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QUESTION Are the following structures vector spaces over
\textbf{R}? If not, which axioms fail to hold?

\begin{description}

\item[(a)]
The set of triples of real numbers $(x,y,z)$ with the operations
$$(x_1,y_1,z_1)+(x_2,y_2,z_2)=(x_1+x_2,y_1+y_2,z_1+z_2)$$
$$\textrm{ and } \lambda (x,y,z)=(0,0,0),\forall \lambda \in
\textbf{R}.$$

\item[(b)]
The singleton set containing the planet Saturn with

Saturn+Saturn=Saturn\ \ and $\lambda$(Saturn)=Saturn, $\forall
\lambda \in \textbf{R}$.

\end{description}


ANSWER
\begin{description}

\item[(a)]
No - the axiom $1(x,y,z)=(x,y,z)$ is violated here for most
$(x,y,z)$

\item[(b)]
Yes.

\end{description}




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