From 54139b1fd21a5f30dc81faa250b8e8053ed31c06 Mon Sep 17 00:00:00 2001 From: Filip Znachor Date: Sat, 31 Dec 2022 13:04:23 +0100 Subject: [PATCH] =?UTF-8?q?P=C5=99id=C3=A1n=C3=AD=20pozn=C3=A1mek=20k=20or?= =?UTF-8?q?togon=C3=A1ln=C3=ADmu=20pr=C5=AFm=C4=9Btu=20v=20LAA?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- KMA LAA/9. Prostory se skalárním součinem.md | 72 ++++++++++++++++++- KMA LAA/_assets/ortogonalni-prumet.png | Bin 0 -> 53166 bytes 2 files changed, 71 insertions(+), 1 deletion(-) create mode 100644 KMA LAA/_assets/ortogonalni-prumet.png diff --git a/KMA LAA/9. Prostory se skalárním součinem.md b/KMA LAA/9. Prostory se skalárním součinem.md index 97174b7..cac2fba 100644 --- a/KMA LAA/9. Prostory se skalárním součinem.md +++ b/KMA LAA/9. Prostory se skalárním součinem.md @@ -10,7 +10,20 @@ Nechť $U$ je lineární vektorový prostor nad $\mathbb{R}$. Zobrazení $(\vec{ se nazývá **skalární součin**. -### Euklidovský prostor +#### Skalární součin v prostorech nad $\mathbb{C}$ + +Nechť $U$ je lineární vektorový prostor nad $C$. Zobrazení $(\vec{x}, \vec{y}) : U \times U \to \mathbb{C}$ splňující vlastnosti +1. $(\vec{x}, \vec{x}) \geq 0$ pro každé $\vec{x} \in U; (\vec{x}, \vec{x}) = 0$, právě když $\vec{x} = \vec{o}$, +2. $(\vec{x}, \vec{y}) = \overline{(\vec{y}, \vec{x})} \space \forall \vec{x}, \vec{y} \in U$, +3. $(k\vec{x}, \vec{y}) = k(\vec{x}, \vec{y}) \space \forall \vec{x}, \vec{y} \in U$ a $\forall k \in \mathbb{C}$ +4. $(\vec{x} + \vec{y}, \vec{z}) = (\vec{x}, \vec{z}) + (\vec{y}, \vec{z}) \space \forall \vec{x}, \vec{y}, \vec{z} \in U$, + +se nazývá **skalární součin**. L.V.P. se skalárním součinem se nazývá **Unitární prostor**. + +Vše zde funguje jako v Eukleidovském prostoru, až na Pythagorovu větu, kde neplatí opačná implikace, tj. +platí-li rovnost $\Vert \vec{x} + \vec{y} \Vert^2 = \Vert \vec{x} \Vert^2 + \Vert \vec{y} \Vert^2$, potom nemusí platit, že $\vec{x} \perp \vec{y}$. + +### Eukleidovský prostor Lineární vektorový prostor se skalárním součinem se nazývá **Eukleidovský prostor**. @@ -89,3 +102,60 @@ V každém Eukleidovském prostoru konečné dimenze existuje ortogonální báz $$ 5. Pak jistě $\vec{g}_{k} \perp \vec{g}_{1}, \vec{g}_{k} \perp \vec{g}_{2}, \dots, \vec{g}_{k} \perp \vec{g}_{k-1}$. + +### Ortogonální průmět + +Mějme Eukleidovský prostor $U$, jeho podprostor $V$ a v něm generátor (ne nutně bázi) $\vec{b}_{1}, \vec{b}_{2}, \dots, \vec{b}_{k}$. Máme určit ortogonální průmět $\overline{\vec{x}}$ prvku $\vec{x} \in U$ do $V$. +- Víme, že $\vec{x} - \overline{\vec{x}} \perp \vec{b}_{i}$ pro každé $i = 1, 2, \dots, k$. +- Dále: $\overline{\vec{x}} \in V$, tedy $\overline{\vec{x}} = a_{1}\vec{b}_{1} + a_{2}\vec{b}_{2} + \dots + a_{k}\vec{b}_{k}$ (je to LK generátorů). + +![[_assets/ortogonalni-prumet.png]] + +Pro každé $i = 1, 2, \dots, k$ platí: + +$$ +0 = (\vec{b}_{i}, \vec{x} - \overline{\vec{x}}) = (\vec{b}_{i}, \vec{x} - a_{1}\vec{b}_{1} + a_{2}\vec{b}_{2} + \dots + a_{k}\vec{b}_{k}) = (\vec{b}_{i}, \vec{x}) - a_{1}(\vec{b}_{i}, \vec{b}_{1}) - a_{2}(\vec{b}_{i}, \vec{b}_{2}) - \dots - a_{k}(\vec{b}_{i}, \vec{b}_{k}). +$$ + +Dostaneme tak soustavu rovnic: + +$$ +\begin{matrix} +(\vec{b}_{1}, \vec{b}_{1})a_{1} + (\vec{b}_{1}, \vec{b}_{2})a_{2} + \dots + (\vec{b}_{1}, \vec{b}_{k})a_{k} = (\vec{b}_{i}, \vec{x}) \qquad i=1 \\ +(\vec{b}_{2}, \vec{b}_{1})a_{1} + (\vec{b}_{2}, \vec{b}_{2})a_{2} + \dots + (\vec{b}_{2}, \vec{b}_{k})a_{k} = (\vec{b}_{2}, \vec{x}) \qquad i=2 \\ +\vdots \qquad\qquad \vdots \qquad\qquad \vdots \qquad\qquad \vdots \\ +(\vec{b}_{k}, \vec{b}_{1})a_{1} + (\vec{b}_{k}, \vec{b}_{2})a_{2} + \dots + (\vec{b}_{k}, \vec{b}_{k})a_{k} = (\vec{b}_{k}, \vec{x}) \qquad i=k +\end{matrix} +$$ + +tedy **Gramovu matici**: + +$$ +\begin{bmatrix} +(\vec{b}_{1}, \vec{b}_{1}) & (\vec{b}_{1}, \vec{b}_{2}) & \dots & (\vec{b}_{1}, \vec{b}_{k}) \\ +(\vec{b}_{2}, \vec{b}_{1}) & (\vec{b}_{2}, \vec{b}_{2}) & \dots & (\vec{b}_{2}, \vec{b}_{k}) \\ +\vdots & \vdots & \vdots & \vdots \\ +(\vec{b}_{k}, \vec{b}_{1}) & (\vec{b}_{k}, \vec{b}_{2}) & \dots & (\vec{b}_{k}, \vec{b}_{k}) +\end{bmatrix} +\begin{bmatrix} +a_{1} \\ +a_{2} \\ +\vdots \\ +a_{3} +\end{bmatrix} += +\begin{bmatrix} +(\vec{b}_{1}, \vec{x}) \\ +(\vec{b}_{2}, \vec{x}) \\ +\vdots \\ +(\vec{b}_{k}, \vec{x}) +\end{bmatrix} +$$ + +- Je-li $\{ \vec{b}_{1}, \vec{b}_{2}, \dots, \vec{b}_{k} \}$ **ortogonální báze**, potom **Gramova matice je diagonální**. +- Gramova matice je regulární, právě když množina vektorů $\vec{b}_{1}, \vec{b}_{2}, \dots, \vec{b}_{k}$ je LN. + +Zřejmě $\overline{\vec{x}}$ je nejbližším vektorem k $\vec{x}$ ve $V$. + +Je-li $V$ podprostorem prostoru $U$ a $\vec{x} \notin V$, potom existuje právě jeden prvek $\overline{\vec{x}}$ takový, že $\vec{x} - \overline{\vec{x}} \perp V$ a $\overline{\vec{x}} \in V$. +- Pro každý vektor $\vec{y} \in V$ platí $\Vert \vec{x} - \vec{y} \Vert \geq \Vert \vec{x} - \overline{\vec{x}} \Vert$ a rovnost nastává, právě když $\vec{y} = \overline{\vec{x}}$. \ No newline at end of file diff --git a/KMA LAA/_assets/ortogonalni-prumet.png b/KMA LAA/_assets/ortogonalni-prumet.png new file mode 100644 index 0000000000000000000000000000000000000000..ac3a4b8a652c660f243c848f078c66224deda58f GIT binary patch literal 53166 zcmeFZWl)^Kwl0h_xVyW%ySqyW?(QD8&u$E|zh> z{l0tE__qA@arYtr!U>~3d->%_%qf5Q@$9!z@8ua6^5&w5enovf$s|eO*8WY-vkH!U 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