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P
lndice general
Capítulo 1
CONCEPTOS PRELIMINARES
1.
2.3.
4.
5.
6.
7.
8.
9.
1o.
11 .
12.
13.
Conjuntos [13
Subconjuntos [15
Operaciones con conjuntos [16
Producto cartesiano 118
Relaciones 120
Funciones [21
Composición de funciones 122
Funciones inyectivas, suprayectivas y
biyectivas [24
Cardinalidad Y conjuntos finitos [27
Inducción matemática [29
El teorema del binomio [31
Relaciones de equivalencia y particiones
Estructuras numéricas 136[33
Capítulo 2
CALCULO COMBINATORIO
1 . Ejemplos ilustrativos 139
2. Funciones [47
3. Funciones inyectivas, suprayectivas y
biyectivas [54
4. Ordenaciones, permutacioncs y combinaciones
5. Problemas [64
[57
13
39
7
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a (NDICE GENERAL
capitulo 3
ESPACIOS VECTORIALES
1. E1 espacio vectorial R1 [73
2. El espacio vectorial Rn [80
3. Subespacios vectoriales 1824. Combinaciones lineales. Dependencia e
independencia lineal 184
5. Bases de subespacios vectoriales. Dimensión r89
Capitulo 4MATRICES Y DETERMINANTES
1.
2.
3.
4.
5.
6.
7.8.
Matrices [97
El rango de una matriz
Permutaciones [lo8
Determinantes [ 13
Propiedades básicas de los determinantes [117
Más propiedades de los determinantes [123
Cálculo de determinantes
Caracterización del rango de una matriz mediante
determinantes [133
[lo1
[13
Capitulo 5
SISTEMAS DE ECUACIONES LINEALES
1. Definiciones [137
2. Existencia de soluciones [140
3. Sistemas de n ecuaciones con n incógnitas [144
4. Sistemas homogéneos [148
5. Sistema homogéneo asociado 1152
6. Resolución de sistemas [154
Capítulo 6
EL ANILLO DELOS
NÚMEROS ENTEROS
73
97
137
163
1. Propiedades básicas de las operaciones en Z 1163
2. Anillos [164
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"DICE GENERAL 9
3. Propiedades de anillos de los enteros [1674. Dominios enteros [170
5. El orden en Z E171
6. Unidades en 2 [173
7. El principio de inducción E1748. El principio de buen orden [177
Capítulo 7
DlVlSl5lLlDAD
1. Definiciones y propiedades elementales [179
2. El algoritmo de la división [184
3. El máximo común divisor [187
4. El algoritmo de Euclides y ecuaciones diofantinas [193
5. Factorización única [198
6. Congruencias 1202
Capitulo 8
LOS N Ú M E R O S R E A L E S
1. Los números racionales, [209
2. El conjunto R de los reales. Orden en R [217
3. Cotas y fronteras [219
4. Suma y producto de reales [222
5. Propiedades de la suma, el producto y el
orden en R [224
6. Racionales y reales [233
7. Raíces de reales positivos. Exponentes
fraccionarios [2388. Valor absoluto [241
9. Aproximación [242
Capítulo 9
EL CAMPO DE LOS N Ú M E R O S COMPLEJOS
179
209
245
1. Módulo y argumento de vectores de R2 [245
2. Los números complejos [2533. Propiedades de las operaciones [259
4. Raíz cuadrada [266
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10 INDlCE GENERAL
5. Raíces n-ésimas de números complejos [271
6. El campo de los números complejos [273
Capítulo 10
POLINOMIOS Y TEORCA DE ECUACIONES
1.
2.
3.
4.
5.
6.
7.
8.
9.
1o.
11.
12.
13.
14.
15.
16.
Polinomios [27 7
Los polinomios como funciones [279
Suma y producto de polinomios [280División con residuo [283
Raíces de polinomios. Teorema del residuo.
Todo polinomio de grado positivo tieneraíces [286
Ecuaciones de segundo grado [288
División sintética. Expresión de un polinomio
en la forma x a i ( ñ - a ) i E290
Cálculo de una raíz aislada en un intervalo en
cuyos extremos el polinomio tiene signos
contrarios C293
Factorización de un polinomio. Raíces
múltiples [297Derivadas y multiplicidad 1300
Coeficientes y raíces [303
Polinomios con coeficientes reales E304
El algoritmo de Euclides con polinomios 1306Aislamiento de las raíces reales de un polinomio
con coeficientes reales (teorema de Strum)
Fracciones racionales. Descomposición en
fracciones parciales [312
Ecuaciones de tercero y cuarto grados con
coeficientes reales [3 18
[308
277
fndice analítico E32 1
índice de símbolos [323
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