<exercise checkit-seed="0012" 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(x_{1} + x_{2},\,y_{1} + y_{2}\right) </me>
<me>c \odot (x,y) = \left(c x,\,c y - 7 \, c + 7\right) .</me>
<p>
(a) Show that scalar multiplication is associative, that is:
</p>
<me>a\odot(b\odot (x,y))=(ab)\odot(x,y).
</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>scalar multiplication does not distribute over vector addition</li>
<li>scalar multiplication does not distribute over scalar addition</li>
</ul>
</answer>
</exercise>
