0007.ptx

<exercise checkit-seed="0007" 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} + 4,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\right) </me>
    <me>c \odot (x,y) = \left(c x,\,c y\right) .</me>
    <p>
      (a) Show that vector addition is associative, that is:
    </p>
    <me>\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).
    </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>there is no additive identity element</li>
      <li>scalar multiplication does not distribute over vector addition</li>
      <li>scalar multiplication does not distribute over scalar addition</li>
    </ul>
  </answer>
</exercise>