Products of linearly independent sets in finite fields Let $\mathbb F_q$ be the finite field with $q$ elements, $q=2^n$. This is a vector space over $\mathbb F_2$. My question is rather general: given two linearly independent sets of vectors of the same size, say $A=\{x_1,…,x_k\}$ and $B=\{y_1,…,y_k\}$, is it possible to say anything useful about the set $C=\{x_iy_j:1\leq i,j \leq k\}$ in terms of independence, i.e., the size of a maximal linearly independent subset?
The question is probably too vague to admit a full answer, so feel free to make any assumptions that might restrict the problem considerably. For example, what if we take $B$ to be the image of $A$ under a certain (injective) map (automorphism, permutation polynomial etc) ?
P.S. I should clarify that I'm not really asking of anyone to spend time on this. What I'm interested in is whether this problem has appeared somewhere in the existing literature in one form or another.
 A: Let $k < n/2$ and let $\alpha$ be a generator of $\mathbb F_q^\times$.  Then, any $n$
consecutive powers of $\alpha$ are a basis for $\mathbb F_q$ regarded as 
a vector space over $\mathbb F_p$.  Take $A = \{1, \alpha, \ldots, \alpha^{k-1}\}$ and $B = \{x^{-1} \colon x\in A\}$ which clearly are linearly independent. Then, $C = \{a_ib_j\}$ 
is a multiset containing $\{\alpha^\ell \colon -k < \ell < k\}$, a set of $2k < n$ distinct elements that are linearly independent. Note that $p$ need not be restricted to have value $2$.
But perhaps you are asking what is the minimum size of the maximal independent subset of $C$ for two arbitrarily chosen linearly independent sets $A$ and $B$
of size $k$?

Added in response to "For instance, suppose $k \leq \sqrt{n}$. Is it possible to choose 
$A$ and $B$ in such a way so that dim(span($C))=k^2$?"
Yes, at least in one instance.  Let $q=16$, $n = 4$, and take $A = \{1,\alpha\}$ and
$B = \{1,\alpha^2\}$ of cardinality $k = \sqrt{4} = {2}$. Then $C = \{1,\alpha, \alpha^2,\alpha^3\}$ is the canonical polynomial basis for $\mathbb F_{16}$ as a $4$-dimensional
vector space over $\mathbb F_2$.
