<question>

<question>
 <title>Unions of vector subspaces</title>
 <statement>
   <p>
       Is the union of subspaces of a vector space always a vector space?
   </p>
 </statement>
 <hint>
   <p>
     Examples are often the key to answering questions like this.
     Pick a nice vector space like <m>\mathbb R^2</m> or
     <m>\mathbb R^3</m> and play.
   </p>
 </hint>
 <answer>
   <p>
     No they are not.  For example, the <m>x</m>-axis and <m>y</m>-axis in
     <m>\mathbb R^2</m> are both subspaces, but their union is not since the
     union is not closed under addition:  <m>(0,1) + (1,0) = (1,1)</m>, and
     the vector <m>(1,1)</m> is not a member of either axis.
   </p>
 </answer>
 <solution>
   <p>
     Show that if <m>U,W</m> are subspaces of a vector space <m>V</m>,
     then <m>U\cup W</m> is a subspace if and only if
     <m>U\subseteq W</m> or <m>W\subseteq U.</m>
   </p>
 </solution>
</question>