From 7919d40873299b90f5cb23adabb8da90ca99d476 Mon Sep 17 00:00:00 2001 From: Petr Velycko Date: Sun, 18 May 2025 15:15:48 +0200 Subject: [PATCH] preklepy --- linearni-rovnice.tex | 4 ++-- skripta.pdf | Bin 227256 -> 227249 bytes 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/linearni-rovnice.tex b/linearni-rovnice.tex index bc4106e..540e789 100644 --- a/linearni-rovnice.tex +++ b/linearni-rovnice.tex @@ -15,7 +15,7 @@ \item $\| A + B \| \leq \|A\| + \|B\|$. \item $\| AB \| \leq \|A\|\|B\|$. \item $|Ax| \leq \|A\| |x|$ pro $x \in \mathbb{R}^n$. - \item Je-li $A$ regulární, pak $Ay \geq \frac{|y|}{\|A^{-1}\|}$ pro $y \in \mathbb{R}^n$. + \item Je-li $A$ regulární, pak $|Ay| \geq \frac{|y|}{\|A^{-1}\|}$ pro $y \in \mathbb{R}^n$. \end{enumerate} \end{theorem} @@ -83,7 +83,7 @@ Použijeme znalosti lineární algebry k tomu, abychom mohli formalizovat postup \end{theorem} \begin{proof} - Jádro lineárního zobrazení $Lx := x' - Ax$ je vektorový prostor. Dokážeme, že má dimenzi $n$. Nechť $i = 1,\dots,n$ a $x(t_0) = e_i$, pro tuto počáteční podmínku dostaneme řešení $x^i$. Potom $\{x^1, \dots, x^n\}$ tvoří bázi prostoru všech řešení. Skutečně, tyto vektory jsou lineárně nezávislé, mějme lineární kombinaci $c_1 x^1 + \dots + c_n x^n = 0$, speciálně v čase $t_0$ máme $c_1 e^1 + \dots + c_n x^n$, což implikuje, že $c_i = 0$ pro každé $i$. Navíc vezmeme libovolné řešení $z' = A(t) z$, opět zkoumejme stav v čase $t_0$. Máme $z(t_0) = d_1e^1 + \dots d_ne^n$ pro vhodná $d_1, \dots, d_n$. Definujme $y(t) := d_1 x^1(t) + \dots + d_n x^n(t)$, tedy $y$ řeší rovnici $y' = Ay$ a $y(t_0) = z(t_0)$, z čehož díky jednoznačnosti řešení dostáváme $y = z$. Nalezli jsme $n$-prvkovou bázi, tedy prostor $\mathcal{R}_H$ má dimenzi $n$. + Jádro lineárního zobrazení $Lx := x' - Ax$ je vektorový prostor. Dokážeme, že má dimenzi $n$. Nechť $i = 1,\dots,n$ a $x(t_0) = e_i$, pro tuto počáteční podmínku dostaneme řešení $x^i$. Potom $\{x^1, \dots, x^n\}$ tvoří bázi prostoru všech řešení. Skutečně, tyto vektory jsou lineárně nezávislé, mějme lineární kombinaci $c_1 x^1 + \dots + c_n x^n = 0$, speciálně v čase $t_0$ máme $c_1 e^1 + \dots + c_n e^n$, což implikuje, že $c_i = 0$ pro každé $i$. Navíc vezmeme libovolné řešení $z' = A(t) z$, opět zkoumejme stav v čase $t_0$. Máme $z(t_0) = d_1e^1 + \dots d_ne^n$ pro vhodná $d_1, \dots, d_n$. Definujme $y(t) := d_1 x^1(t) + \dots + d_n x^n(t)$, tedy $y$ řeší rovnici $y' = Ay$ a $y(t_0) = z(t_0)$, z čehož díky jednoznačnosti řešení dostáváme $y = z$. Nalezli jsme $n$-prvkovou bázi, tedy prostor $\mathcal{R}_H$ má dimenzi $n$. \end{proof} \begin{definition} diff --git a/skripta.pdf b/skripta.pdf index 5957848fdc9d3fdc8433b88d71a06d97f1a75de8..ed11959d0dec34bf0ce8774b216db665fd7e641c 100644 GIT binary patch delta 12960 zcmai4MNpj!vkVFD7Tg^U4#C~sB{+fL?#>t79fG@aa0w0x?(XjHZU_Fme7jdQRa0Fx zo0;iVceh-k)Lo*~CN)EGG$c3c!vL>x4z0n)BMg`7FUnvQ>H8DilHA-+h@3Jv)Vww+ z!TiGA23ClXdlZNXxeJ&i@H2l7+Uf2;QHMfzW$#L+6d_UVpzWOx!O`z(x-~K1F&ySEfw`T#;*Y66Fvj3g?ZWULk2l7J@(&z+*Ux)|}N z-vLwAa6OVY?Molz9QA7Dl+QHY=~y6(?-5?KoCEm`j78ICHQ3JCiU6tocg2=DNlM!s zme_D!|J2^UGSK7r2^;PqWFB>!!;n$6<30&}e^R5}o90Yklw-;&N-s%SkqXtK<)CFZ zvErH6BTrjx(M@b%rL}tVBTt;a8k6vht?jxr% zvM?k`y)XLzy}T2bPBKrgRTp(~3;?%^k2=^J#3vT3cB^TJvyr}^EaGnuPKXMG^`kv&J7vOTzb4;QUY%fK(6Fe(FT|vVf0Veht>CA-S(OMv z4*;9nUglIMH&YCzMC#L%zu@ZyL%Sa;MbzhcwH`>0MfUkj4=CV!$INi-HA(SQ2QXte z^gS@+&S+W#j&dz2brVdb^TH0365@~uIG=>Sw}b&_X{JA77Ag1 zEF9552q--8SNu4~A%EPv#ek2){Q{47bJTYs8%3x+UQdgSrQF7M~!nCJEn 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