= Solution
The rows of the <Jacobian matrix> $Dg(a)$ are $\nabla g_1(a)^T,\ldots,\nabla g_k(a)^T$. Rank $k$ means precisely that these $k$ <gradients> are <linearly independent>. The <Lagrange multiplier> condition is that real numbers $\lambda_1,\ldots,\lambda_k$ exist with
$$
\boxed{\nabla f(a)=\sum_{j=1}^k\lambda_j\nabla g_j(a)}.
$$
Geometrically, $\nabla f(a)$ 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>.
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