# A fundamental fact about Linear Algebra

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Hello MHB,
This is probably my first challenge problem which falls in the 'University Math' category.

$V$ is a vector space over an infinite field $F$, prove that $V$ cannot be written as a set theoretic union of a finite number of proper subspaces.

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Nobody participated

Here's my solution:

Assume contradictory to the problem. Let $n$ be the minimum integer such that $V$ can be written as $V=V_1\cup\cdots\cup V_n$ where each $V_i$ is a proper subspace of $V$. Thus, \begin{equation*}\forall i,\exists x_i\in V \text{ such that } x_i\in V_j\iff j=i\tag{1}\end{equation*}Now consider $S=\{f_1x_1+\cdots+f_nx_n:f_i\in F\}$. Clearly this set is infinite, thus, by PHP, there is a $k$ such that $a,b\in V_k$ for distinct $a$ and $b$ in $S$. This contradicts $(1)$. Hence we achieve the required contradiction and the proof is complete.