# What does the theorem of Linear maps and basis of domain actually mean?

"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?

In layman's terms, a linear map is determined by what it does to a basis. If you specify the values of $$T$$ on a basis, then you can recover the value of $$T$$ on any vector you like by writing that vector as an appropriate linear combination of basis elements.
The uniqueness comes from the fact that if $$T'$$ is any other linear map that agrees with $$T$$ on the basis elements, then $$T' = T$$ because we can write any vector $$v$$ as a linear combination of basis elements: $$v = \sum a_iv_i$$ and then observe that $$T'(v) = T'\big(\sum a_iv_i\big) = \sum a_i T'(v_i) = \sum a_i T(v_i) = T\big(\sum a_iv_i\big) = T(v)$$.