Let V be a finite-dimensional vector space over the field F and let $\{\alpha_1,\ldots,\alpha_n\}$ be an ordered basis for V. Let W be a vector space over the same field F and let $\beta_1,\ldots,\beta_n$ be any vectors in W. Then there is precisely one linear transformation T from V into W such that $$\text{T}\alpha_j=\beta_j,\quad j=1,\ldots,n$$
I think the theorem is perfectly fine without the notion of an ordered basis. Is there any specific reason?