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Na pervuyu stranicu << A.1 Matrichnaya algebra | Oglavlenie | A.3 Dekartovy pryamougol'nye i >>

A.2 Lineinaya algebra

Sistema lineinyh uravnenii

$\displaystyle \sum_{j=1}^n a_{ij}x_j = b_i,\quad i=1,2,\ldots,m $

ekvivalentna matrichnomu uravneniyu

$\displaystyle Ax=B \quad \textrm{ili} \quad \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix}. $

Esli matrica $ A$ ne vyrozhdena, to matrichnoe uravnenie imeet edinstvennoe reshenie:

$\displaystyle x=A^{-1}B. $

Esli matrica $ A$ yavlyaetsya ortogonal'noi i ee determinant raven edinice $ (\textrm{det}\,A=1)$, to lineinoe preobrazovanie $ y=Ax$ nazyvaetsya vrasheniem.

$ m$ uravnenii

$\displaystyle f_i(x_1,\ldots,x_n)=\sum_{j=1}^n a_{ij}x_j - b_i=0 \quad (i=1,2,\ldots,m) $

lineino nezavisimy, esli iz usloviya $ \sum_{i=1}^m \lambda_i f_i(x_1,\ldots,x_n)\equiv 0$ pri vseh znacheniyah $ x_1,\ldots,x_n$ sleduet, chto $ \lambda_1=\lambda_2=\ldots=\lambda_m=0$. V protivnom sluchae eti $ m$ uravnenii lineino zavisimy, t.e. po krainei mere odno iz uravnenii mozhet byt' predstavleno v vide lineinoi kombinacii ostal'nyh.

Vektory $ {\mathbf{a}}_1,{\mathbf{a}}_2, {\mathbf{a}}_3$ lineino nezavisimy, esli iz uravneniya

$\displaystyle \lambda_1{\mathbf{a}}_1+\lambda_2{\mathbf{a}}_2+ \lambda_3{\mathbf{a}}_3 =0 $

sleduet, chto $ \lambda_1=\lambda_2=\lambda_3=0$. V protivnom sluchae vektory $ {\mathbf{a}}_1,{\mathbf{a}}_2, {\mathbf{a}}_3$ lineino zavisimy i po krainei mere odin iz nih, naprimer, $ {\mathbf{a}}_1$ mozhet byt' vyrazhen v vide lineinoi kombinacii $ {\mathbf{a}}_1=\xi_2{\mathbf{a}}_2+ \xi_3{\mathbf{a}}_3$ ostal'nyh vektorov.

V trehmernom prostranstve kazhdoe mnozhestvo treh lineino nezavisimyh vektorov yavlyaetsya bazisom.

Lyuboi vektor $ \mathbf{a}$ v trehmernom prostranstve mozhet byt' predstavlen v vide razlozheniya:

$\displaystyle {\mathbf{a}} = a_1{\mathbf{e}}_1+a_2{\mathbf{e}}_2+ a_3{\mathbf{e}}_3 $

otnositel'no bazisnyh vektorov $ {\mathbf{e}}_1,{\mathbf{e}}_2, {\mathbf{e}}_3$. V trehmernom prostranstve chisla $ a_1,a_2,a_3$ yavlyayutsya koordinatami vektora $ \mathbf{a}$ v sisteme koordinat, opredelyaemyh bazisnymi vektorami $ {\mathbf{e}}_1,{\mathbf{e}}_2, {\mathbf{e}}_3$.


<< A.1 Matrichnaya algebra | Oglavlenie | A.3 Dekartovy pryamougol'nye i >>

Publikacii s klyuchevymi slovami: astrometriya - sfericheskaya astronomiya - sistemy koordinat - shkaly vremeni
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