Consider the binary finite field $\mathbb{F}_{2^q}$ for some $q$. consider a set of elements $r_i$, $1\leq i \leq m$, in $\mathbb{F}_{2^q}$ where $m<2^q$.
My question: How to check that elements $r_i$'s are linear linear independence over $\mathbb{F}_{2^q}$.
Is there a matrix method such as the method that we use for vectors. In fact, is it true that we consider $r_i$'s as a vector in $\mathbb{F}_{2}$ and construct a matrix by these vectors. In the rest, if determinant of this matrix over $\mathbb{F}_{2}$ be non-zero then we conclude that $r_i$'s are linear linear independence over $\mathbb{F}_{2^q}$.
There is a similar question here but I can not understand how to apply for my question.
Thanks for your suggestions.
There is nothing peculiar about these fields as far as linear independence is concerned. Consider, for instance, the vectors $(1,1,0)$, $(0,1,1)$, and $(1,0,1)$. Are they linearly independent over $\mathbb{F}_{2^q}$? In order to know, compute the determinant$$\begin{vmatrix}1&0&1\\1&1&0\\0&1&1\end{vmatrix}.$$In turns out that it is equal to $0$. Therefore, the vectors are linearly dependent (for any $q$).