1.E: Ejercicios para el Capítulo 1
- Page ID
- 114915
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Ejercicios de cálculo
1. Resolver los siguientes sistemas de ecuaciones lineales y caracterizar sus conjuntos de soluciones. (Es decir, determinar si existe una solución única, no hay solución, etc.) Además, escriba cada sistema de ecuaciones lineales como una sola función\(f : \mathbb{R}^n \rightarrow \mathbb{R}^m\) para las elecciones apropiadas de\(m, n \in \mathbb{Z}_+ .\)
(a) Sistema de 3 ecuaciones en las incógnitas\(x, y, z, w:\)
\[ x + 2y − 2z + 3w = 2 \\ 0.2x + 4y − 3z + 4w = 5 \\ 5x + 10y − 8z + 11w = 12 \]
b) Sistema de 4 ecuaciones en las incógnitas\(x, y, z:\)
\[ x + 2y − 3z \\ x + 3y + z \\ 2x + 5y − 4z \\ 2x + 6y + 2z \]
c) Sistema de 3 ecuaciones en las incógnitas\(x, y, z:\)
2. Encuentra todos los pares de números reales\(x_1\) y\(x_2\) que satisfagan el sistema de ecuaciones
\[ x_1 + 3x_2 = 2, \;\;\; \tag{1.12} \]
\[ x_1 − x_2 = 1. \;\;\; \tag{1.13} \]
Ejercicios de prueba de escritura
1. Dejar\(a, b, c,\) y\(d\) ser números reales, y considerar el sistema de ecuaciones dado por
\[ ax_1 + bx_2 = 0,\;\;\; \tag{1.14} \]
\[ cx_1 + dx_2 = 0 \;\;\; \tag{1.15} \]
Tenga en cuenta que\(x_1 = x_2 = 0\) es una solución para cualquier elección de\(a, b, c,\) y\(d.\) Demostrar que si\(ad − bc = 0\), entonces\(x_1 = x_2 = 0\) es la única solución.