 Solution

The rows of the Jacobian matrix $Dg(a)$ are $\nabla g_1(a{)}^T,\dots ,\nabla g_k(a{)}^T$. Rank $k$ means precisely that these $k$ gradients are linearly independent. The Lagrange multiplier condition is that real numbers ${\lambda }_1,\dots ,{\lambda }_k$ exist withGeometrically, $\nabla f(a)$ is perpendicular to the tangent space of the constraint level set, whose tangent vectors form $\ker Dg(a)$. The full-rank hypothesis ensures this level set is a smooth local constraint manifold.