 Solution

For each fixed finite tuple length $n$, use the finite conjunctionIt asserts that no nontrivial scalar combination vanishes, exactly linear independence. It is an $L$-formula because the field is finite and every scalar multiplication is named. Scalar coefficients are indices of function symbols, not variables ranging over a separate field sort. There is no single finite formula with arbitrarily many free variables; the assertion is a formula for each prescribed finite $n$.