I can't get my head around this introductory theorem from Lang's Linear Algebra.
Let $V$ and $W$ be vector spaces and $\{ v_1,\cdots v_n\}$ be a basis of $V$ and $w_1,\cdots , w_n$ be arbitrary elements of $W$. Then there exists a unique mapping $T:V\rightarrow W$ such that $T(v_i) = w_i $ for all $i$ from $1$ to $n$.
Let $V$ and $W$ be $\mathbb{R}^3$ and $w_1, w_2 ,w_3$ be three collinear points, then how can we have a mapping which maps the basis to these three points? Or am I missing something?