0002.tex

\begin{exercise}{V1}{Vector spaces}{0002}
\begin{exerciseStatement}

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

\[(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \]\[c \odot (x,y) = \left(c x,\,c y - 3 \, c + 3\right) .\]

 (a) Show that scalar multiplication is associative, that is: 

\[a\odot(b\odot (x,y))=(ab)\odot(x,y).
    \]

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

    \end{exerciseAnswer}
\end{exercise}