Proposición 4.2.1
Sea \(V\) un \(\mathbb{K}\)- espacio vectorial, \(X=V\) y
\begin{equation*}
\begin{array}{cccc}
\cdot : \amp V \times V \amp \rightarrow \amp V \\
\amp \left( \overrightarrow{v},\overrightarrow{w}\right) \amp \rightsquigarrow \amp \overrightarrow{v}\cdot \overrightarrow{w}=\overrightarrow{v}+ \overrightarrow{w} \\
\end{array}
\end{equation*}
entonces \((V,V,\cdot)\) es un espacio afín.
Veamos si \(\cdot\) cumple con las propiedades anteriores
- \(\left( \forall \overrightarrow{v}\in V \right) \left( \overrightarrow{0}\cdot \overrightarrow{
v}=\overrightarrow{v}\right)\)
Sea \(\overrightarrow{v}\in V\text{,}\)
\begin{equation*}
\begin{array}{rcl}
\overrightarrow{0} \cdot \overrightarrow{v} \amp = \amp \overrightarrow{0}+\overrightarrow{v} \\
\amp = \amp \overrightarrow{v} \\
\end{array}
\end{equation*}
- \(\left( \forall \overrightarrow{v},\overrightarrow{w}, \overrightarrow{x} \in V \right) \left( \overrightarrow{v} \cdot \left( \overrightarrow{w}\cdot \overrightarrow{x}\right)=\left(\overrightarrow{v}+\overrightarrow{w}\right)\cdot \overrightarrow{x}\right)\)
Sean \(\overrightarrow{v},\overrightarrow{w}, \overrightarrow{x} \in V \text{,}\)
\begin{equation*}
\begin{array}{rcl}
\overrightarrow{v}\cdot \left( \overrightarrow{w}\cdot \overrightarrow{x} \right) \amp = \amp \overrightarrow{v} \cdot \left( \overrightarrow{w}+\overrightarrow{x} \right) \\
\amp = \amp \overrightarrow{v}+\left( \overrightarrow{w}+\overrightarrow{x} \right) \\
\amp = \amp \left( \overrightarrow{v}+ \overrightarrow{w} \right)+ \overrightarrow{x} \\
\amp = \amp \left( \overrightarrow{v}+ \overrightarrow{w} \right) \cdot \overrightarrow{x}
\end{array}
\end{equation*}
- \(\left( \forall \overrightarrow{x},\overrightarrow{y}\in V \right) \left(\exists ! \text{ } \overrightarrow{w}\in V )( \overrightarrow{w}\cdot \overrightarrow{x}= \overrightarrow{y}\right)\)
Sean \(\overrightarrow{x},\overrightarrow{y}\in V\text{,}\)
\begin{equation*}
\begin{array}{rcl}
\overrightarrow{u}\cdot \overrightarrow{x} \amp = \amp \overrightarrow{y} \\
\overrightarrow{u}+ \overrightarrow{x} \amp = \amp \overrightarrow{y} \\
\overrightarrow{u} \amp = \amp \overrightarrow{y}-\overrightarrow{x}
\end{array}
\end{equation*}
Además, \(\left( \overrightarrow{y}-\overrightarrow{x}\right)\in V\)
\begin{equation*}
\begin{array}{rcl}
\left( \overrightarrow{y}- \overrightarrow{x} \right) \cdot \overrightarrow{x} \amp = \amp \overrightarrow{y}- \overrightarrow{x}+\overrightarrow{x}\\
\amp = \amp \overrightarrow{y}
\end{array}
\end{equation*}
Luego al tomar \(V=X\text{,}\) entonces \((V,X,\cdot)\) es un Espacio Afín.
