16-00-CartesianProductOfVectorSpaces.tex

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\pmtitle{Cartesian product of vector spaces}
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\pmtype{Definition}
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\begin{document}
Suppose $V_1,\ldots, V_N$ are vector spaces over a field $\F$.
Then the Cartesian product $V_1\times \cdots \times V_N$ is a vector space
when addition and scalar multiplication is  defined as follows
\begin{eqnarray*}
   (u_1,\ldots, u_N) +    (v_1,\ldots, v_N) &=&    (u_1+v_1,\ldots, u_N+v_N), \\
   k (u_1,\ldots, u_N) &=& (k u_1,\ldots, k u_N)
\end{eqnarray*}
for $u_i, v_i \in V_i$, $k\in \F$.

For example, the vector space structure of $\R^n$ if defined as above.

\subsubsection*{Properties}
\begin{enumerate}
\item If $V_i$ are vector spaces and $W_i\subset V_i$ are subspaces,
    then $W_1\times \cdots \times W_N$ is a vector subspace of 
    $V_1\times \cdots \times V_N$.
\item The dimension of $V_1\times \cdots \times V_N$ is 
    $\dim V_1+ \cdots +\dim V_N$.
\end{enumerate}
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