# Given any linear map from basis to a space W, it can be expanded to a linear map from whole space to a space W

I'm reading through Linear Algebra notes and came across something I don't quite get.

Every linear map $T:V \rightarrow W$ is determined by its values on a basis $\mathcal{B}$ for $V$.
Every map $T: \mathcal{B} \rightarrow W$ can be expanded to a linear map $T:V \rightarrow W$.

I think I understand the first sentence; it makes sense because $\mathcal{B}$ is spanning. However, I have trouble understand the second sentence. Could anyone please explain how to expand it to a bigger linear map?

Thank you.

$T(\sum \alpha _k b_k)=\sum \alpha_k T(B_k)$. Remember that every vector in $V$ is uniquely of the form $\sum \alpha _k b_k$, a finite sum, where the $\alpha _k$ are scalars and $b_k$ are basis vectors. The expansion I gave is well-defined because of the uniqueness, and is linear by definition. It is clearly an extension of $T$.