ВВЕДЕНИЕ В ЛИНЕЙНУЮ АЛГЕБРУ / Lla05

\section*{\it ‹ҐЄжЁп 5.}

{\bf ‚ҐЄв®ал. ‹Ё­Ґ©­лҐ ®ЇҐа жЁЁ ­ ¤ ўҐЄв®а ¬Ё. Џа®ҐЄжЁп ўҐЄв®а .
„ҐЄ ав®ўл Є®®а¤Ё­ вл ўҐЄв®а®ў. ‘Є «па­®Ґ Їа®Ё§ўҐ¤Ґ­ЁҐ ўҐЄв®а®ў,
ҐЈ® ®б­®ў­лҐ бў®©бвў . ‚ҐЄв®а­®Ґ Ё ᬥ蠭­®Ґ Їа®Ё§ўҐ¤Ґ­ЁҐ
ўҐЄв®а®ў.}

…йҐ а § ЇҐаҐзЁб«Ё¬ ®б­®ў­лҐ бў®©бвў  б®ў®ЄгЇ­®бвЁ ўбҐе
ўҐЄв®а-бв®«Ўж®ў $\bar{x}$ а §¬Ґа  $n\times 1$ (ҐЈ® ®Ў®§­ з ов
$\RR^n$): Ґб«Ё $\bar{x}, \bar{y}, \bar{z}\in\RR^n$, $\alpha,
\beta\in\RR$, в®

$0^{\circ}$ ¤«п $\bar{x}$, $\bar{y}$ ®ЇаҐ¤Ґ«Ґ­л ®ЇҐа жЁЁ б«®¦Ґ­Ёп
$(\bar{x}+\bar{y})\in\RR^n$ Ё г¬­®¦Ґ­Ёп ­  зЁб« 
$\alpha\bar{x}\in\RR^n$,

$1^{\circ}$ $\bar{x}+\bar{y}=\bar{y}+\bar{x}$,

$2^{\circ}$ $(\bar{x}+\bar{y})+\bar{z}=\bar{x}+(\bar{y}+\bar{z})$,

$3^{\circ}$ $\exists\bar{0}\in\RR^n$:
$\forall\bar{x}\in\RR^n\Rightarrow$ $\bar{0}+\bar{x}=\bar{0}$,

$4^{\circ}$  $\forall\bar{x}\in\RR^n$ $\exists(-\bar{x})\in\RR^n$:
$\bar{x}+(-\bar{x})=\bar{0}$,

$5^{\circ}$ $0\cdot\bar{x}=\bar{0}$,

$6^{\circ}$ $1\cdot\bar{x}=\bar{x}$,

$7^{\circ}$ $\alpha\cdot(\beta\cdot\bar{x})=(\alpha
\beta)\cdot\bar{x}$,

$8^{\circ}$ $(\alpha+\beta)\cdot\bar{x}=\alpha\cdot\bar{x}+
\beta\cdot\bar{x}$,

$9^{\circ}$ $\alpha\cdot(\bar{x}+\bar{y})=\alpha\cdot\bar{x}+
\alpha\cdot\bar{y}$.\\ ќвЁ бў®©бвў  ўлЇ®«­повбп в Є¦Ґ Ё ¤«п
ўҐЄв®а-бва®Є. „«п $n=2$ Ё $n=3$ १г«мв в б«®¦Ґ­Ёп ¤ўге ўҐЄв®а®ў Ё
१г«мв в г¬­®¦Ґ­Ёп ўҐЄв®а  ­  зЁб«® ¬®Јгв Ўлвм Ё§®Ўа ¦Ґ­л ­ 
зҐа⥦Ґ. €­®Ј¤  ¤«п Ў®«ҐҐ Ї®«­®Ј® Ї®пб­Ґ­Ёп, ® Є ЄЁе Ё¬Ґ­­®
ўҐЄв®а е Ё¤Ґв аҐзм, Ј®ў®апв Ї®¤а®Ў­ҐҐ, ­ ЇаЁ¬Ґа: $n$-¬Ґа­ п
ўҐЄв®а-бва®Є  Ё в.Ї.

‘ Є ¦¤л¬ ўҐЄв®а®¬ бўп§ ­ ­ Ў®а зЁбҐ« $(x_i)_{i=1}^n$, Ё§ Є®в®але Ё
б®бв®Ёв б ¬ ўҐЄв®а: $\bar{x}=(x_i)_{i=1}^n$. —Ёб«  $x_i$ ЇаЁ н⮬
­ §лў ов ¤ҐЄ ав®ўл¬Ё Є®®а¤Ё­ в ¬Ё $\bar{x}$, $x_1$ -- ЇҐаў®©
Є®®а¤Ё­ в®©, $x_2$ -- ўв®а®© Ё в.¤.

\noindent ЋЇаҐ¤Ґ«Ґ­ЁҐ. ‘Є «па­л¬ Їа®Ё§ўҐ¤Ґ­ЁҐ¬ ¤ўге $n$-¬Ґа­ле
ўҐЄв®а®ў $\bar{x}=(x_i)_{i=1}^n$ Ё $\bar{y}=(y_i)_{i=1}^n$
­ §лў ов ўҐ«ЁзЁ­г $$(\bar{x},\bar{y})=\sum_{i=1}^nx_iy_i.$$
‘ў®©бвў  бЄ «па­®Ј® Їа®Ё§ўҐ¤Ґ­Ёп:

1) $(\bar{x},\bar{y})=(\bar{y},\bar{x})$,

2) $(\bar{x},\alpha \bar{y})=\alpha (\bar{x},\bar{y})$, $\alpha$
--- зЁб«® Ё, §­ зЁв, $(\alpha \bar{x},\beta \bar{y})=\alpha\beta
(\bar{x},\bar{y})$

3)
$(\bar{x},\bar{y}_1+\bar{y}_2)=(\bar{x},\bar{y}_1)+(\bar{x},\bar{y}_2)$,

4) $(\bar{x},\bar{x})>0$, Ґб«Ё $\bar{x}\neq 0$.

\noindent ќвЁ бў®©бвў  пў«повбп б«Ґ¤бвўЁҐ¬ ®ЇаҐ¤Ґ«Ґ­Ёп бЄ «па­®Ј®
Їа®Ё§ўҐ¤Ґ­Ёп ¤ўге ўҐЄв®а®ў.

ЌҐа ўҐ­бвў® Љ®иЁ-Ѓг­пЄ®ўбЄ®Ј®-ў аж :\quad
$(\bar{x},\bar{y})^2\leqslant
(\bar{x},\bar{x})(\bar{y},\bar{y})$.\\ $\lhd:$ ЏаЁ ўбҐе $t\in \RR$
$$0\leqslant(\bar{x}+t\cdot \bar{y},\bar{x}+t\cdot \bar{y})=
(\bar{x},\bar{x})+t(\bar{y},\bar{x})+t(\bar{x},\bar{y})+
t^2(\bar{y},\bar{y})=(\bar{x},\bar{x})+2t(\bar{x},\bar{y})
+t^2(\bar{y},\bar{y}).$$ Џ®н⮬㠤ЁбЄаЁ¬Ё­ ­в нв®Ј® Єў ¤а в­®Ј® Ї®
ЇҐаҐ¬Ґ­­®© $t$ ўла ¦Ґ­Ёп ­ҐЇ®«®¦ЁвҐ«Ґ­:
$$(\bar{x},\bar{y})^2-(\bar{x},\bar{x})(\bar{y},\bar{y})\leqslant
0.\quad\rhd$$ ‚Ґ«ЁзЁ­г $|\bar{x}|=\sqrt{(\bar{x},\bar{x})}$
­ §лў ов ¤«Ё­®© Ё«Ё ¬®¤г«Ґ¬ ўҐЄв®а  $\bar{x}$,   ®в­®иҐ­ЁҐ
(б®Ј« б­® ­Ґа ўҐ­бвў  Љ®иЁ-Ѓг­пЄ®ўбЄ®Ј®-ў аж  ЇаЁ­ ¤«Ґ¦ йҐҐ
®в१Єг $[-1,1]$) $$\cos \phi=
\frac{(\bar{x},\bar{y})}{|\bar{x}|\,|\bar{y}|}\in [-1,1]$$
--- Є®бЁ­гᮬ гЈ«  ¬Ґ¦¤г ўҐЄв®а ¬Ё $\bar{x}$ Ё $\bar{y}$. ќвЁ
ўҐЄв®а  ­ §лў ов Є®««Ё­Ґ а­л¬Ё, Ґб«Ё $\cos \phi=\pm 1$; Ёе
­ §лў ов ®ав®Ј®­ «м­л¬Ё (ЇҐаЇҐ­¤ЁЄг«па­л¬Ё), Ґб«Ё $\cos \phi=0$.

‚ Є зҐб⢥ б«Ґ¤бвўЁп Ї®«гз Ґ¬ е®а®и® Ё§ўҐбв­®Ґ а ўҐ­бвў® ¤«п
бЄ «па­®Ј® Їа®Ё§ўҐ¤Ґ­Ёп ўҐЄв®а®ў: $(\bar{x},\bar{y})=
|\bar{x}|\,|\bar{y}|\cos \phi$.

\noindent ЋЇаҐ¤Ґ«Ґ­ЁҐ. Џа®ҐЄжЁҐ© ўҐЄв®а  $\bar{x}=(x_i)_{i=1}^n$
­  ўҐЄв®а $\bar{y}=(y_i)_{i=1}^n$ ­ §лў ов Є®««Ё­Ґ а­л© $\bar{y}$
ўҐЄв®а $${\rm Їа}_{{y}} \bar{x}=\frac
{\bar{y}}{|\bar{y}|}\cdot|\bar{x}|\cos \phi=\bar{y}\frac
{(\bar{x},\bar{y})}{(\bar{y},\bar{y})}.$$ ‘«Ґ¤бвўЁҐ.
$(\bar{x},\bar{y})=({\rm Їа}_y \bar{x},\bar{y})$.\\ ‚ Є зҐб⢥
ЇаЁ¬Ґа  ўўҐ¤Ґ­­ле Ї®­пвЁ© ®в¬ҐвЁ¬ бЁб⥬г $\{\bar{e}_i\}_{i=1}^n$
®ав®Ј®­ «м­ле ўҐЄв®а®ў, Ё¬ҐойЁе ўЁ¤ $\bar{e}_i=(\delta_{i,j})$,
Ј¤Ґ $\delta_{i,j}$ -- бЁ¬ў®« Ља®­ҐЄҐа :
$$\bar{e}_1=(1,0,\dots,0),\quad
\bar{e}_2=(0,1,0,\dots,0),\dots\quad, \bar{e}_n=(0,\dots,0,1).$$
ќвг бЁб⥬㠢ҐЄв®а®ў ЇаЁ­пв® ­ §лў вм бв ­¤ ав­л¬ ®ав®Ј®­ «м­л¬
Ў §Ёб®¬ Їа®бва ­бвў  $\RR^n$. Ѓ®«ҐҐ ®Ўй®, ®ав®Ј®­ «м­л¬ Ў §Ёб®¬ ў
Їа®бва ­б⢥ $\RR^n$ ­ §лў ов ўбпЄго бЁб⥬г, б®бв®пйго Ё§ $n$
­Ґ­г«Ґўле ®ав®Ј®­ «м­ле ¤агЈ ¤агЈг ўҐЄв®а®ў. ЌҐваг¤­® ўЁ¤Ґвм, зв®
«оЎлҐ ¤ў  а §­ле ўҐЄв®а  $\bar{e}_i$ Ё $\bar{e}_j$ ®ав®Ј®­ «м­л
¤агЈ ¤агЈг, Ёе ®ЎйҐҐ зЁб«® а ў­® $n$ Ё зв® Є ¦¤л© ўҐЄв®а
$\bar{e}_i$ Ё¬ҐҐв Ґ¤Ё­Ёз­го ¤«Ё­г. ‘в ­¤ ав­л© ®ав®Ј®­ «м­л© Ў §Ёб
®Ў« ¤ Ґв ҐйҐ ®¤­Ё¬ ў ¦­л¬ Ё ­ Ј«п¤­л¬ бў®©бвў®¬: Є ¦¤л© ўҐЄв®а
$\bar{x}=(x_i)_{i=1}^n$ ®¤­®§­ з­л¬ ®Ўа §®¬ § ЇЁблў Ґвбп ў ўЁ¤Ґ
$$\bar{x}=\sum_{i=1}^nx_i\bar{e}_i.$$ ’ Є®Ґ ЇаҐ¤бв ў«Ґ­ЁҐ ўҐЄв®а 
$\bar{x}$ ­ §лў ов а §«®¦Ґ­ЁҐ¬ $\bar{x}$ Ї® Ў §Ёбг (ў ¤ ­­®¬
б«гз Ґ Ї® Ў §Ёбг $\{\bar{e}_i\}_{i=1}^n$).

Џ®­пвЁҐ ўҐЄв®а­®Ј® Їа®Ё§ўҐ¤Ґ­Ёп ¤ўге ўҐЄв®а®ў ¤ Ґвбп ў ваҐе¬Ґа­®¬
Їа®бва ­б⢥. Џгбвм $\bar{x}$ Ё $\bar{y}$ --- Ї а  ўҐЄв®а®ў ў н⮬
Їа®бва ­б⢥. ‚ҐЄв®а­л¬ Їа®Ё§ўҐ¤Ґ­ЁҐ¬ $[\bar{x},\bar{y}]$ ­ §лў ов
ўҐЄв®а, Є®в®ал©

--- а ўҐ­ Ї® ¤«Ё­Ґ Ї«®й ¤Ё Ї а ««Ґ«®Ја ¬¬ , Ї®бв஥­­®Ј® ­ 
ўҐЄв®а е $\bar{x}$, $\bar{y}$:
$$|[\bar{x},\bar{y}]|=|\bar{x}|\,|\bar{y}|\,\sin\phi,\quad\phi\in
[0,\pi] {\rm\ -\ гЈ®«\ ¬Ґ¦¤г}\ \bar{x}\ {\rm Ё}\ \bar{y},$$

--- ЇҐаЇҐ­¤ЁЄг«п७ $\bar{x}$ Ё $\bar{y}$,

--- Ґб«Ё $\bar{x}$ Ё $\bar{y}$ ­ҐЄ®««Ё­Ґ а­л, ®Ў« ¤ Ґв бў®©бвў®¬ в®Ј®, зв® ва®©Є  $\bar{x}$, $\bar{y}$, $[\bar{x},\bar{y}]$
®Ўа §гҐв Їа ўго ва®©Єг ўҐЄв®а®ў.

‘ў®©бвў  ўҐЄв®а­®Ј® Їа®Ё§ўҐ¤Ґ­Ёп

1) $[\bar{y},\bar{x}]=-[\bar{x},\bar{y}]$,

2) $[\bar{x},\bar{x}]=\bar{0}$,

3) $[\lambda \bar{x},\bar{y}]=[\bar{x},\lambda \bar{y}]=\lambda
[\bar{x},\bar{y}]$,

4)
$[\bar{x}+\bar{y},\bar{z}]=[\bar{x},\bar{z}]+[\bar{y},\bar{z}]$.

\noindent „®Є § вҐ«мбвў® ЇҐаўле ваҐе бў®©бвў пў«пҐвбп Їа®бвл¬
б«Ґ¤бвўЁҐ¬ ®ЇаҐ¤Ґ«Ґ­Ёп; ¤®Є § вҐ«мбвў® Ї®б«Ґ¤­ҐЈ® вॡгҐв
Їа®ўҐ¤Ґ­Ёп  ЄЄга в­ле ўлЄ« ¤®Є. Њл ­Ґ Ўг¤Ґ¬ ҐЈ® Їа®ў®¤Ёвм. Џгбвм
$\bar{e}_1,\bar{e}_2,\bar{e}_3$ --- бв ­¤ ав­л© Ў §Ёб ў $\RR^3$,
$\bar{x}=x_1\cdot \bar{e}_1+x_2\cdot \bar{e}_2+x_3\cdot
\bar{e}_3$, $\bar{y}=y_1\cdot \bar{e}_1+y_2\cdot
\bar{e}_2+y_3\cdot \bar{e}_3$. ’®Ј¤ 
$$[\bar{x},\bar{y}]=y_1[\bar{x},\bar{e}_1]+y_2[\bar{x},\bar{e}_2]
+y_3[\bar{x},\bar{e}_3]=y_1(x_1[\bar{e}_1,\bar{e}_1]+
x_2[\bar{e}_2,\bar{e}_1]+x_3[\bar{e}_3,\bar{e}_1])
+y_2[\bar{x},\bar{e}_2]+y_3[\bar{x},\bar{e}_3]=$$ $$=y_1(x_2\cdot
(-\bar{e}_3)+x_3\cdot \bar{e}_2) +y_2(x_1\cdot \bar{e}_3+x_3\cdot
(-\bar{e}_1))+y_3(x_1\cdot (-\bar{e}_2)+x_2\cdot
\bar{e}_1)=\begin{vmatrix}\bar{e}_1 & \bar{e}_2 & \bar{e}_3\\x_1 &
x_2 & x_3\\y_1 & y_2 & y_3\end{vmatrix}.$$

ЋЇаҐ¤Ґ«Ґ­ЁҐ. ‘¬Ґи ­­л¬ Їа®Ё§ўҐ¤Ґ­ЁҐ¬ $\langle \bar{x}, \bar{y},
\bar{z} \rangle$ ўҐЄв®а®ў $\bar{x}$, $\bar{y}$ Ё $\bar{z}$
­ §лў ов зЁб«® $([\bar{x},\bar{y}],\bar{z})$.

€§ ®ЇаҐ¤Ґ«Ґ­Ёп ўҐЄв®а­®Ј® Їа®Ё§ўҐ¤Ґ­Ёп б«Ґ¤гҐв, зв®
$[\bar{x},\bar{y}]$ -- ўҐЄв®а, Ї® ¤«Ё­Ґ а ў­л© Ї«®й ¤Ё
Ї а ««Ґ«®Ја ¬¬ , Ї®бв஥­­®Ј® ­  ўҐЄв®а е $\bar{x}$, $\bar{y}$, Ё
®Ўа §го饣® Їа ўго ва®©Єг ўҐЄв®а®ў ў Є®¬ЎЁ­ жЁЁ $\bar{x}$,
$\bar{y}$, $[\bar{x},\bar{y}]$. …б«Ё бзЁв вм нв®в Ї а ««Ґ«®Ја ¬¬
®б­®ў ­ЁҐ¬ Ї а ««Ґ«ҐЇЁЇҐ¤ , Ї®бв஥­­®Ј® ­  ўҐЄв®а е $\bar{x}$,
$\bar{y}$, $\bar{z}$, в® ҐЈ® ®ЎкҐ¬, в ЄЁ¬ ®Ўа §®¬, а ўҐ­ $\langle
\bar{x}, \bar{y}, \bar{z} \rangle$, ¤«п Їа ў®© ва®©ЄЁ ўҐЄв®а®ў
$\bar{x}$, $\bar{y}$, $\bar{z}$, Ё«Ё а ўҐ­ $(-\langle \bar{x},
\bar{y}, \bar{z} \rangle)$ ¤«п «Ґў®© ва®©ЄЁ ўҐЄв®а®ў $\bar{x}$,
$\bar{y}$, $\bar{z}$. ‘ў®©бвў  ᬥ蠭­®Ј® Їа®Ё§ўҐ¤Ґ­Ёп

$\langle \bar{x}, \bar{y}, \bar{z} \rangle=\langle \bar{y},
\bar{z}, \bar{x} \rangle=\langle \bar{z}, \bar{x}, \bar{y}
\rangle= -\langle \bar{y}, \bar{x}, \bar{z} \rangle=-\langle
\bar{x}, \bar{z}, \bar{y} \rangle=-\langle \bar{z}, \bar{y},
\bar{x} \rangle$.

\noindent …б«Ё $x_1,x_2,x_3$, $\ y_1,y_2,y_3$, $\ z_1,z_2,z_3$ ---
Є®®а¤Ё­ вл ᮮ⢥вб⢥­­® ўҐЄв®а®ў $\bar{x}$, $\bar{y}$ Ё
$\bar{z}$ ў бв ­¤ ав­®¬ Ў §ЁбҐ ў $\RR^3$, в® $$\langle \bar{x},
\bar{y}, \bar{z} \rangle=([\bar{x},\bar{y}],\bar{z})=(z_1\cdot
\bar{e}_1 +z_2\cdot \bar{e}_2 +z_3\cdot \bar{e}_3,
\begin{vmatrix}\bar{e}_1 & \bar{e}_2 & \bar{e}_3\\x_1 & x_2 & x_3\\y_1 & y_2 &
y_3\end{vmatrix})=
\begin{vmatrix}z_1 & z_2 & z_3\\x_1 & x_2 & x_3\\y_1 & y_2 &
y_3\end{vmatrix}=
\begin{vmatrix}x_1 & x_2 & x_3\\y_1 & y_2 &
y_3\\z_1 & z_2 & z_3\end{vmatrix}.$$