I know that for an equality-constrained optimization problems the optimality conditions are that the gradient must be orthogonal to the constraint vectors.
So, say we have a problem like this:
$$\min_x \frac{1}{2}\, x^T Q\, x + c^T x$$ $$s.t. \;A x = b$$
since [edited] $\nabla f = Qx+c$, the optimality condition is $\nabla f = Qx+c \in \{v|A^Tw=v\},$ for some $w\in \mathbb{R}^n$, which can be said like so: $Qx+c = A^Tw$.
But, according to resources like this one, https://ee227c.github.io/notes/ee227c-lecture13.pdf, there is an additional constraint, which is that the minimizer lies in the affine set described by the equality constraint: $Ax=b$
I don't see how this constraint comes about from the FONC of optimality. What is it about the FONC that give rise satisifying $Ax=b$? Isn't that taken care of by the gradient being orthogonal to the constraint vectors by saying $Qx+c = A^Tw$? or no?
related: How do I find KKT Conditions for the Quadratic Function?