TITLE: Can a vector space over finite field be written as union of finite number of proper subspaces?
QUESTION [6 upvotes]: Recently, I solved a problem that says-
If $V$ is a vector space over an infinite field. Prove that, V cannot be written as set-thoretic union of a finite number of proper subspaces.
But is this result true in case of finite field? . I can't get such an example where a vector space over finite field can be written as union of finite number of proper subspaces.
Can anybody give such an example? Thanks for assistance in advance.

REPLY [3 votes]: It seems to me that the case of an infinite dimensional space is creating extra headache. Spelling out an example to show what can be done.
Consider the field $\Bbb{F}_2$. Let $V=\Bbb{F}_2[x]$ be the space of univariate polynomials. Let $U$ be the subspace of polynomials with zero constant and linear terms.
Consider the following three subspaces of $V$:
$$
V_1=\langle 1\rangle\oplus U=\{a_0+a_1x+a_2x^2+\cdots +a_nx^n\mid n\in\Bbb{N}, a_i\in\Bbb{F}_2, a_1=0\},
$$
$$
V_2=\langle x\rangle\oplus U=\{a_0+a_1x+a_2x^2+\cdots +a_nx^n\mid n\in\Bbb{N}, a_i\in\Bbb{F}_2, a_0=0\},
$$
and
$$
V_3=\langle 1+x\rangle\oplus U=\{a_0+a_1x+a_2x^2+\cdots +a_nx^n\mid n\in\Bbb{N}, a_i\in\Bbb{F}_2, a_0=a_1\}.
$$
It is clear that every polynomial of $V$ belongs to at least one of the subspaces $V_1,V_2,V_3$ according to what its constant and linear terms look like. Therefore
$$V=V_1\cup V_2\cup V_3.$$

What just happened?
Consider the quotient space $V/U$. It is 2-dimensional. Its non-zero elements are the cosets $1+U,x+U$ and $1+x+U$. So we can write $V/U$ as a union of three 1-dimensional subspaces following the recipe of Servaes' answer each spanned by one of those cosets. The resulting subspaces are exactly the spaces $V_i/U, i=1,2,3$. No wonder that $V_1,V_2,V_3$ cover all of $V$!