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zkB$C+ItY0GA1(s&Xd8kb9lW6Y7^u~>ba_B;@hlj<@GLA4`ad~Hvp`l~qCdKP1%7m8 zfl>1hG~J}(h-tDr$xaHAf|_5FS+2InE^eCiI#92juK=7Q;MD)8(Hx!h9h}@8e$B+h Q#>T=3NlYv(D+2jH022lvZ2$lO literal 0 HcmV?d00001 diff --git a/program-1.tex b/program-1.tex new file mode 100644 index 0000000..5321b41 --- /dev/null +++ b/program-1.tex @@ -0,0 +1,127 @@ +\documentclass[12pt]{article} +\usepackage[T2A]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[russian]{babel} +\usepackage{mdwlist} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{paralist} +\usepackage{ccfonts,eulervm,euler} +\renewcommand{\bfdefault}{sbc} +%\usepackage{ulem} +\usepackage[margin=0.7in,bmargin=0.7in]{geometry} + + +\pagestyle{empty} + +\DeclareMathOperator{\Hom}{Hom} + +\newcommand\glava[1]{{\medskip\bf\hfill #1}} + +\begin{document} + +\begin{center} +{\Large Вопросы экзамена по алгебре} +\par\medskip +Группы 151, 153 (лектор Александр Лузгарев) +\par\medskip +Первый семестр, осень 2016 +\end{center} + +\glava{Глава 3. Комплексные числа} +\begin{compactenum} +\item Комплексные числа: определение, алгебраическая форма записи. +\item Комплексное сопряжение и модуль. Деление комплексных чисел. +\item Неравенство треугольника. Тригонометрическая форма записи + комплексного числа. +\item Перемножение комплексных чисел в тригонометрической + форме. Формула Муавра. +\item Корни $n$-ой степени из комплексного числа. Свойства корней из + единицы. +\item Первообразные корни из единицы, их количество. +\item Экспоненциальная форма записи комплексного числа и логарифм. +\suspend{compactenum} +\glava{Глава 4. Кольцо многочленов} +\resume{compactenum} +\item Кольцо многочленов над кольцом. +\item Теорема о степени произведения многочленов + над областью целостности и ее следствия. +\item Делимость в кольце многочленов. Теорема о делении с остатком. +\item Многочлен как функция. Лемма Безу. +\item Выделение линейных множителей и число различных корней + многочлена над полем. Формальное и функциональное равенство + многочленов. +\item Алгебраическая замкнутость. Разложение многочленов над полями + комплексных и вещественных чисел. +\item Определение и свойства производной. Связь между корнями + многочлена и его производной. +\item Характеристика поля. Поведение кратности корня при взятии + производной над полем характеристики $0$. +\item Интерполяционная задача, единственность ее решения. + Интерполяционные формулы Лагранжа и Ньютона. +\item Наибольший общий делитель многочленов: существование и линейное + представление. +\item Алгорифм Эвклида для многочленов. Оценка на степень + коэффициентов в линейном представлении НОД. +\item Неприводимые многочлены. Основная теорема арифметики в кольце + многочленов. +\item Конструкция поля частных области целостности: эквивалентность + дробей, введение операций и проверка аксиом поля. +\item Поле рациональных функций. Правильные дроби, их + свойства. Выделение многочлена из дроби. +\item Простейшие дроби: две леммы о разложении знаменателя. +\item Теорема о представлении правильной дроби в виде суммы + простейших. +\item Простейшие дроби над полями комплексных и вещественных + чисел. Нахождение коэффициентов в случае простых корней. +\suspend{compactenum} +\glava{Глава 5. Вычислительная линейная алгебра} +\resume{compactenum} +\item Системы линейных уравнений и матрицы. Элементарные + преобразования и связь с множеством решений. +\item Метод Гаусса решения систем линейных уравнений. +\item Матрицы, свойства сложения и умножения. +\item Транспонирование и его свойства. Матричные единицы. +\item Матрицы элементарных преобразований. Элементарные преобразования + строк и столбцов как умножения на матрицы. +\item Приведение матрицы к окаймленному единичному виду элементарными + преобразованиями. Связь с обратимостью. +\item Блочные матрицы и операции над ними. +\item Группа перестановок. Табличная запись перестановки. +\item Разложение перестановки в произведение [элементарных] + транспозиций. +\item Число инверсий и знак перестановки. Изменение знака при + домножении на транспозицию. +\item Знак как число транспозиций в разложении. Мультипликативность + знака. +\suspend{compactenum} +\glava{Глава 6. Векторные пространства} +\resume{compactenum} + \item Определение векторного пространства, простейшие свойства, + примеры. +\item Подпространства: определение и примеры. Пересечение + и сумма подпространств. +\item Прямая сумма подпространств. Критерии разложения пространства + в прямую сумму подпространств. +\item Системы образующих и линейно независимые системы. Примеры. +\item Лемма о линейной зависимости. Следствие о добавлении вектора +к линейно независимой системе. +\item Количество элементов в линейно независимой и порождающей +системах. +\item Конечномерность подпространства конечномерного пространства. +\item Базис, определение и эквивалентная переформулировка. +Любая система образующих содержит базис. +\item Любая линейно независимая система содержится в базисе. +Дополнение пространства до прямой суммы. +\item Равномощность всех базисов. Размерность подпространства. +\item Теорема Грассмана о размерности суммы подпространств. +\item Если длина порождающей системы равна размерности, то это +базис; то же для линейно независимой системы. +Размерность прямой суммы подпространств. Критерий разложения +в прямую сумму, использующий размерность. +\end{compactenum} + +\end{document} +