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. 

Consider the quotient space $V/W.$ Then the given condition says that the image of the vectors $v_1, v_2, \cdots , v_n \in V,$ under the natural map $V \rightarrow V/W$ is also linearly ondependent.
