"Suppose $v_1...v_n$ is a basis for $V$ and $w_1...w_n \in W$. Then there exists a unique linear map $T: V \implies W$ such that: $T(v_j) = w_j$ for each $j = 1...n$."
I understand that because $v_1...v_n$ is a basis and spans $V$, it can generate any value in $V$ that can be mapped to $W$ by $T$.
So, by using $T$ and some linear combination of the basis of $V$, we can map to any value in the range/image of $T$. However, why does the linear map $T$ have to be unique as the theorem says? And, what is the true point of the theorem in layman's terms?