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"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?

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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 $T'(v) = T'\big(\sum a_iv_i\big) = \sum a_i T'(v_i) = \sum a_i T(v_i) = T\big(\sum a_iv_i\big) = T(v)$.

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