Proof of Linear Independence involving vector spaces Consider a vector space V , vectors v1,...,vn ∈ V and 

w = $α_1v_1$ + $α_2v_2$ +···+ $α_nv_n$, where $α_1$,...,$α_n$ ∈R and $α_1$ $\neq$ 0. 

Prove that:
a) If $v_1$,$v_2$,...,$v_n$ are linearly independent, then w,$v_2$,...,$v_n$ are linearly independent

How would I a formulate proof for this or even the starting step?
HINT

Let apply the definition to the set $w,v_2,\ldots,v_n$ to obtain

$$c_1w+c_2v_2+\ldots+c_nv_n=c_1(α_1v_1 + α_2v_2 +\cdots+ α_nv_n)+c_2v_2+\ldots+c_nv_n=\ldots=0$$

Can you continue form here? What condition do we find for $c_1,c_2,\ldots,c_n$.
Suppose $a_1w+a_2v_2+\cdots + a_nv_n = 0$. If $a_1 \neq 0$, then $\alpha_1v_1+\alpha_2v_2+\cdots + \alpha_nv_n = w = -\dfrac{a_2}{a_1}v_2 - \dfrac{a_3}{a_1}v_3 - \cdots - \dfrac{a_n}{a_1}v_n\implies \alpha_1v_1+\displaystyle \sum_{k=2}^n \left(\alpha_k+\dfrac{a_k}{a_1}\right)v_k = 0\implies v_1,v_2,..., v_n$ are linearly dependent, since $\alpha_1 \neq 0$ - a contradiction. Thus $a_1 = 0$ and since $v_2,v_3,...,v_n$ are linearly independent, we have $a_2=a_3=...=a_n = 0$, proving that $w, v_2,v_3,...,v_n$ are linearly independent.