,w_{n}$ be a family of vectors in ${}W$.

a) Prove that there is at most one linear map

$\varphi \colon V\longrightarrow W$

such that ${}\varphi (v_{i})=w_{i}$ for all ${}i$.

b) Give an example of such a situation, where there is no linear mapping with ${}\varphi (v_{i})=w_{i}$

for all ${}i$.

Create a solution