I'm, trying to get an intuitive feel for the various constraint qualifications for KKT points. Most of them seem to rely on the linear independence of $\nabla g_i(x^*)$ where $g_i$ are the equality constraints. The book doesn't really state why.

The first KKT condition states

$\nabla f(x^*) + \sum\mu_i\nabla g_i(x^*) + \sum \lambda_j\nabla h_j(x^*) = \textbf{0}$

A hazy first guess is that if the gradients were to be linearly dependent, then any choice of $\lambda$ could potentially satisfy the condition, thus producing 'trivial' KKT points. We need to ensure that the term associated with the equality constraints only vanishes for $\lambda_j \equiv 0 $.

I think this is somewhat in analogue to the situation with the $\mu$ multiplier potentially being zero for $\nabla f(x^*)$ in the Fritz-John conditions.

Self-studying is hard :) Am I anywhere close here?