Let $V$ be a vector space over an algebraically closed field $k$ of characteristic $p>0$, and denote by $V_q$ the vector space obtained from $V$ by restricting scalars to $\mathbb{F}_q$, where $q=p^d$ for some positive integer $d$. Given a finite-dimensional subspace $W\subset V_p$, we can consider $W$ as a subset of $V_q$, and define $W_q$ to be the subspace generated by $W$ in $V_q$. It is possible that $W=W_q$, but how much bigger (in terms of dimension as $\mathbb{F}_p$-spaces) can $W_q$ be?
Writing down a few examples, I think that we may have the inequality
$$\dim_{\mathbb{F}_p}(W)\le\dim_{\mathbb{F}_p}(W_q)\le d\cdot\dim_{\mathbb{F}_p}(W)$$
Can anyone verify this?
Also, its interesting to note that $W_q$ is a sort of $q$-linear envelope for $W$, that is, it is the smallest $\mathbb{F}_p$-subspace of $V_p$ containing $W$ that is also an $\mathbb{F}_q$-subspace of $V_q$. Has this been studied before? Can anyone point me to a reference? There could be some interesting questions to ask concerning such a construction and its iterates (i.e. do chains $W\subset W_q\subset (W_q)_{q'}\subset\ldots$ always stabilize?).
Suppose $V$ is a $K$-vector space, and we have subfields $M\subseteq L\subseteq K$.
Let $W\subseteq V$ be an $M$-subspace and let $W_L$ be the smallest $L$-subspace of $V$ which contains $W$. Then $\dim_LW_L\leq\dim_MW$; indeed, any set of $M$ which generates $W$ over $M$ also generates $W_L$ over $L$.
On the other hand, we have $\dim_MW_L=[L:M]\dim_LW_L$, so that $$[L:M]^{-1}\dim _MW\leq[L:M]^{-1}\dim_MW_L=\dim_LW_L.$$ We thus get the inequalities $$\frac{1}{[L:M]}\dim_MW\leq\dim_LW_L\leq\dim_MW.$$
This is best possible, in that both equalities do occur. For example, if $V=K$, and $W=M$, then $W_L=L$ and the right equality holds. On the other hand, if $W=L$, then $W_L=L$ and the left equality holds.