Proving there exists a linear transformation between the basis of one subspace and the vectors of another

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If $\{v_1, ..., v_n\}$ is a basis for V and $w_1, ..., w_2$ are vectors in $W$, not necessarily distinct, then there exists a linear transformation $T: V \to W$ such that $T(v_1) = w_1, ..., T(v_n) = w_n$.

I'm not sure how to go about solving this. I try proving that such a linear transformation exists, but I run into issues. For example, in this case, I'm finding it troublesome in proving that $T(kv_i) = kT(v_i)$, as $kv_i$ will generate a value elsewhere in $V$, and not necessarily within the basis, and we are not guaranteed that the transformation of this value is within the set $w_1, ..., w_n$. Could someone give me some help on this one? Just a hint though.

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1 Answer

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Note that if $v\in V$, then there exists unique scalars $\alpha_i$ such that $$v=\alpha_1 v_1 +\alpha_2 v_2+\dotsb+\alpha_n v_n.$$ Define $T:V\to W$ by $$T(v)=\alpha_1 w_1 +\alpha_2 w_2+\dotsb+\alpha_n w_n.$$ Note that this map is well-defined and meets the criteria you are looking for. Check that $T$ is linear.

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