3.7: Suplementos - Espacio Vectorial
- Page ID
- 113091
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Hace tiempo que se da por sentado que el conjunto de números reales,\(\mathbb{R}\), se cierra bajo suma y multiplicación, que cada número tiene una inversa aditiva única, y que las leyes conmutativas, asociativas y distributivas tenían razón como lluvia. El conjunto\(\mathbb{C}\), de números complejos también disfruta de cada una de estas propiedades, al igual que los conjuntos\(\mathbb{R}^{n}\) y\(\mathbb{C}^n\) de columnas de n números reales y complejos, respectivamente.
Para ser más precisos, escribimos\(\textbf{x}\) y\(\textbf{y}\) en\(\mathbb{R}^{n}\) como
\(\textbf{x} = (x_{1}, x_{2}, \cdots, x_{n})^{T}\)
\(\textbf{y} = (y_{1}, y_{2}, \cdots, y_{n})^{T}\)
y definir su suma vectorial como la suma por elementos
\[\textbf{x}+\textbf{y} = \begin{pmatrix} {x_{1}+y_{1}}\\ {x_{2}+y_{2}}\\ {\vdots}\\ {x_{n}+y_{n}} \end{pmatrix} \nonumber\]
y de manera similar, el producto de un escalar complejo,\(\textbf{z} \in \mathbb{C}\) con\(\textbf{x}\) como:
\[\textbf{zx} = \begin{pmatrix} {zx_{1}}\\ {zx_{2}}\\ {\vdots}\\ {zx_{n}} \end{pmatrix} \nonumber\]
Espacio vectorial
Estas nociones conducen naturalmente al concepto de espacio vectorial. Se dice que un conjunto\(V\) es un espacio vectorial si
- \(\textbf{x}+\textbf{y}=\textbf{y}+\textbf{x}\)para cada uno\(\textbf{x}\) y\(\textbf{y}\) en\(V\).
- \(\textbf{x}+\textbf{y}+\textbf{z} = \textbf{y}+\textbf{x}+\textbf{z}\)para cada uno\(\textbf{x}\),\(\textbf{y}\) y\(\textbf{z}\) en\(V\).
- Hay un “vector cero” único tal que\(\textbf{x}+\textbf{0} = \textbf{x}\) para cada uno\(\textbf{x}\) en\(V\).
- Para cada uno\(\textbf{x}\) en\(V\) hay un vector único\(-\textbf{x}\) tal que\(\textbf{x}+ -\textbf{x} = \textbf{0}\).
- \(1 \textbf{x} = \textbf{x}\).
- \((c_{1}c_{2}) \textbf{x} = c_{1}(c_{2} \textbf{x})\)para cada uno\(\textbf{x}\) en\(V\)\(c_{1}\) y\(c_{2}\) en\(\mathbb{C}\).
- \(c(\textbf{x}+\textbf{y}) = c\textbf{x}+c\textbf{y}\)para cada uno\(\textbf{x}\) y\(\textbf{y}\) en\(V\) y c. en\(\mathbb{C}\).
- \((c_{1}+c_{2}) \textbf{x} = c_{1} \textbf{x}+c_{2} \textbf{x}\)para cada uno\(\textbf{x}\) en\(V\)\(c_{1}\) y\(c_{2}\) en\(\mathbb{C}\).