= Solution
A <basis> provides unique coordinates. Define
$$
\boxed{\alpha(a_1e_1+a_2e_2+a_3e_3)=a_1v_1+a_2v_2+a_3v_3.}
$$
The uniqueness of the coordinates makes this a well-defined <function>. Adding coordinates and multiplying them by a real <scalar> shows directly that $\alpha(au+bw)=a\alpha(u)+b\alpha(w)$, so it is a <linear map>. Substituting $e_i$ gives $\alpha(e_i)=v_i$.
Conversely, every <linear map> with those three values must send $\sum_i a_ie_i$ to $\sum_i a_iv_i$, by preservation of <linear combinations>. This determines its value on every element of $V$, proving uniqueness. No <linear independence> of $v_1,v_2,v_3$ is needed.
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