ALG2_Q1

Suppose $\mathbb{F}$ is a infinite field. Let $V$ be a vector space over $\mathbb{F}$. Show that $V$ cannot be written as a union of finitely many proper subspace. Does the same statement hold if $\mathbb{F}$ is a finite field?

When F is an infinite field

FTSOC, Let $V={\bigcup }_{i=1}^n{V}_i$, where $V_i$ are proper subspaces of $V$. Pick a non-zero vector $\mathbf{x}\in V_1$. Pick another non zero vector $\mathbf{y}\in V-V_1$. Note that there are infinite vectors of the form $\mathbf{x}+\alpha \mathbf{y}$, $\alpha \in \mathbb{F}\setminus \{0\}$, and none of them are in $V_1$. Thus, some $V_j$, $2\le j\le n$, must contain an infinite number of these vectors. $V_j$ must thus contain $\mathbf{y}$, and thus must contain $\mathbf{x}$. Since $\mathbf{x}$ was arbitrary to begin with, we have shown that every $\mathbf{x}\in V_1$ can be found in ${\bigcup }_{i=2}^n{V}_i$, i.e, $V={\bigcup }_{i=2}^n{V}_i$. We can proceed inductively to reach $V=V_n$, which is impossible since $V_n$ is a proper subspace of $V$. Thus, $V$ cannot be a finite union of proper subspaces.

When F is a finite field

A finite dimensional vector space over a finite field has a finite number of elements. Suppose the field  $\mathbb{F}$  is finite, with  |$\mathbb{F}$| = $q$ , where  $q$  is the number of elements in the field. Let  $V$  be a vector space over  $\mathbb{F}$  with dimension  $n$ . This means that  $V$  has a basis consisting of  $n$  linearly independent vectors. Each element of  $V$  is a linear combination of the basis vectors. Specifically, if  $\{{\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n\}$  is a basis for  $V$, then every element of  $V$  can be written uniquely as:

$$a_1{\mathbf{v}}_1+a_2{\mathbf{v}}_2+\cdots +a_n{\mathbf{v}}_n$$

where  $a_1,a_2,\dots ,a_n\in \mathbb{F}$.

Since  $\mathbb{F}$  has  $q$  elements, each coefficient  $a_i$  in the linear combination has  $q$  possible values. Therefore, the total number of elements in  $V$  is $|V|=q^n$.

Thus, it is easy to see that $V$ can be expressed as a finite union of proper subspaces.