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Предположим, что $B(T(v),v) = 0$ для всех $v\in V$. Тогда $T = 0$. \end{lemma} @@ -1198,7 +1198,7 @@ $$ \begin{remark} Замечание~\ref{rem:complex-unitary-counterexample} показывает, -что на эвклидовом пространстве оператор $T$ может удовлетворять +что на эвклидовом пространстве ненулевой оператор $T$ может удовлетворять тождеству $B(T(v),v)=0$ для всех $v\in V$. Однако, этого не может случиться для самосопряженного оператора. \end{remark} @@ -1257,7 +1257,7 @@ $||T(v)|| = ||T^*(v)||$ для всех $v\in V$. \begin{proof} Заметим, что оператор $T^*\circ T - T\circ T^*$ самосопряжен. По лемме~\ref{lem:selfadjoint-zero-characterisation} -равенство $T^*\circ T - T\circ T^*$ равносильно тому, +равенство $T^*\circ T - T\circ T^*$ нулю равносильно тому, что $B((T^*\circ T - T\circ T^*)(v),v) = 0$ для всех $v\in V$, что равносильно равенству $B(T^*(T(v)),v) = B(T(T^*(v)),v)$ для всех $v\in V$. @@ -1292,7 +1292,7 @@ $T$, соответствующие различным собственным ч По предложению~\ref{prop:normal-operator-adjoint-eigenvalues} теперь $T^*(u) = u\ol\lambda$. Поэтому $(\lambda-\mu)B(u,v) = B(u\ol\lambda,v) - B(u,v\mu) -= B(T(u),v) - B(u,T^*(v)) = 0$. += B(T^*(u),v) - B(u,T(v)) = 0$. Поскольку $\lambda\neq\mu$, из этого равенства следует, что $B(u,v)=0$, что и требовалось. \end{proof} @@ -1318,7 +1318,7 @@ $V$ диагональна. \begin{proof} Очевидно, что $(2)\Leftrightarrow(3)$ (см. также доказательство теоремы~\ref{thm:diagonalizable-equivalent}). -Покажем, что из (3) следует (1). Пусть матрица $t$ в некотором +Покажем, что из (3) следует (1). Пусть матрица $T$ в некотором ортонормированном базисе $\mc B$ диагональна. По предложению~\ref{prop:adjoint_matrix} матрица $T^*$ тогда получается из матрицы $T$ транспонированием @@ -1505,7 +1505,7 @@ $V$ диагональна. диагональна. Но диагональная матрица совпадает со своей транспонированной, поэтому $T=T^*$, откуда следует $(1)$. -Теперь мы докажем. что из $(1)$ следует $(2)$ индукцией по размености +Теперь мы докажем, что из $(1)$ следует $(2)$ индукцией по размерности пространства $V$. Если $\dim(V)=1$, утверждение очевидно. Пусть теперь $\dim(V) > 1$, и оператора $T$ самосопряжен. @@ -1518,7 +1518,7 @@ $V$ диагональна. и оператор $T|_{U^\perp}$ самосопряжен. По предположению индукции у $U^\perp$ есть ортонормальный базис, состоящий из собственных векторов оператора $T|_{U^\perp}$. -Присоединив к нему $u$, получаем ортонормальный базис $U^\perp$, +Присоединив к нему $u$, получаем ортонормальный базис $V$, состоящий из собственных векторов оператора $T$. \end{proof} @@ -1673,6 +1673,7 @@ $(2)\Rightarrow (1)$: несложно проверить, что матрица \literature{[F], гл. XIII, \S~5; [K2], гл. 3, \S~3, пп. 3, 6; [KM], ч. 2, \S~7, пп. 1--2, 4; \S~8, пп. 2--6.} +\nopagebreak Сейчас мы применим знания, полученные при изучении нормальных операторов, к некоторым частным случаям. 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Пусть $e_1,e_2$~--- ортонормальный базис пространства $V$, и пусть матрица $T$ в этом базисе имеет вид $$ -\begin{pmatrix}a & c\\b & d\end{pmatrix}. +\begin{pmatrix}\alpha & \gamma\\\beta & \delta\end{pmatrix}. $$ -Тогда $||T(e_1)||^2 = a^2 + b^2$, $||T^*(e_1)||^2 = a^2 + c^2$. +Тогда $||T(e_1)||^2 = \alpha^2 + \beta^2$, $||T^*(e_1)||^2 = \alpha^2 + \gamma^2$. По предложению~\ref{prop:normal-operator-equiv} эти числа равны, -откуда $c = \pm b$. Если $c=b$, то $T$ самосопряжен (его матрица -симметричны), поэтому $c = -b$, при этом $b\neq 0$. +откуда $\gamma = \pm \beta$. Если $\gamma=\beta$, то $T$ самосопряжен (его матрица +симметрична), поэтому $\gamma = -\beta$, при этом $\beta\neq 0$. Перемножим теперь матрицы $T$ и $T^*= T^T$ в одном и в другом порядке. Результаты должны -совпасть, но в правом верхнем углу у одной матрицы стоит $bd$, а у -другой $ab$. Значит, $a=d$, и мы получили матрицу нужного вида. +совпасть, но в правом верхнем углу у одной матрицы стоит $\beta\delta$, а у +другой $\alpha\beta$. Значит, $\alpha=\delta$, и мы получили матрицу нужного вида. -$(2)\Rightarrow (3)$. Если в нашем базисе уже $b>0$, то все доказано, +$(2)\Rightarrow (3)$. Если в нашем базисе уже $\beta>0$, то все доказано, а если нет~--- поменяем знак у второго базисного вектора. $(3)\Rightarrow (1)$. Если $T$ имеет указанный вид, то видно, что $T$ @@ -1626,9 +1626,9 @@ $T|_{U^\perp}$ нормален. матрица оператора $T$ блочно-диагональна, причем каждый блок имеет либо размер $1\times 1$, либо размер $2\times 2$ и вид $$ -\begin{pmatrix} a & -b \\ b & a\end{pmatrix}, +\begin{pmatrix} \alpha & -\beta \\ \beta & \alpha\end{pmatrix}, $$ -где $b > 0$. +где $\beta > 0$. \end{enumerate} \end{theorem} \begin{proof} @@ -1717,7 +1717,7 @@ $V$, в котором матрица оператора $a$ диагональ \end{enumerate} \end{theorem} \begin{proof} -Если оператор самосопряженный, кососимметрический, нормальный, то по +Если оператор самосопряженный, кососимметрический, унитарный, то по теореме~\ref{thm:spectral-unitary} существует базис, в котором его матрица диагональна. Если он самосопряжен, то каждый диагональный блок $1\times 1$ самосопряжен, поэтому в нем стоит комплексное число @@ -1740,14 +1740,14 @@ $V$, в котором матрица оператора $a$ диагональ \item Оператор $a$ является кососимметрическим тогда и только тогда, когда существует ортонормированный базис пространства $V$, в котором матрица оператора $a$ имеет блочно-диагональный -вид, и каждый блок выглядит как $(0)$ или $\begin{pmatrix} 0 & -b - \\ b & 0\end{pmatrix}$ для $b\in\mb R$, $\beta > 0$. +вид, и каждый блок выглядит как $(0)$ или $\begin{pmatrix} 0 & -\beta + \\ \beta & 0\end{pmatrix}$ для $\beta\in\mb R$, $\beta > 0$. \item Оператор $a$ является ортогональным тогда и только тогда, когда существует ортонормированный базис пространства $V$, в котором матрица оператора $a$ имеет блочно-диагональный вид, и каждый блок выглядит как $(1)$, $(-1)$ -или $\begin{pmatrix}a&-b\\ b & a\end{pmatrix}$ для -$a,b\in\mb R$, $b > 0$, $a^2 + b^2 = 1$. +или $\begin{pmatrix}\alpha&-\beta\\ \beta & \alpha\end{pmatrix}$ для +$\alpha,\beta\in\mb R$, $\beta > 0$, $\alpha^2 + \beta^2 = 1$. \end{enumerate} \end{theorem} \begin{proof} @@ -1756,18 +1756,18 @@ $a,b\in\mb R$, $b > 0$, $a^2 + b^2 = 1$. матрица блочно-диагональна, с блоками вида $$ \begin{pmatrix} -a & -b\\ -b & a +\alpha & -\beta\\ +\beta & \alpha \end{pmatrix}, $$ где $b>0$. Если он самосопряжен, то каждый диагональный блок самосопряжен, что -для блока $2\times 2$ указанного вида означает, что $b=-b$, +для блока $2\times 2$ указанного вида означает, что $\beta=-\beta$, что невозможно. Поэтому остаются только блоки размера $1\times 1$, что означает диагональность матрицы. Аналогично, из кососимметричности -для блока $2\times 2$ следует, что $a=0$, а для блока $(\lambda)$ +для блока $2\times 2$ следует, что $\alpha=0$, а для блока $(\lambda)$ размера $1\times 1$~--- что $\lambda = 0$. Наконец, из ортогональности -для блока $2\times 2$ следует, что $s^2+b^2=1$, а для блока +для блока $2\times 2$ следует, что $\alpha^2+\beta^2=1$, а для блока $(\lambda)$~--- что $\lambda^2=1$, откуда следует, что $\lambda=\pm 1$. Обратно, если матрица оператора состоит из блоков указанного вида, @@ -1820,7 +1820,7 @@ $||a(v)|| = ||v||$ для всех $v\in V$. \lambda B(u,v) + \ol\lambda\lambda B(v,v). \end{align*} Воспользуемся равенствами $B(a(x),a(x)) = B(x,x)$ и $B(x,y) = - \ol{B(x,y)}$: + \ol{B(y,x)}$: $$ \lambda B(a(u),a(v)) + \ol{\lambda B(a(u),a(v))} = \lambda B(u,v) + \ol{\lambda B(u,v)}. @@ -1838,7 +1838,7 @@ $||a(v)|| = ||v||$ для всех $v\in V$. Пусть $V = \mb R^3$~--- трехмерное вещественное пространство со стандартным эвклидовым скалярным произведением, $a\colon\mb R^3\to\mb R^3$~--- изометрия на $\mb R^3$. Тогда в некотором -ортогональном базисе матрица оператора $a$ имеет вид +ортонормированном базисе матрица оператора $a$ имеет вид $$ \begin{pmatrix} \pm 1 & 0 & 0\\ @@ -1852,7 +1852,7 @@ $$ \end{corollary} \begin{proof} По лемме~\ref{lem:isometry_equiv} оператор $a$ ортогонален. По -теореме~\ref{thm:euclidean_canonical_forms} найдется ортогональный +теореме~\ref{thm:euclidean_canonical_forms} найдется ортонормированный базис $V$, в котором матрица оператора $a$ имеет блочно-диагональный вид, и блоки имеют вид $(\pm 1)$ или $\begin{pmatrix}\cos(\ph)&\sin(\ph)\\-\sin(\ph)&\cos(\ph)\end{pmatrix}$. Если @@ -1866,17 +1866,17 @@ $\begin{pmatrix}\cos(\ph)&\sin(\ph)\\-\sin(\ph)&\cos(\ph)\end{pmatrix}$ \begin{corollary}[Приведение вещественной квадратичной формы к диагональному виду при помощи ортогонального преобразования] Пусть $(V,B)$~--- эвклидово пространство, и пусть -$q\colon V\times V\to B$~--- симметрическая билинейная +$q\colon V\times V\to \mb R$~--- симметрическая билинейная форма. Существует ортогональный базис пространства $V$, в котором матрица Грама формы $q$ имеет диагональный вид. \end{corollary} \begin{proof} Выберем некоторый ортонормированный базис $\mc B$ пространства $V$; пусть $Q$~--- матрица Грама формы $q$ в этом базисе. -Поскольку форма $q$ симметрична, матрица $Q$ является симметричной +Поскольку форма $q$ симметрична, матрица $Q$ является симметрической матрицей: $Q^T = Q$. Рассмотрим $Q$ как матрицу некоторого оператора $a$ на пространстве $V$; по предложению~\ref{prop:adjoint_matrix} -оператор $q$ самосопряжен. +оператор $a$ самосопряжен. По теореме~\ref{thm:euclidean_canonical_forms} существует ортонормированный базис $\mc C$ пространства $V$, в котором матрица оператора $a$ диагональна. Это означает, что @@ -1932,6 +1932,7 @@ $\lambda_1,\dots,\lambda_m$, и кратность $\lambda_i$ равна раз \subsection{Положительно определенные операторы} \literature{[F], гл. XIII, \S~4, п. 4; [K2], гл. 3, \S~3, пп. 8, 9.} +\nopagebreak Пусть $(V,B)$~--- эвклидово или унитарное пространство, $a\colon V\to V$~--- самосопряженный оператор на нем. @@ -1984,11 +1985,11 @@ $\lambda_1,\dots,\lambda_n$ на диагонали. По теоремам~\ref{thm:unitary_canonical_forms} и~\ref{thm:euclidean_canonical_forms} мы уже знаем, что $a$ самосопряжен. Разложим произвольный вектор $v$ по базису $\mc B$: -$v = \sum_i c_i e_i$. -Тогда $a(v) = \sum_i c_i a(e_i) = \sum_i c_i\lambda_i e_i$. +$v = \sum_i e_i c_i$. +Тогда $a(v) = \sum_i a(e_i) c_i = \sum_i e_i c_i\lambda_i$. Поэтому $$ -B(a(v),v) = B(\sum_i c_i\lambda_i e_i,\sum_j c_i e_j) +B(a(v),v) = B(\sum_i e_i c_i\lambda_i,\sum_j e_j c_i) = \sum_{i,j}\overline{c_i}\lambda_i c_j B(e_i,e_j) = \sum_i\lambda_i \overline{c_i}c_i B(e_i,e_i) = \sum_i\lambda_i |c_i|^2 \geq 0. @@ -2120,7 +2121,7 @@ p^{-1}p^2 p^{-1} = \id$, что и требовалось. Наконец, если $pu = a = p'u'$, то $(pu)^* = (p'u')^*$, откуда $u^* p = (u')^*p'$. Из этого следует, что -$(pu)(u^*p) = (p'u')((u')^*p^*)$, откуда $p^2 = (p')^2$, и в силу +$(pu)(u^*p) = (p'u')((u')^*p')$, откуда $p^2 = (p')^2$, и в силу единственности извлечения квадратного корня (теорема~\ref{thm:square_root_positive}), получаем, что $p=p'$, и, стало быть, $u=u'$. 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Множество левых смежных классов $G$ по $H$ по модулю подпространства (см. определение~\ref{def:quotient_space}); однако, отсутствие коммутативности приводит к тому, что необходимо рассматривать два варианта обобщения: условие $v_1-v_2\in U$ из -определения~\ref{def:quotient_space} мы заменяем на $v_1v_2^{-1}$ в -одном варианте и на $v_2^{-1}$ в другом. Если группа $G$ абелева, то +определения~\ref{def:quotient_space} мы заменяем на $v_1v_2^{-1}\in U$ в +одном варианте и на $v_2^{-1}v_1\in U$ в другом. Если группа $G$ абелева, то $gH = Hg$ для всех $g\in G$, и отношения $\sim_H$, ${}_H{\sim}$ совпадают. \end{remark} @@ -481,6 +481,7 @@ $ghg^{-1} = {}^gh$. абелевой группы нормальны. \end{remark} +\hspace{0em} \begin{examples}\label{examples:normal_subgroups} \hspace{1em} \begin{enumerate} @@ -573,7 +574,7 @@ $\ph(x)^{-1} = \ph(x^{-1})$. \begin{definition} Пусть $\ph\colon G\to H$~--- гомоморфизм групп. \dfn{Ядром} гомоморфизма $\ph$ называется множество $\Ker(\ph)=\{x\in G\mid -f\ph(x) = e_H\}$ (полный прообраз единицы). \dfn{Образом} гомоморфизма +\ph(x) = e_H\}$ (полный прообраз единицы). \dfn{Образом} гомоморфизма $\ph$ называется его теоретико-множественный образ: $\Img(\ph) = \{y\in H\mid y = \ph(x)\text{ для некоторого }x\in G\}$. \end{definition} @@ -621,14 +622,14 @@ $\ph$ сюръективно тогда и только тогда, когда $ \begin{lemma}\label{lem:injective_homo} Пусть $\ph\colon G\to H$~--- гомоморфизм групп. Он инъективен тогда и -только тогда, когда $\Ker(\ph) = e$. +только тогда, когда $\Ker(\ph) = \{e\}$. \end{lemma} \begin{proof} Если $\ph$ инъективен, то есть только один элемент $g\in G$ такой, что $\ph(g) =e$, и мы знаем, что $\ph(e)=e$. -Обратно, если $\Ker(\ph)=e$ и $g,g'\in G$ таковы, что +Обратно, если $\Ker(\ph)=\{e\}$ и $g,g'\in G$ таковы, что $\ph(g)=\ph(g')$, то $\ph(g^{-1}g') = \ph(g)^{-1}\ph(g') = e$, поэтому -$g^{-1}g'\in\Ker(\ph)=e$, откуда $g = g'$. +$g^{-1}g'\in\Ker(\ph)=\{e\}$, откуда $g = g'$. \end{proof} \begin{definition} diff --git a/program-4.pdf b/program-4.pdf index 1b581686bce9a18ad19cc30da6800c0cb0973ad5..0032a1e267ee18e34b61610d9024004cceb9e278 100644 GIT binary patch delta 658 zcmdn@oN4cKrU|Ny7Lx@xYi#rjWUbfS;pI>$^sr>f?MX$ijpMVqc(_dC!_yg?`={w)-oKZ}!b+ zp_%SaY@AOy-Z0y|A*tC}pgEOY($cu(``n7DUI8r^n64<-UdUZ(5W~Dn?&vJ1?1?J% zzZd2CP87P>lPdE5YF_r#%Z)FY9m1?uM6oV!=8@WeuhvpDDt>3=>^R#!Wx)%AP4l!p z@2W}f{`+EEt9j!NEw*)~=F_MC+T6fYbo201ZPz<&+v>Z8(idu~v&VO8hVC!-^U*S& zz4x|ZN7Tg4ug*s|-Lui68JI`@&{R(?H?jrA7oi}JR!h(7vOm3DgNsa>(A9KKWqC|x>7SC=7m*QtZ=t>Vx!wE{^Kiuv^w01 z+kPg9RZDiGsZy4sxCiS_#*eS!_GDjvv3mQuiPxVn`TlJe-xIfWa<57Ju2Zc#B5%)l zf8F~h!s3Xp*~1HNzrUR~NnV~(9L=9*mN)Hf`_wcsj;}2q(P!p{s~i7~cf2Eg;rO{Q z`Ou2}e}8>^D?d@NJ9x4qukhwH_NK+`mIj6<=Eeq-->;H^u{R5?UdQfaV(DyQYGPq% zY3X9@>S$!>WNcw>Xy{~WZeZ@{YGG=pU}H)^$z-O@PC<9KC~`Y;IPnB+5NYJlVQHAl zvU2~9=YJUwh$}8`;0TC*`D2dRcOma1H5+ZGDsbNA`o*~>eRujKZ^hmzK^r6AKDUSq iJD4rU|NymXifGYi#rjWUXKMNchR4T>&pv%U#vU-fsWdbUDMr%?vB`Opm9k z`Y+s}ce-q=sd>Dn)9mcxT#K^#0{5bSavyr@r084zX6J5?HM{aNmIbbwbz$qPwd$Es zr_J7bSt~u44&pnsZDp&dv`mwvj`gV%a~JOZeq^;oRLcdXD~oF{yj^J!!@f)I=q#`7 zjrA(O7v%*`6uQ{+ROJ2Dyxh5~mifqYY`Wz5Rm(y)<G;FTXC>FZ z;-(a)R$+OnLCzAMi&V4ul!r!Sl@NjQlxaZ>I#AF*Nvn7Lbpo$ z>G-(6k}Ft~HNRj=S?%%P#+@g3Usm^6S6}+gI`>`X{}bMdkDadXd^Ia_P8X|GrjyL_ zN8(JG)o+cwMOTy@xi(=%YpjCk6SX6&e}7yTyLkPBU2SobxlO*x$9<`~8fJE><)Yk4 zv97XbOMlhX>Bu}XUsjN@xc2w*SsRl#?cB<4ENy=J+hH$b?v|>cCA!bft)4vd-+P63 zf-fGYukP2{bHDEIk8k{jCwm1p2eD6G%x-LGZfI$2Jo()!85n!B;OcekPHwKo7S7Iw z#x9o522N(?&IZoTj^+mDuI6Trj+TxVb_zD81e8o>-0T$OyIqmnlf#K;Zqr;Imjp$I zGNC2^9?kpC&Jf?!#H8f#HuXXAO&c#Q&eIvydX7fBcyq)zM8A#m(41Paa_PeRZ=PG+ jzc%f6;@gx*wN3d)_{FujL?=((tiWN(rK;-c@5TiH?dT~O diff --git a/program-4.tex b/program-4.tex index 23ad4d8..40f227e 100644 --- a/program-4.tex +++ b/program-4.tex @@ -63,7 +63,7 @@ \glava{Теория групп} \resume{compactenum} \item Группы: определение, примеры. -\item Подгруппы: определение, примеры. Подгруппы циклической группы. +\item Подгруппы: определение, примеры. Подгруппы аддитивной группы. \item Подгруппа, порожденная множеством: две конструкции \item Классы смежности, разбиение на классы и соответствующие отношения эквивалентности. 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Действ $gh=gh'$, то $h=h'$, и в силу определения смежного класса это отображение сюръективно. Поэтому в каждом смежном классе столько же элементов, сколько в подгруппе $H$. Таким образом, элементы $G$ -разбиваются на $|G:H|$ смежных классов, в каждом по $H$ +разбиваются на $|G:H|$ смежных классов, в каждом по $|H|$ элементов. Отсюда сразу следует требуемое равенство. \end{proof} \begin{corollary}\label{cor:order_divides} @@ -911,7 +911,7 @@ $gh=gh'$, то $h=h'$, и в силу определения смежного к \end{proof} \begin{corollary}\label{cor:power_order} -Пусть $G$~--- конечная группа. Тогда $g^{|G|} = 1$ для любого $g\in G$. +Пусть $G$~--- конечная группа. Тогда $g^{|G|} = e$ для любого $g\in G$. \end{corollary} В качестве примера приложения теоремы Лагранжа выведем из нее теорему @@ -989,7 +989,7 @@ i_2\colon H\to G\times H,&\;\; h\mapsto (e,h),\\ $\Img(i_2)=\Ker(\pi_1)=\{e\}\times H$~--- нормальные подгруппы в $G\times H$; \item $\pi_1\circ i_1 = \id_G$, $\pi_2\circ i_2 = \id_H$; - $\pi_1\circ i_2 = 0$, $\pi_2\circ i_1 = 0$; + $\pi_1\circ i_2 = e$, $\pi_2\circ i_1 = e$; \end{enumerate} \end{proposition} \begin{proof} @@ -1052,7 +1052,7 @@ $g'^{-1}g = e = h'h^{-1}$, откуда $g=g'$ и $h=h'$. Проверим, что $\ph$~--- гомоморфизм групп. Возьмем $y\in F$ и запишем его в виде $y = g'h'$, где $g',h'\in H$. Тогда $xy = (gh)(g'h') = g(hg')h' = (gg')(hh')$ (по -свойству~\ref{item:they_commute}. По определению $\ph$ теперь +свойству~\ref{item:they_commute}). По определению $\ph$ теперь $\ph(xy) = (gg',hh')$, в то время как $\ph(x) = (g,h)$, $\ph(y) = (g',h')$, и, стало быть, $\ph(x)\ph(y) = (g,h)(g',h') = (gg', hh')$. @@ -1118,7 +1118,7 @@ $(i_1\;\;i_2\;\;\dots\;\;i_{k-1}\;\;i_k) = \end{theorem} \begin{proof} Будем вести индукцию по числу $i\in\{1,\dots,n\}$ таких, что -$\pi(i)\neq i$, то есть, по $n-\Fix(\pi)$. +$\pi(i)\neq i$, то есть, по $n-|\Fix(\pi)|$. Если это число равно $0$, то перестановка $\pi$ тождественна и, таким образом, есть произведение пустого множества циклов. Это база индукции. Докажем переход. @@ -1146,7 +1146,7 @@ $j\in\{1,\dots,n\}\setminus\{i_1,\dots,i_k\}$. Это значит, что к $\pi'$ можно применить предположение индукции: действительно, $\Fix(\pi') = \Fix(\pi)\cup\{i_1,\dots,i_k\}$, поэтому -мощность множества $\{i\in\{1,\dots,n\}\mid \pi'(i)\neq i$ на $k$ +мощность множества $\{i\in\{1,\dots,n\}\mid \pi'(i)\neq i\}$ на $k$ меньше, чем мощность аналогичного множества для $\pi$. По предположению индукции $\pi'$ можно записать в виде произведения независимых циклов, носители которых не пересекаются с $\Fix(\pi')$: @@ -1314,6 +1314,7 @@ $\Img(\pi)$ изоморфна $G$ и является подгруппой в $ \subsection{Диэдральная группа} \literature{[K3], гл. 1, \S~4, п. 5.} +\nopagebreak Рассмотрим на эвклидовой плоскости правильный $n$-угольник с вершинами $A_1,\dots,A_n$ и центром в начале координат (точке $O$). 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\usepackage{multicol} -\usepackage[colorlinks=true,pagebackref=true]{hyperref} +\usepackage[colorlinks=false,pagebackref=true]{hyperref} \usepackage{mathabx} @@ -102,7 +102,7 @@ \begin{document} \title{Алгебра и теория чисел\footnote{Конспект - лекций для механиков, 2014--2015 учебный год; предварительная + лекций для механиков, 2014--2016; предварительная версия}} \author{Александр Лузгарев} \date{} From e8226da2229a80c19a065b0341bc3cbba7412c03 Mon Sep 17 00:00:00 2001 From: broadwaylamb Date: Fri, 17 Jun 2016 22:17:21 +0300 Subject: [PATCH 02/11] =?UTF-8?q?Corrected=20errata=20in=209.1=E2=80=939.3?= =?UTF-8?q?=20subsections?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- algebra.pdf | Bin 1295728 -> 1295725 bytes euclidean-spaces.tex | 16 ++++++++-------- 2 files changed, 8 insertions(+), 8 deletions(-) diff --git a/algebra.pdf b/algebra.pdf index fc96b39dd2a7ee6d768c420bc7c00338cf18bd17..09cb96aa1768d8e830084d4ce2dcfb7684b07677 100644 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a/euclidean-spaces.tex +++ b/euclidean-spaces.tex @@ -158,10 +158,10 @@ $B(iv,iv) = -B(v,v)$ для всех $v\in V$. $B(u,v) = \ol{u}^Tv$. Нетрудно видеть, что эта форма полуторалинейная \begin{align*} -&B(u,v_1+v_2) = u^T(v_1+v_2) = \ol{u}^Tv_1 + \ol{u}^Tv_2 = B(u,v_1) + +&B(u,v_1+v_2) = \ol{u}^T(v_1+v_2) = \ol{u}^Tv_1 + \ol{u}^Tv_2 = B(u,v_1) + B(u,v_2)\\ &B(u,v\lambda)=\ol{u}^T(v\lambda)=\lambda(\ol{u}^Tv)=\lambda B(u,v)\\ -&B(u_1+u_2,v) = \ol{(u_1+u_2)}^Tv = \ol{u_1}^tv + \ol{u_2}^Tv = B(u_1,v) +&B(u_1+u_2,v) = \ol{(u_1+u_2)}^Tv = \ol{u_1}^Tv + \ol{u_2}^Tv = B(u_1,v) + B(u_2,v)\\ &B(u\lambda,v)=\ol{(u\lambda)}^Tv=\ol\lambda(\ol{u}^Tv)=\ol\lambda B(u,v)\\ \end{align*} @@ -227,7 +227,7 @@ $||v|| = \sqrt{B(v,v)}$ \dfn{длиной}\index{длина вектора} $v$. \begin{lemma}\label{lem:triangle_inequality} Пусть $(V,B)$~--- эвклидово или унитарное пространство, $u,v,\in V$. Тогда \begin{enumerate} -\item ({\it Однородность нормы}). $||\lambda v|| = |\lambda|\cdot +\item ({\it Однородность нормы}). $||v\lambda|| = |\lambda|\cdot ||v||$ для любого $\lambda\in k$. \item ({\it Теорема Пифагора}). Если $B(u,v)=0$, то $||u+v||^2 = ||u||^2 + ||v||^2$. @@ -243,7 +243,7 @@ $|B(u,v)|\leq ||u||\cdot ||v||$, причем равенство достига Однородность нормы следует из полуторалинейности: $$ -||\lambda v||^2 = B(\lambda v,\lambda v) = +||v\lambda||^2 = B(v\lambda, v\lambda ) = \lambda\ol{\lambda}B(v,v) = |\lambda|^2\cdot ||v||^2. $$ @@ -253,15 +253,15 @@ $$ Для доказательства неравенства Коши--Буняковского--Шварца положим $$ -w = u - \frac{B(u,v)}{B(v,v)}v +w = u - v\frac{B(u,v)}{B(v,v)} $$ -и заметим, что $$B(w,v) = B(u-\frac{B(u,v)}{B(v,v)}v,v) +и заметим, что $$B(w,v) = B(u-v\frac{B(u,v)}{B(v,v)},v) = B(u,v) - \frac{B(u,v)}{B(v,v)}B(v,v) = 0.$$ Это означает, что векторы $v$ и $w$ ортогональны. Поэтому и вектор -$\frac{B(u,v)}{B(v,v)}v$ ортогонален вектору $w$. Применим к этой паре +$v\frac{B(u,v)}{B(v,v)}$ ортогонален вектору $w$. Применим к этой паре векторов теорему Пифагора: $$ -||u||^2 = ||w||^2 + ||\frac{B(u,v)}{B(v,v)}v||^2 = ||w||^2 + +||u||^2 = ||w||^2 + ||v\frac{B(u,v)}{B(v,v)}||^2 = ||w||^2 + \frac{|B(u,v)|^2}{||v||^2} \geq \frac{|B(u,v)|^2}{||v||^2}, $$ откуда $|B(u,v)|\leq ||u||\cdot ||v||$. 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zN$y{Tj1!nAiU@?98J(-glDC0l1SA5^>$G7btH(C{yTw=7uYF=rT8ZYaFr9{ zk|)2jAaDhHagmoOIyOP}29o*(a8=g~r`qho3nETlkW>mU-zLxNA`2pUyCBno6PKT7 zbVikj4DX}kMEds8@hJafa9?aYnm(5M!(+E`ayen1vVXPdC&K@svxD~)_RFDk(ZT~k|pKi(=yT~{Xe_23+Vs= diff --git a/euclidean-spaces.tex b/euclidean-spaces.tex index 1956582..36cac6d 100644 --- a/euclidean-spaces.tex +++ b/euclidean-spaces.tex @@ -359,7 +359,7 @@ $$ Таким образом, мы показали, что матрица Грама симметрической билинейной формы является симметрической, а матрица Грама эрмитовой -билинейной формы является эрмитовой. +полуторалинейной формы является эрмитовой. Обратно, по любой симметрической матрице над $\mb R$ можно построить симметрическую билинейную форму, а по любой эрмитовой матрице над $\mb @@ -479,7 +479,7 @@ $e_i\perp e_j$ при $i\neq j$. Этот базис называется \begin{lemma}\label{lem:orthogonality_implies_independency} Пусть $(V,B)$~--- эвклидово или унитарное пространство. Если ненулевые векторы $e_1,\dots,e_n\in V$ попарно ортогональны, -то они линейно независимо. Если, кроме того, $\dim V=n$, то векторы +то они линейно независимы. Если, кроме того, $\dim V=n$, то векторы $e_1,\dots,e_n$ образуют ортогональный базис. \end{lemma} \begin{proof} @@ -678,7 +678,7 @@ $\ol{C}=C$ для $C\in M(n,\mb R)$. \begin{theorem} Пусть $(V,B)$~--- эвклидово или унитарное пространство. -Пусть $\mc E$, $\mc F$~--- ортогональные базисы $V$, и +Пусть $\mc E$, $\mc F$~--- ортонормированные базисы $V$, и $C=(\mc E\rsa\mc F)$~--- матрица перехода между ними. Тогда матрица $C$ ортогональна в случае эвклидова пространства и унитарна в случае унитарного пространства. @@ -687,7 +687,7 @@ $C$ ортогональна в случае эвклидова простран По теореме~\ref{thm:Gram_matrix_change_of_coordinates} выполнено $G_{\mc F} = \ol{C}^T\cdot G_{\mc E}\cdot C$, где $G_{\mc E}$, $G_{\mc F}$~--- матрицы Грама формы $B$ в базисах $\mc E$, -$\mc F$ соответственно. Но базисы $\mc E$, $\mc F$ ортогональны, +$\mc F$ соответственно. Но базисы $\mc E$, $\mc F$ ортонормированы, поэтому $G_{\mc E} = G_{\mc F} = E$. Значит, $E = \ol{C}^T\cdot C$, и матрица $C$ ортогональна в эвклидовом случае и унитарна в унитарном случае. 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--- a/euclidean-spaces.tex +++ b/euclidean-spaces.tex @@ -750,7 +750,7 @@ $$ Применяя к этому равенству отображение $\ph$ и пользуясь его линейностью, получаем \begin{align*} \ph(v) &= \ph(e_1 B(e_1,v) + e_2 B(e_2, v) + \dots + e_n B(e_n,v)) \\ -&= \ph(e_1)B(e_1,v) + \ph(e_2)B(e_2,v) + \dots + \ph(e_n B(e_n) \\ +&= \ph(e_1)B(e_1,v) + \ph(e_2)B(e_2,v) + \dots + \ph(e_n)B(e_n) \\ &= B(e_1\overline{\ph(e_1)} + e_2\overline{\ph(e_2)} + \dots + e_n\overline{\ph(e_n)},v). \end{align*} Заметим, что первый аргумент полученного выражения не зависит от $v$. @@ -830,7 +830,7 @@ $U\leq V$~--- конечномерное подпространство в $V$. \begin{enumerate} \item\label{num:orth-comp-prop-findim-1} $V = U\oplus U^\perp$; \item если, кроме того, $V$ конечномерно, то $\dim (U^\perp) = \dim (V) - \dim (U)$; -\item $(U\perp)^\perp = U$. +\item $(U^\perp)^\perp = U$. \end{enumerate} \end{proposition} \begin{proof} @@ -927,7 +927,7 @@ $U\leq V$~--- конечномерное подпространство, $v\in V $w_1+w_2\in U^\perp$. По определению $\pr_U(v_1) = u_1$, $\pr_U(v_2) = u_2$ и $\pr_U(v_1+v_2) = u_1 + u_2 = \pr_U(v_1) + \pr_U(v_2)$. -Мы показали аддитивность отображения $\pr_U$. Если $v\in U$ +Мы показали аддитивность отображения $\pr_U$. Если $v\in V$ и $v = u + w$ для $u\in U$, $w\in U^\perp$, то $v\lambda = u\lambda + w\lambda$, откуда следует и однородность $\pr_U$. @@ -1039,7 +1039,7 @@ $(\eta\circ\ph)^* = \ph^*\circ\eta^*$ \begin{align*} B(v,(\ph+\psi)^*(v')) &= B'((\ph+\psi)(v),v') \\ &= B'(\ph(v) + \psi(v),v') \\ -&= B'(\ph(v),v') + B(\psi(v),v') \\ +&= B'(\ph(v),v') + B'(\psi(v),v') \\ &= B(v,\ph^*(v')) + B(v,\psi^*(v')) \\ &= B(v,\ph^*(v')+\psi^*(v')), \end{align*} From 297e2ceeba1bffa718e4e135cf0e7935a3d74947 Mon Sep 17 00:00:00 2001 From: broadwaylamb Date: Sat, 18 Jun 2016 22:41:04 +0300 Subject: [PATCH 05/11] =?UTF-8?q?Corrected=20errata=20in=209.10=E2=80=939.?= =?UTF-8?q?12=20subsections?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- algebra.pdf | Bin 1296340 -> 1296321 bytes euclidean-spaces.tex | 15 ++++++++------- 2 files changed, 8 insertions(+), 7 deletions(-) diff --git a/algebra.pdf b/algebra.pdf index 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