Suppose that $\quad \vec{a}^T\vec{d}>0\quad$ for a $j\in I-W_k\quad$ and $\quad \vec{d}\neq 0$ is the optimum solution of the system $$\text{min}_{\vec{d}} (\frac{1}{2}d^TGd+d^T\vec{g}_k)$$ $$\vec{a}_i\vec{d}=0, \quad(i\in W_k)$$ with $$g_k=Gx_k+c$$
Show that $\vec{a}_j$ is linear independent with respect to $\{\vec{a}_i:i\in W_k\}$.
My try:
Let's suppose that $\vec{a}_j$ is linear dependent of $\{\vec{a}_i:i\in W_k\}$. This is: for at least one $\alpha_i\neq 0$
$$a_j=\alpha_1\vec{a}_1+...+\alpha_n\vec{a}_n$$
Since $\vec{d}$ is an optimum solution then for $j\in W_k$ and $\alpha\in\mathbb{R}$ $$\vec{a}^T_j(a+\alpha\vec{d})=\vec{b}_j$$
Then if we substitute:
$$(\alpha_1\vec{a}_1+...+\alpha_n\vec{a}_n)^T_j(a+\alpha\vec{d})=\vec{b}_j$$
I'm not sure if I'm not the right track. Any suggestions would be great!