From dcb83acf211153715876c61d7662029306af6678 Mon Sep 17 00:00:00 2001 From: Semyon Savkin Date: Tue, 6 Apr 2021 19:23:38 +0300 Subject: [PATCH 1/3] Fix typos --- Lectures/lecture24.tex | 4 ++-- Lectures/lecture25.tex | 2 +- main.pdf | Bin 1569510 -> 1275499 bytes 3 files changed, 3 insertions(+), 3 deletions(-) diff --git a/Lectures/lecture24.tex b/Lectures/lecture24.tex index 032c09d..52871e1 100644 --- a/Lectures/lecture24.tex +++ b/Lectures/lecture24.tex @@ -259,7 +259,7 @@ \section{Билинейные формы на одном пространств Аналогичная ситуация обстоит и с билинейными формами. В случае, когда билинейная форма живет на одном пространстве у нас на много больше характеристик и поведение ее изучать несколько сложнее. Ниже я буду рассказывать о билинейных формах на одном пространстве. -Окажется, что от части ситуация с билинейными формами технически сильно проще, чем случай линейных операторов. +Окажется, что отчасти ситуация с билинейными формами технически сильно проще, чем случай линейных операторов. \subsection{Симметричность и кососимметричность} @@ -348,7 +348,7 @@ \subsection{Матричные характеристики билинейной Действительно, $\det(B') = \det(B) c^2$. В поле $\mathbb R$ число равно квадрату тогда и только тогда, когда оно положительно. -\item Пусть $F = \mathbb Q$. +\item Пусть $F = \mathbb Z$. Тут ничего особенно сказать нельзя, кроме явной формулировки. Пусть $\det(B) = p_1^{k_1}\ldots p_r^{k_r}$, где $p_i\in \mathbb N$ -- простые числа, а $k_i\in \mathbb Z$ -- степени (вообще говоря положительные или отрицательные). Тогда четности чисел $k_i$ определены однозначно независимо от базиса. diff --git a/Lectures/lecture25.tex b/Lectures/lecture25.tex index 92d2ba2..65bd25e 100644 --- a/Lectures/lecture25.tex +++ b/Lectures/lecture25.tex @@ -348,7 +348,7 @@ \subsection{Метод Якоби} \[ \beta(e_{k+1}', e_i') = \beta(e_{k+1}, e_i') - \frac{\beta(e_{k+1}, e_1')}{\beta(e_1',e_1')}\beta(e_1', e_i') - \ldots - \frac{\beta(e_{k+1}, e_{k}')}{\beta(e_{k}',e_{k}')}\beta(e_{k}', e_i') \] -Так как все построенные вектры $e_1',\ldots,e_k'$ были ортогональны, то справа выживает лишь одно слагаемое, то есть +Так как все построенные векторы $e_1',\ldots,e_k'$ были ортогональны, то справа выживает лишь одно слагаемое, то есть \[ \beta(e_{k+1}', e_i') = \beta(e_{k+1}, e_i') - \frac{\beta(e_{k+1}, e_i')}{\beta(e_i',e_i')}\beta(e_i', e_i') = 0 \] diff --git a/main.pdf b/main.pdf index 3379c91a0cc06ff5bf7750fbfc9c3f2b630985c2..d04876c623e511a19e2da22f2be7f8be6628f983 100644 GIT binary patch delta 546343 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--git a/Lectures/lecture24.tex b/Lectures/lecture24.tex index 52871e1..d92e472 100644 --- a/Lectures/lecture24.tex +++ b/Lectures/lecture24.tex @@ -348,7 +348,7 @@ \subsection{Матричные характеристики билинейной Действительно, $\det(B') = \det(B) c^2$. В поле $\mathbb R$ число равно квадрату тогда и только тогда, когда оно положительно. -\item Пусть $F = \mathbb Z$. +\item Пусть $F = \mathbb Q$. Тут ничего особенно сказать нельзя, кроме явной формулировки. Пусть $\det(B) = p_1^{k_1}\ldots p_r^{k_r}$, где $p_i\in \mathbb N$ -- простые числа, а $k_i\in \mathbb Z$ -- степени (вообще говоря положительные или отрицательные). Тогда четности чисел $k_i$ определены однозначно независимо от базиса. From d9ae4bbce111db8dc138d8e3bd39aa2aa894e3fb Mon Sep 17 00:00:00 2001 From: Semyon Savkin Date: Tue, 27 Apr 2021 23:32:26 +0300 Subject: [PATCH 3/3] Fix a typo --- Lectures/lecture27.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Lectures/lecture27.tex b/Lectures/lecture27.tex index 6acd04d..2471b18 100644 --- a/Lectures/lecture27.tex +++ b/Lectures/lecture27.tex @@ -364,7 +364,7 @@ \subsection{Классификация Евклидовых пространст \item $\phi$ -- изоморфизм векторных пространств. \item Для любых $v,u\in V$ выполнено $(v, u) = (\phi(v), \phi(u))$.% -\footnote{Здесь слева скалярное произведение в пространстве $V$, а с права в пространстве $U$.} +\footnote{Здесь слева скалярное произведение в пространстве $V$, а справа в пространстве $U$.} \end{enumerate} При наличии изоморфизма между евклидовыми пространствами $V$ и $U$ они называются изоморфными. \end{definition}