Find a field K where the vectors are linearly dependent in the vector space K^3

Question

My task is to find a field $K$ where the vectors $$ \begin{pmatrix} 2 \\ 3 \\ 1 \\ \end{pmatrix} , \begin{pmatrix} 0 \\ 4 \\ 0 \\ \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 2 \\ \end{pmatrix} $$are linearly dependent in the vector space $K^3$ So far I thought about that its working with $Z_2$, but I'm not sure if its valid or if there are other solutions.

Answer 1

Hint: These vectors are linearly dependent iff the determinant of the matrix formed by their coordinates is zero. This will depend on the characteristic of $K$.

Solution:

The determinant is $16$ and so the vectors are linearly dependent iff the characteristic of $K$ is $2$. Note that when the characteristic is $2$, you don't need the determinant because the second vector is the zero vector and so the three vectors are certainly linearly dependent.