Hi guys I have been reading Numerical optimization by Nocedal. While I was on the chapter on quasi- Newton methods it was mentioned that $g_{k+1}^Ts_k=0$ where $g_k = Ax_k -b^Tx_k$ if we use exact line search on a quadratic (The claim is in chapter 7.2 Limited quasi-Newton methods pg 181 in the second edition of the book).I think I understand why but I wanted to check and see if I am correct.
$\phi(x_k) = x_k^TAx_k-b^Tx_k$
if we take the directional derivative in direction $p$ we get $\nabla \phi(x_k)p = (Ax_k-b)^Tp$. Now if we are trying to minimize the function we have the optimality condition that $\nabla \phi(x_k)p =0$. This leads me to $g_k^T p_k =0$ which will imply that for $s_k = \alpha_k p_k$ we have $g_k^Ts_k = 0$ (it seems as if I am an index off). So my question is am I making a small mistake or I am really not understanding why we have $g_{k+1}^Ts_k =0$ in the book.