Coordinates with Respect to a Basis

In the setting of Linear Algebra: Standing Notation, let $K$, with $+$, $\cdot$, $0$ and $1$, be a field, let $V$, with its addition and scalar multiplication, be a vector space over $K$, let $n\in\mathbb{N}_{0}$, and let $v=(v_{1},\dots,v_{n})$ be an $n$-tuple in $V$.

(i) Coordinates. $v$ is a basis of $V$ if and only if for every $u\in V$ there is exactly one $n$-tuple $(\lambda_{1},\dots,\lambda_{n})$ in $K$ with

$$u=\sum_{i=1}^{n}\lambda_{i}v_{i}.$$