Understanding linearly independent vectors modulo $W$ We've learned in class:  

Let $W \subseteq V$, a subspace. $v_1, \ldots, v_k \in V$ are said to be linearly independent modulo $W$ if for all $\alpha_1, \ldots, \alpha_k: \sum_{k=1}^n \alpha_i v_i \in W \implies \alpha_1 = \ldots = \alpha_k = 0$

Can you explain it to me / give some intuitions or an example?
Thanks. 

Example
Let $V=\Bbb R[x]$ the vector space of the real polynomials and $W=(x^3)=\{x^3P(x)\mid P(x)\in V\}$. Then using the Euclidean division we see that $(1,x,x^2)$ is a basis for $V/W$ where we mean by $1,x,x^2$ the cosets $1+W,x+W,x^2+W$ respectively.
