I'm now self-studying the book Linear Algebra Done Right. In chapter 3, I learned the theorem 3.5, Linear maps and basis of domain: Suppose $v_1,...,v_n$ is a basis of V and $w_1,...,w_n∈W$. Then there exists a unique linear map $T:V→W$ such that $$Tv_j=w_j$$ for each $j=1,...,n$.
"The existence part of the next result means that we can find a linear map that takes on whatever values we wish on the vectors in a basis. The uniqueness part of the next result means that a linear map is completely determined by its values on a basis." -- a paragraph before this theorem.
I'm still wondering how to make use of this theorem. It confused me quite a while, so would you please help me, really really appreciate it.