Simple question about changing bases in a vector space

Question

Let V be a vector space of dimension n. Consider bases $v_1, v_2 ... v_n$ and $w_1, w_2, ... w_n$. Is there a linear map $f : V \rightarrow V$ such that $f(v_i) = cw_i$ for $1 \le i \le n$ and some scalar c? What about $f(v_i) = w_i$?

Answer 1

Yes. In fact, for any choice of scalars ${c_i}_{i=1}^n$, there is a unique linear map $f$ which satisfies this. Particularly, for $c_i=1$.