= Solution
The vectors are <linearly independent> over $F$ if $\sum_{j=1}^n c_j\mathbf v_j=0$ with $c_j\in F$ implies $c_j=0$ for every $j$. Otherwise there is a nontrivial <linear dependence>. Forming a matrix with the displayed vectors as columns gives
$$
\det\begin{pmatrix}\alpha&2&-1\\3&4&0\\2&0&4\end{pmatrix}=16\alpha-16.
$$
Thus
$$
\boxed{\alpha=1}
$$
is the unique dependent case. At that value the explicit relation $2\mathbf v_1-\tfrac32\mathbf v_2-\mathbf v_3=0$ verifies the dependence.
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