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b/program-4.tex @@ -0,0 +1,108 @@ +\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]{{\bf\hfill #1}} + +\begin{document} + +\begin{center} +{\Large Вопросы экзамена по алгебре} +\par\medskip +Группы 251, 253 (лектор Александр Лузгарев) +\par\medskip +Четвертый семестр, весна 2016 +\end{center} + +\glava{Эвклидовы и унитарные пространства} +\begin{compactenum} +\item Эвклидовы и унитарные пространства: определения и первые + примеры. +\item Норма и угол. +\item Матрица Грама. +\item Поведение матрицы Грама при замене базиса, ее обратимость. +\item Ортогонализация Грама--Шмидта. +\item Ортогональные и унитарные матрицы, равносильные определения. +\item Ортонормированные базисы. Теорема Риса. +\item Ортогональное дополнение, его свойства. +\item Ортогональная проекция. +\item Сопряженное отображение: существование и единственность. +\item Свойства сопряжения. Матрица сопряженного отображения. +\item Самосопряженные операторы. Критерий равенства нулю самосопряженного оператора. +\item Нормальные операторы, их собственные числа и векторы. +\item Спектральная теорема для нормальных операторов в унитарном пространстве. +\item Спектральная теорема для самосопряженных операторов в эвклидовом пространстве. +\item Спектральная теорема для нормальных операторов в эвклидовом пространстве. +\item Самосопряженные, кососимметрические, унитарные операторы в + унитарных и эвклидовых пространствах. +\item Изометрии. +\item Теорема Эйлера о вращениях, приведение квадратичной формы к + диагональному виду, разложение пространства в ортогональную прямую + сумму собственных подпространств. +\item Положительно определенные операторы. +\item Извлечение квадратного корня из положительно определенного + оператора. +\item Полярное разложение. +\suspend{compactenum} +\glava{Теория групп} +\resume{compactenum} +\item Группы: определение, примеры. +\item Подгруппы: определение, примеры. Подгруппы циклической группы. +\item Подгруппа, порожденная множеством: две конструкции +\item Классы смежности, разбиение на классы и соответствующие + отношения эквивалентности. +\item Нормальные подгруппы: определение и равносильные + переформулировки. +\item Гомоморфизмы групп: определение, примеры. +\item Ядро и образ гомоморфизма. +\item Ядро и инъективность, изоморфизм и биективность. +\item Конструкция фактор-группы. +\item Теорема о гомоморфизме. +\item Циклические группы: определение и классификация. Порядок + элемента. +\item Равномощность множеств левых и правых смежных классов. Теорема + Лагранжа. +\item Следствия теоремы Лагранжа. +\item Прямое произведение групп +\item Критерий разложения группы в прямое произведение подгрупп. +\item Разложение перестановки в произведение независимых циклов. +\item Описание классов сопряженности в симметрической группе. +\item Теорема Кэли. +\item Диэдральная группа. +\suspend{compactenum} +\glava{Полилинейная алгебра} +\resume{compactenum} +\item Тензорное произведение двух пространств. +\item Тензорное произведение нескольких пространств. Ассоциативность, + коммутативность. Тензорный базис. +\item Двойственное пространство. Изоморфизм пространства с дважды + двойственным. +\item Выражение $\Hom$ через $\otimes$. +\item Двойственность и $\otimes$. +\item Сопряженность $\otimes$ и $\Hom$. +\item Тензорное произведение линейных отображений. Его свойства, + тензорное произведение $\Hom$-пространств. +\item Кронекерово произведение матриц. Матрица тензорного + произведения. +\item Тензорные пространства. Координаты тензора. +\item Преобразование координат тензора при замене базиса. +\end{compactenum} + +\end{document} +