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\begin{document}
\section{Aritmetick\'{e} priestory $\mathbf{R}^{n}$ a $\mathbf{C}^{n}$}
\subsection{D\^{o}kaz vety}
Nech $\mathbf{x}=(x_{1},x_{2},\ldots ,x_{n}),\ \mathbf{y}=(y_{1},y_{2},%
\ldots ,y_{n}),\ \mathbf{z}=(z_{1},z_{2},\ldots ,z_{n})$, potom
\begin{enumerate}
\item $\mathbf{x}+\mathbf{y}=(x_{1},x_{2},\ldots ,x_{n})+(y_{1},y_{2},\ldots
,y_{n})=(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n})=$\newline
$=(y_{1}+x_{1},y_{2}+x_{2},\ldots ,y_{n}+x_{n})=(y_{1},y_{2},\ldots
,y_{n})+(x_{1},x_{2},\ldots ,x_{n})=\mathbf{y}+\mathbf{x}$
\item $(\mathbf{x}+\mathbf{y})+\mathbf{z}=(x_{1}+y_{1},x_{2}+y_{2},\ldots
,x_{n}+y_{n})+(z_{1},z_{2},\ldots ,z_{n})=\newline
=(x_{1}+y_{1}+z_{1},x_{2}+y_{2}+z_{2},\ldots ,x_{n}+y_{n}+z_{n})=\newline
=(x_{1},x_{2},\ldots ,x_{n})+(y_{1}+z_{1},y_{2}+z_{2},\ldots ,y_{n}+z_{n})=%
\mathbf{x}+(\mathbf{y}+\mathbf{z})$
\item $\mathbf{x}+\mathbf{0}=(x_{1},x_{2},\ldots ,x_{n})+(0,0,\ldots
,0)=(x_{1}+0,x_{2}+0,\ldots ,x_{n}+0)=\newline
=(x_{1},x_{2},\ldots ,x_{n})=\mathbf{x}$
\item $\mathbf{x}+(-\mathbf{x})=(x_{1},x_{2},\ldots
,x_{n})+(-x_{1},-x_{2},\ldots ,-x_{n})=(x_{1}-x_{1},x_{2}-x_{2},\ldots
,x_{n}-x_{n})=\newline
=(0,0,\ldots ,0)=\mathbf{0}$
\item $1\cdot \mathbf{x}=(1\cdot x_{1},1\cdot x_{2},\ldots ,1\cdot
x_{n})=(x_{1},x_{2},\ldots ,x_{n})=\mathbf{x}$
\item $(\alpha \beta )\cdot \mathbf{x}=(\alpha \beta x_{1},\alpha \beta
x_{2},\ldots ,\alpha \beta x_{n})=\alpha \cdot (\beta x_{1},\beta
x_{2},\ldots ,\beta x_{n})=\newline
=\alpha \cdot \lbrack \beta \cdot (x_{1},x_{2},\ldots ,x_{n})]=\alpha \cdot
(\beta \cdot \mathbf{x})$
\item $\alpha \cdot (\mathbf{x}+\mathbf{y})=\alpha \cdot
(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n})=(\alpha x_{1}+\alpha
y_{1},\alpha x_{2}+\alpha y_{2},\ldots ,\alpha x_{n}+\alpha y_{n})=\newline
=(\alpha x_{1},\alpha x_{2},\ldots ,\alpha x_{n})+(\alpha y_{1},\alpha
y_{2},\ldots ,\alpha y_{n})=\alpha \cdot \mathbf{x}+\beta \cdot \mathbf{y}$
\item $(\alpha +\beta )\cdot \mathbf{x}=\left( \alpha +\beta \right) \cdot
(x_{1},x_{2},\ldots ,x_{n})=(\left( \alpha +\beta \right) x_{1},\left(
\alpha +\beta \right) x_{2},\ldots ,\left( \alpha +\beta \right) x_{n})=%
\newline
=(\alpha x_{1},\alpha x_{2},\ldots ,\alpha x_{n})+(\beta x_{1},\beta
x_{2},\ldots ,\beta x_{n})=\alpha \cdot \mathbf{x}+\beta \cdot \mathbf{x}$
\item $\alpha \cdot \mathbf{x}=(\alpha x_{1},\alpha x_{2},\ldots ,\alpha
x_{n})$ je nulov\'{a} $n$-tica pr\'{a}ve vtedy, ke\v{d} $\alpha
x_{1}=0,\alpha x_{2}=0,\ldots ,\alpha x_{n}=0$, \v{c}o nastane pr\'{a}ve
vtedy, ke\v{d} $\alpha =0$ alebo $x_{1}=x_{2}=\cdots =x_{n}=0$ t.j. $\alpha
=0$ alebo $\mathbf{x}=\mathbf{0}$.
\end{enumerate}
\hfill $\blacksquare $
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