My textbook asks this:
Suppose that $K$ is a finite field with $k$ elements, and that $V$ is an $r$-dimensional vector space over $K$. Show that if $V = \bigcup_{i=1}^n U_i$, where $U_1,\dotsc,U_n$ are proper subspaces of $V$, then $n\geq (k^r - 1)/(k-1)$.
Struggling to prove this for a while, I did some googling and found this paper which claims to show $n = k+1$ is possible, a result which is independent of the dimension of $V$. Which is correct?
Consider the linear equations $$\tag{eq} \begin{cases} x_{1} - \lambda_{j} x_{r} = 0\\ x_{r} = 0. \end{cases} $$ Here $K = \{\lambda_{1}, \dots, \lambda_{k} \}$, so there are $k+1$ equations in (eq), which define $k+1$ maximal subspaces $W_{j}$ of $V$.
Let $v \in V$. If the $r$-th component $v_{r}$ of $v$ is zero, then $v \in W_{k+1}$. If $v_{r} \ne 0$, then $v_{1} = \lambda_{j} v_{r}$ for $\lambda_{j} = v_{1} v_{r}^{-1}$, so $v \in W_{j}$. Thus $V = \bigcup_{j=1}^{k+1} W_{j}$.
So it can indeed be done with $k+1$ subspaces. The answer mentioned above proves that you cannot do better than this.