Covering of a vector space over a finite field

Question

Let $k$ be a finite field and $V$ a finite-dimensional vector space over $k$.

Let $d$ be the dimension of $V$ and $q$ the cardinal of $k$.

Construct $q+1$ hyperplanes $V_1,\ldots,V_{q+1}$ such that $V=\bigcup_{i=1}^{q+1}V_i$

I tried induction on $d$, to no avail.

There has been discussion about this topic (here), but it doesn't answer my question.

Answer 1

Note that it suffices to do this for a vector-space of dimension $2$. You can then simply "blow up" each line to a hyperplane in a larger space.

For dimension two, check that $q+1$ is in fact equal to the number of hyperplanes, that is lines in this case.

Answer 2

It is useful in this case to consider the dual. In the dual space, your collection of hyperplanes covering every vector corresponds to a set of $(q+1)$ $1$-dimensional subspaces such that each hyperplane contains at least one of these $1$-spaces. If we fix a $2$-dimensional subspace $\pi$, then $\pi$ contains $q+1$ $1$-dimensional subspaces, and every hyperplane meets $\pi$ in a $1$-dimensional subspace.

This shows that by taking all of the $1$-dimensional subspaces in a common plane, we have the desired property. Then we can replace each of these $1$-dimensional subspaces with its dual (through any mapping, sending them to the orthogonal complement under the dot product will work), and we obtain a set of $q+1$ hyperplanes covering all of the points of $V$.