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Let $V$ and $W$ are vector spaces over a field $F$. Let $B$ be a basis of $V$, then prove that every set function $f:B→W$ extends uniquely to a linear transformation $T:V→W$. I can't conclude how to show this. please help me.

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Since $B$ is a basis, any element of $V$ can be written as a linear combination of elements of $B$, i.e. if $v \in V$ and $B=\{x_1,\ldots , x_n\}$, then there exist scalars $c_i$ such that $$v = c_1 x_1 + \cdots + c_n x_n.$$

You have a function $f$ that sends each $x_i$ to some element of $W$. Since we want $T$ to extend $f$, it must be that $T(x_i)=f(x_i)$ for each $x_i \in B$. Then, what does linearity of $T$ tell you about $T(v)$?

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