Linear Maps Are Determined by Their Values on a Basis

In the setting of Linear Algebra: Standing Notation, let $K$ be a field, let $V$ and $W$ be vector spaces over $K$, let $n\in\mathbb{N}_{0}$, let $(v_1,\dots,v_n)$ be a basis of $V$, and let $w_1,\dots,w_n\in W$.

(i) Linear extension. There is exactly one linear map $T$ from $V$ to $W$ such that

$$Tv_i=w_i\qquad\text{for each } i\in[n].$$