Suppose $\{e\_1, . . . , e\_n\}$ is a basis of a real vector space $V$ . 

Let 
$$e\_{n+1} = −e\_1 − ... − e\_n$$ 

Prove that any vector $v$ of $V$ can be expressed in a unique way as a
linear combination

$$v = a\_1e\_1 + · · · + a\_ne\_n + a\_{n+1}e\_{n+1}$$

with coefficients $a\_1, . . . , a\_n, a\_{n+1} \in \mathbb{R}$ satisfying $a\_1 + · · · + a\_n + a\_{n+1} = 0$

could anyone help with this or even where to start as it is confusing me already!