0007.tex

\begin{exercise}{V1}{Vector spaces}{0007}
\begin{exerciseStatement}

 Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations: 

\[(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2} + 4,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\right) \]\[c \odot (x,y) = \left(c x,\,c y\right) .\]

 (a) Show that vector addition is associative, that is: 

\[\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).
    \]

 (b) Explain why \(V\) nonetheless is not a vector space. 

\end{exerciseStatement}
\begin{exerciseAnswer}

\(V\) is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold: 



\begin{itemize}
\item there is no additive identity element
\item scalar multiplication does not distribute over vector addition
\item scalar multiplication does not distribute over scalar addition
\end{itemize}

    \end{exerciseAnswer}
\end{exercise}