<exercise checkit-seed="0008" checkit-slug="V1" checkit-title="Vector spaces">
<statement>
<p>
Let <m>V</m> be the set of all pairs <m>(x,y)</m> of real numbers
together with the following operations:
</p>
<me>(x_1,y_1)\oplus (x_2,y_2)= \left(5 \, x_{1} + 5 \, x_{2},\,3 \, y_{1} + 3 \, y_{2}\right) </me>
<me>c \odot (x,y) = \left(c x,\,c y\right) .</me>
<p>
(a) Show that scalar multiplication distributes over vector addition, that is:
</p>
<me>c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).
</me>
<p>
(b) Explain why <m>V</m> nonetheless is not a vector space.
</p>
</statement>
<answer>
<p><m>V</m> is not a vector space, which may be shown by demonstrating that
any one of the following properties do not hold:
</p>
<ul>
<li>vector addition is not associative</li>
<li>scalar multiplication does not distribute over scalar addition</li>
</ul>
</answer>
</exercise>
