each coset $u+W$ intersects the subspace generated for the basis of the quotient space $V/W$ How prove the next property of a quotient subspace?
If $\dim V$ = n, $W\subset V$ is a subspace such that $\dim W = m < n$ with $\beta_{V} = \{ v_1,\dots,v_n \}$, $\beta_{W} = \{ v_1,\dots,v_m \}$ and $ \beta_{V/W} = \{ v_{m+1}+W,\dots,v_n+W \}$. Then if $U = \langle v_{m+1},\dots, v_n \rangle $ each $v+W$ intersects U in a one vector.

Pick $v\in V$ arbitrarily, then establish $c_1,...,c_n$ such that $$v=c_1v_1+\dots+c_mv_m+c_{m+1}v_{m+1}+\dots +c_nv_n$$ Evidently, $$v+W=(c_{m+1}v_{m+1}+\dots +c_nv_n)+W$$ Since $W$ contains the zero vector, $$\begin{eqnarray*}c_{m+1}v_{m+1}+\dots + c_nv_n &=& c_{m+1}v_{m+1}+\dots +c_nv_n+0 \\ &\in& (c_{m+1}v_{m+1}+\dots +c_nv_n)+W \\&=&v+W \end{eqnarray*}$$ This shows $U\cap \Big(v+W\Big)\neq \emptyset$. We will show that $c_{m+1}v_{m+1}+\dots +c_nv_n$ is the only vector in this intersection as follows. Suppose we can find $w\in W$ such that $$c_{m+1}v_{m+1}+\dots +c_nv_n+w\in U$$ Find scalars $\alpha_{m+1},...,\alpha_n$ such that $$c_{m+1}v_{m+1}+\dots +c_nv_n+w=\alpha_{m+1}v_{m+1}+\dots + \alpha_n v_n$$ But this is equivalent to saying $$(\alpha_{m+1}-c_{m+1})v_{m+1}+\dots +(\alpha_n - c_n)v_n=w \in W$$ Since $\beta_{W}=\{v_1,...,v_m\}$ is a basis for $W$, we can find $q_1,...,q_m$ such that $$(\alpha_{m+1}-c_{m+1})v_{m+1}+\dots +(\alpha_n - c_n)v_n=q_1v_1+\dots +q_mv_m$$ We can write this as $$(-q_1)v_1+\dots +(-q_m)v_m+(\alpha_{m+1}-c_{m+1})v_{m+1}+\dots +(\alpha_n - c_n)v_n=0$$ Because $\beta_V=\{v_1,...,v_n\}$ is a basis for $V$ we must have all of these scalars vanishing. In particular, $c_{m+1}=\alpha_{m+1},...,c_n=\alpha_n$. But this immediately implies that $w=0$ which is was we needed.
