0004.tex

\begin{exercise}{V1}{Vector spaces}{0004}
\begin{exerciseStatement}

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

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

 (a) Show that scalar multiplication distributes over vector addition, that is: 

\[c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).
    \]

 (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 vector addition is not associative
\item scalar multiplication does not distribute over scalar addition
\end{itemize}

    \end{exerciseAnswer}
\end{exercise}