We have a linear subspace $L_j := L[s_0,s_1,...,s_j]$
and a linear variety: $x_0 + L_j := [x_0 + y : y \in L_j]$
and a standard quadratic cost function $V(x) = a + b^Tx + 0.5x^TCx, \ \ \ C^T = C > 0$
I want to prove that $x^*$ minimizes $V$ globally on $x_0 + L_j$ if (there's an "only if" too but I don't care about it):
i) $x^* \in x_0 + L_j$
and
ii) $\nabla V(x^*)^Ts_i = 0, \ \ \ i =0,1,...,j$
I can see it geometrically but can't seem to put it down into math.