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\rput[b](0.5, 1.55){$(x_2, y_2)$}
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\begin{definition}[$+, \cdot$ in $\mathbb R^2$]
Let $\fcv{u}=(x_1, y_1)$ and $\fcv{v}=(x_2, y_2)$ be pairs of numbers and let $c$ be a number. Define

\hfil \hfil $
\begin{array}{rcl}
\fcv{u} +\fcv{v} &=& (x_1,  y_1) + (x_2, y_2 ) = ( x_1+x_2, y_1+ y_2) \\
c\cdot \fcv{u}&=& c \cdot (  x_1, y_1 ) =(cx_1, cy_1 ).
\end{array}
$
\end{definition}
\end{columns}
\begin{itemize}
\item Fix a Cartesian coordinate system in the plane.
\item<2-> The correspondence between points in the plane and pairs of numbers depends on the choice of Cartesian coordinate system.
\item<3-> If we change the coordinate system we change $+, \cdot$.
\item<4-> The points in the plane, equipped with the operations $+,\cdot$ form a mathematical object which we call a vector space. 
\end{itemize}
\vskip 10cm
\end{frame}









