Аналитическая геометрия и линейная алгебра. Учебное пособие

Ɋɚɫɫɦɨɬɪɢɦ ɭɪɚɜɧɟɧɢɟ

z2 (2i 3)z 5 i 0.

ɉɨ ɮɨɪɦɭɥɟ ɞɥɹ ɤɨɪɧɟɣ ɤɜɚɞɪɚɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɨɥɭɱɚɟɦ

ȼɵɱɢɫɥɢɦ 15 8i, ɧɚɦ ɧɭɠɧɨ ɧɚɣɬɢ ɬɚɤɨɟ ɱɢɫɥɨ w a ib, ɱɬɨ

w2 15 8i :

a2 b2 2iab 15 8i. ɉɪɢɪɚɜɧɢɜɚɹ ɜɟɳɟɫɬɜɟɧɧɭɸ ɢ ɦɧɢɦɭɸ ɱɚɫɬɢ ɩɨɥɭɱɚɟɦ

61

ɂɡɭɱɢɦ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɨɩɟɪɚɰɢɣ ɫɥɨɠɟɧɢɹ ɢ ɭɦɧɨɠɟɧɢɹ ɤɨɦ ɩɥɟɤɫɧɵɯ ɱɢɫɟɥ ɉɪɢ ɞɨɛɚɜɥɟɧɢɢ ɤ ɱɢɫɥɭ z x iy ɱɢɫɥɚ z0 x0 iy0 ɦɵ ɩɨ ɥɭɱɚɟɦ ɱɢɫɥɨ z z0 (x x0) i( y y0) ɬ ɟ ɩɪɨɢɫɯɨɞɢɬ ɩɚɪɚɥɥɟɥɶɧɵɣ ɩɟ ɪɟɧɨɫ ɩɥɨɫɤɨɫɬɢ ɧɚ ɜɟɤɬɨɪ aG(x0, y0) ɉɭɫɬɶ ɤɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ z ɭɦɧɨɠɚɟɬɫɹ ɧɚ z0, z0 1 Ɂɚɩɢɲɟɦ ɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ ɮɨɪɦɟ

z r(cosK isin K), z0 cosR isin R.

Ɍɨɝɞɚ z ¸z0 r(cos(KR) isin(KR)) , ɬ. ɟ ɩɪɨɢɫɯɨɞɢɬ ɩɨɜɨɪɨɬ ɩɥɨɫɤɨ ɫɬɢ ɧɚ ɭɝɨɥ R ɜɨɤɪɭɝ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɉɭɫɬɶ z0 S(cosR isin R) ɦɵ ɩɨɥɭ ɱɚɟɦ

z ¸z0 rS(cos(KR) isin(KR)), ɩɨɜɨɪɨɬ ɧɚ ɭɝɨɥ R ɢ ɪɚɫɬɹɠɟɧɢɟ (ɫɠɚɬɢɟ ɜ S ɪɚɡ

62

ɉɪɢɦɟɪ ɂɫɫɥɟɞɭɟɦ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɤɨɦ ɩɥɟɤɫɧɨɣ ɩɥɨɫɤɨɫɬɢ

z z 1 i.

ɉɪɢ ɬɚɤɨɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɢ ɤɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ z x iy ɩɟɪɟɯɨɞɢɬ ɜ ɤɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ

u (x 1) i( y 1),

ɬ ɟ ɩɪɨɢɫɯɨɞɢɬ ɩɚɪɚɥɥɟɥɶɧɵɣ ɩɟɪɟɧɨɫ ɩɥɨɫɤɨɫɬɢ ɧɚ ɜɟɤɬɨɪ aG(1,1). ȼɫɟ ɩɪɹ ɦɵɟ ɩɥɨɫɤɨɫɬɢ ɩɟɪɟɯɨɞɹɬ ɜ ɩɚɪɚɥɥɟɥɶɧɵɟ ɩɪɹɦɵɟ ȼɫɟ ɨɤɪɭɠɧɨɫɬɢ ɩɟɪɟɯɨɞɹɬ ɜ ɨɤɪɭɠɧɨɫɬɢ ɬɨɝɨ ɠɟ ɪɚɞɢɭɫɚ

ɂɫɫɥɟɞɭɟɦ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɤɨɦɩɥɟɤɫɧɨɣ ɩɥɨɫ ɤɨɫɬɢ

z z(1 i). Ɂɚɩɢɲɟɦ ɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ ɮɨɪɦɟ

z r(cosKisin K),

1 i 2(cos(Q/ 4) isin(Q/ 4)), z(1 i) 2r(cos(K Q/ 4) isin(K Q/ 4)).

ɉɪɨɢɫɯɨɞɢɬ ɪɚɫɬɹɠɟɧɢɟ ɩɥɨɫɤɨɫɬɢ ɜ 2 ɪɚɡ ɢ ɩɨɜɨɪɨɬ ɧɚ ɭɝɨɥ Q/ 4 ɩɨ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɟ ɜɨɤɪɭɝ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ

z z.

Ʉɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ z x iy ɩɟɪɟɯɨɞɢɬ ɜ ɤɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ u x i( y),

ɫɢɦɦɟɬɪɢɱɧɨɟ ɟɦɭ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɬ ɟ ɬɚɤɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɩɥɨɫɤɨɫɬɢ ɟɫɬɶ ɰɟɧɬɪɚɥɶɧɚɹ ɫɢɦɦɟɬɪɢɹ ɫ ɰɟɧɬɪɨɦ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ

Ʉɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ z x iy ɩɟɪɟɯɨɞɢɬ ɜ ɤɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ

z x i( y),

ɫɢɦɦɟɬɪɢɱɧɨɟɟɦɭ ɨɬɧɨɫɢɬɟɥɶɧɨɨɫɢ Ox ɬ ɟ ɬɚɤɨɟɩɪɟɨɛɪɚɡɨɜɚɧɢɟɩɥɨɫɤɨɫɬɢ ɟɫɬɶ ɨɫɟɜɚɹ ɫɢɦɦɟɬɪɢɹ

z iz 1.

Ⱦɚɧɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɟɫɬɶ ɤɨɦɩɨɡɢɰɢɹ ɩɨɜɨɪɨɬɚGɧɚ ɭɝɨɥ Q/ 2 ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ ɢ ɩɚɪɚɥɥɟɥɶɧɨɝɨ ɩɟɪɟɧɨɫɚ ɧɚ ɜɟɤɬɨɪ a(1,0) ɇɚɣɞɟɦ ɧɟɩɨ ɞɜɢɠɧɭɸ ɬɨɱɤɭ ɞɚɧɧɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɉɭɫɬɶ iz 1 z Ɍɨɝɞɚ

63

Ɍɟɤɫɬ ɡɚɞɚɧɢɹ ɂɫɫɥɟɞɨɜɚɬɶ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɟɨɛɪɚɡɨɜɚ ɧɢɹ ɧɚɣɬɢ ɧɟɩɨɞɜɢɠɧɭɸ ɬɨɱɤɭ ɟɫɥɢ ɨɧɚ ɫɭɳɟɫɬɜɭɟɬ.

ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɹ ʋ

3.2 DzȏȔȌȐȔȢȌ ȖȗȕȘșȗȇȔȘșȉȇ

Ɉɩɪɟɞɟɥɟɧɢɟ Ɇɧɨɠɟɫɬɜɨ L ɷɥɟɦɟɧɬɨɜ ɥɸɛɨɣ ɩɪɢɪɨɞɵ ɧɚɡɵɜɚɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɟɫɥɢ ɧɚ ɷɬɨɦ ɦɧɨɠɟɫɬɜɟ ɜɜɟɞɟɧɵ ɞɜɟ ɨɩɟɪɚɰɢɢ

ɋɥɨɠɟɧɢɟ

x, y L z L : z x y;

64

ɍɦɧɨɠɟɧɢɟ ɧɚ ɱɢɫɥɨ

x L M R z L : z Mx.

ȼɜɟɞɟɧɧɵɟ ɨɩɟɪɚɰɢɢ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɫɥɟɞɭɸɳɢɦ ɫɜɨɣɫɬɜɚɦ

1)x, y L x y y x;

2)x, y, z L (x y) z x ( y z);

3)ɋɭɳɟɫɬɜɭɟɬ ɧɭɥɟɜɨɣ ɷɥɟɦɟɧɬ

0 L : x L x 0 x ;

4)x L xa L : x xa ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɣ ɷɥɟɦɟɧɬ

5)x L 1¸x x;

6) x L,M,N R M(Nx) (MN)x;

7)x L,M,N R (MN)x Mx Nx;

8)x, y L,M R M(x y) Mx My.

ɗɥɟɦɟɧɬɵ ɥɢɧɟɣɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɧɚɡɵɜɚɸɬ ɜɟɤɬɨɪɚɦɢ ɷɬɨɝɨ ɩɪɨɫɬɪɚɧ ɫɬɜɚ

ȿɫɥɢ ɨɩɪɟɞɟɥɢɬɶ ɨɩɟɪɚɰɢɸ ɭɦɧɨɠɟɧɢɹ ɧɚ ɤɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ ɫ ɬɟɦɢ ɠɟ ɫɜɨɣɫɬɜɚɦɢ ɱɬɨ ɞɚɧɵ ɜ ɨɩɪɟɞɟɥɟɧɢɢ ɞɥɹ ɜɟɳɟɫɬɜɟɧɧɨɝɨ ɫɥɭɱɚɹ ɬɨ ɦɵ ɩɨɥɭ ɱɢɦ ɤɨɦɩɥɟɤɫɧɨɟ ɥɢɧɟɣɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ

ɉɪɢɦɟɪɵ ɥɢɧɟɣɧɵɯ ɩɪɨɫɬɪɚɧɫɬɜ

^ `. Ɇɧɨɠɟɫɬɜɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɜɟɤɬɨɪɨɜ ɩɥɨɫɤɨɫɬɢ ɬɪɟɯɦɟɪɧɨɝɨ ɩɪɨ ɫɬɪɚɧɫɬɜɚ ɫ ɨɩɟɪɚɰɢɹɦɢ ɫɥɨɠɟɧɢɹ ɜɟɤɬɨɪɨɜ ɩɨ ɩɪɚɜɢɥɭ ɬɪɟɭɝɨɥɶɧɢɤɚ ɢ ɭɦɧɨ ɠɟɧɢɹ ɜɟɤɬɨɪɚ ɧɚ ɜɟɳɟɫɬɜɟɧɧɨɟ ɱɢɫɥɨ ɪɚɫɫɦɨɬɪɟɧɧɵɦɢ ɜ ɪɚɡɞɟɥɟ ɹɜɥɹ ɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ

^ `. Ɇɧɨɠɟɫɬɜɨ ɦɧɨɝɨɱɥɟɧɨɜ ɫ ɜɟɳɟɫɬɜɟɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ {Pn (x)}

ɫɬɟɩɟɧɢ ɧɟ ɜɵɲɟ ɡɚɞɚɧɧɨɝɨ ɱɢɫɥɚ n . Ɉɩɪɟɞɟɥɢɦ ɥɢɧɟɣɧɵɟ ɨɩɟɪɚɰɢɢ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ ɩɭɫɬɶ

Ʌɟɝɤɨ ɩɪɨɜɟɪɢɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɦɧɨɝɨɱɥɟɧɨɜ ɫɬɟɩɟɧɢ ɧɟ ɜɵɲɟ n ɫ ɜɜɟ ɞɟɧɧɵɦɢ ɜɵɲɟ ɥɢɧɟɣɧɵɦɢ ɨɩɟɪɚɰɢɹɦɢ ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ Ɂɚ ɧɭɥɟɜɨɣ ɦɧɨɝɨɱɥɟɧ ɜɨɡɶɦɟɦ ɦɧɨɝɨɱɥɟɧ ɜɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɤɨɬɨɪɨɝɨ ɪɚɜɧɵ ɧɭɥɸ. ɉɪɨɬɢɜɨɩɨɥɨɠɧɵɣ ɷɥɟɦɟɧɬ ɞɥɹ ɦɧɨɝɨɱɥɟɧɚ

Pn (x) anxn an 1xn 1 .... a1x a0

65

ɟɫɬɶ

Pna(x) anxn an 1xn 1 .... a1x a0.

^ ` Ɇɧɨɠɟɫɬɜɨ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɜɟɤɬɨɪɨɜ Ⱥɥɝɟɛɪɚɢɱɟɫɤɢɣ ɜɟɤɬɨɪ – ɭɩɨ ɪɹɞɨɱɟɧɧɵɣ ɧɚɛɨɪ ɢɡ n ɜɟɳɟɫɬɜɟɧɧɵɯ ɱɢɫɟɥ ɜɢɞɚ

u1 ¬­­­

u u2­­­­.

...­­­­

un­®

Ⱦɜɚ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɜɟɤɬɨɪɚ

ɫɱɢɬɚɟɦ ɪɚɜɧɵɦɢ u v ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ ɤɨɝɞɚ ui vi , i 1, ..., n. Ɉɩɪɟɞɟɥɢɦ ɥɢɧɟɣɧɵɟ ɨɩɟɪɚɰɢɢ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦɉɭɫɬɶ

Ɍɨɝɞɚ

u1 v1 ¬­­­

u v u2 v2­­­­.

u ... v ­­­­­

n n®

Mu1 ¬­­­

Mu Mu2­­­­.

... ­­­­

Mun­®

66

Ɂɚ ɧɭɥɟɜɨɣ ɷɥɟɦɟɧɬ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜɨɡɶɦɟɦ ɧɚɛɨɪ ɢɡ n ɧɭɥɟɣ

0¬

0­­­­­­­.

...­­­­

0®

ɉɪɨɬɢɜɨɩɨɥɨɠɧɵɣ ɷɥɟɦɟɧɬ

u1 ¬­­­

ua u2­­­­.

... ­­­

un­­®

Ʌɟɝɤɨ ɜɢɞɟɬɶ ɱɬɨ ɞɥɹ ɜɜɟɞɟɧɧɵɯ ɨɩɟɪɚɰɢɣ ɜɵɩɨɥɧɹɸɬɫɹ ɜɫɟ ɫɜɨɣɫɬɜɚ 1)– ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɟ ɦɧɨɠɟɫɬɜɨ ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨ ɫɬɪɚɧɫɬɜɨɦ

Ɉɩɪɟɞɟɥɟɧɢɟ ɉɭɫɬɶ x1, x2,..., xn – ɜɟɤɬɨɪɚ ɧɟɤɨɬɨɪɨɝɨ ɥɢɧɟɣɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ L , B1,B2,...,Bn – ɜɟɳɟɫɬɜɟɧɧɵɟ ɱɢɫɥɚ Ʌɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɟɣ ɜɟɤɬɨɪɨɜ x1, x2,..., xn ɫ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ B1,B2,...,Bn ɧɚɡɵɜɚɟɬɫɹ ɜɟɤɬɨɪ ɜɢɞɚ

B1x1 B2x2 ... Bnxn.

Ɉɩɪɟɞɟɥɟɧɢɟ ɋɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ x1, x2,..., xn ɧɚɡɵɜɚɟɬɫɹ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɨɣ ɟɫɥɢɢɯɥɢɧɟɣɧɚɹɤɨɦɛɢɧɚɰɢɹɪɚɜɧɚɧɭɥɟɜɨɦɭɜɟɤɬɨɪɭɩɪɨɫɬɪɚɧ ɫɬɜɚ ɬɨɥɶɤɨ ɩɪɢ ɧɭɥɟɜɵɯ ɤɨɷɮɮɢɰɢɟɧɬɚɯ B1,B2,...,Bn :

B1x1 B2x2 ... Bnxn B1 0,B2 0,...,Bn 0.

ɋɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ x1, x2,..., xn ɧɚɡɵɜɚɟɬɫɹ ɥɢɧɟɣɧɨ ɡɚɜɢɫɢɦɨɣ ɟɫɥɢ ɫɭ ɳɟɫɬɜɭɟɬ ɧɚɛɨɪ ɤɨɷɮɮɢɰɢɟɧɬɨɜ B1,B2,...,Bn ɧɟ ɜɫɟ ɢɡ ɤɨɬɨɪɵɯ ɪɚɜɧɵ ɧɭɥɸ ɬɚɤɨɣ ɱɬɨɥɢɧɟɣɧɚɹ ɤɨɦɛɢɧɚɰɢɹ ɫ ɷɬɢɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɪɚɜɧɚ ɧɭɥɟɜɨɦɭɜɟɤ ɬɨɪɭ

B1,B2,...,Bn; Bi v0 : B1x1 B2x2 ... Bnxn 0.

ɍɬɜɟɪɠɞɟɧɢɟ ȿɫɥɢ ɫɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ ɫɨɞɟɪɠɢɬ ɧɭɥɟɜɨɣ ɜɟɤɬɨɪ ɬɨ ɨɧɚ ɥɢɧɟɣɧɨ ɡɚɜɢɫɢɦɚ [5].

ɍɬɜɟɪɠɞɟɧɢɟ ɋɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ ɥɢɧɟɣɧɨ ɡɚɜɢɫɢɦɚ ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ

ɬɨɝɞɚ ɤɨɝɞɚ ɯɨɬɹ ɛɵ ɨɞɢɧ ɜɟɤɬɨɪ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɜ ɜɢɞɟ ɥɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚ ɰɢɢ ɨɫɬɚɥɶɧɵɯ ɜɟɤɬɨɪɨɜ [5].

ɉɪɢɦɟɪ Ɋɚɫɫɦɨɬɪɢɦ ɥɢɧɟɣɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɢɡ ɩɪɢɦɟɪɚ {1}.

1)ȼ ɥɢɧɟɣɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɜɟɤɬɨɪɨɜ ɩɥɨɫɤɨɫɬɢ ɥɸ ɛɵɟ ɞɜɚ ɧɟɤɨɥɥɢɧɟɚɪɧɵɯ ɜɟɤɬɨɪɚ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵ Ʌɸɛɵɟ ɞɜɚ

67

ɤɨɥɥɢɧɟɚɪɧɵɯ ɜɟɤɬɨɪɚ ɫɱɢɬɚɟɦ ɧɭɥɟɜɨɣ ɜɟɤɬɨɪ ɤɨɥɥɢɧɟɚɪɧɵɦ ɥɸ ɛɨɦɭ ɜɟɤɬɨɪɭ ɥɢɧɟɣɧɨ ɡɚɜɢɫɢɦɵ Ʌɸɛɵɟ ɬɪɢ ɜɟɤɬɨɪɚ ɩɥɨɫɤɨɫɬɢ ɥɢ ɧɟɣɧɨ ɡɚɜɢɫɢɦɵ

2) Ʌɸɛɵɟ ɬɪɢ ɧɟɤɨɦɩɥɚɧɚɪɧɵɯ ɜɟɤɬɨɪɚ ɬɪɟɯɦɟɪɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɥɢ ɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵ Ʌɸɛɵɟ ɱɟɬɵɪɟ ɜɟɤɬɨɪɚ ɩɪɨɫɬɪɚɧɫɬɜɚ ɥɢɧɟɣɧɨ ɡɚ ɜɢɫɢɦɵ

ɉɪɢɦɟɪ Ɋɚɫɫɦɨɬɪɢɦ ɥɢɧɟɣɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ {2} ɦɧɨɝɨɱɥɟɧɨɜ ɫɬɟ ɩɟɧɢ ɧɟ ɜɵɲɟ ɱɟɦ ɞɜɚ

1) Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɦɧɨɝɨɱɥɟɧɨɜ f1 1, f2 x, f3 x2 ɢ ɩɨɤɚɠɟɦ ɱɬɨ ɨɧɚ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɚ

ɋɨɫɬɚɜɢɦ ɥɢɧɟɣɧɭɸ ɤɨɦɛɢɧɚɰɢɸ ɷɬɢɯ ɦɧɨɝɨɱɥɟɧɨɜ ɢ ɩɪɢɪɚɜɧɹɟɦ ɟɟ ɤ ɧɭɥɸ

B1 f1 B2 f2 B3 f3 B1 B2x B3x2 0.

ɗɬɨ ɪɚɜɟɧɫɬɜɨ ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ ɩɪɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ x, ɫɥɟɞɨɜɚ ɬɟɥɶɧɨ ɜɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɪɚɜɧɵ ɧɭɥɸ B1 0,B2 0,B3 0 ɋɢɫɬɟɦɚ ɜɟɤɬɨ ɪɨɜ f1 f2 f3 ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɚ

2) Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɦɧɨɝɨɱɥɟɧɨɜ f1 x2 x 1 f2 x 1 f3 x2 ɢ ɩɨɤɚɠɟɦ ɱɬɨ ɨɧɚ ɥɢɧɟɣɧɨ ɡɚɜɢɫɢɦɚ

ɋɨɫɬɚɜɢɦ ɥɢɧɟɣɧɭɸ ɤɨɦɛɢɧɚɰɢɸ ɷɬɢɯ ɦɧɨɝɨɱɥɟɧɨɜ ɢ ɩɪɢɪɚɜɧɹɟɦ ɟɟ ɤ ɧɭɥɸ

B1 f1 B2 f2 B3 f3 B1(x2 x 1) B2(x 1) B3x2 0 ɢɥɢ

(B1 B3)x2 (B1 B2)x B1 B2 0.

ɗɬɨ ɪɚɜɟɧɫɬɜɨ ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ ɩɪɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ x, ɩɪɢɪɚɜɧɢɜɚɹ ɤ ɧɭɥɸ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ ɨɞɢɧɚɤɨɜɵɯ ɫɬɟɩɟɧɹɯ x, ɩɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟ ɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɷɮɮɢɰɢɟɧɬɨɜ

ɗɬɚ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɛɟɫɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɧɟɧɭɥɟɜɵɯ ɪɟɲɟɧɢɣ ɧɚɩɪɢ ɦɟɪ B1 1,B2 1,B3 1 ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɫɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ f1 f2 f3 ɥɢ ɧɟɣɧɨ ɡɚɜɢɫɢɦɚ

ɉɪɢɦɟɪɊɚɫɫɦɨɬɪɢɦ ɥɢɧɟɣɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ^ ` ɩɪɢ n 3 ɦɧɨɠɟ ɫɬɜɨ ɭɩɨɪɹɞɨɱɟɧɧɵɯ ɧɚɛɨɪɨɜ ɜɟɳɟɫɬɜɟɧɧɵɯ ɱɢɫɟɥ ɜɢɞɚ

68

u1¬­­­­ u u2­­.

u3­­®

ɉɨɤɚɠɟɦ ɱɬɨ ɫɢɫɬɟɦɚ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɜɟɤɬɨɪɨɜ

ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɚ ɉɪɢɪɚɜɧɢɜɚɟɦ ɤ ɧɭɥɸ ɥɢɧɟɣɧɭɸ ɤɨɦɛɢɧɚɰɢɸ ɷɬɢɯ ɜɟɤ ɬɨɪɨɜ

ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɷɮɮɢɰɢɟɧɬɨɜ

Ʌɟɝɤɨ ɜɢɞɟɬɶ ɱɬɨ ɷɬɚ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɬɨɥɶɤɨ ɧɭɥɟɜɨɟ ɪɟɲɟɧɢɟ ɋɥɟɞɨɜɚ ɬɟɥɶɧɨ ɫɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ u1 u2 u3 ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɚ

2 ɉɨɤɚɠɟɦ ɱɬɨ ɫɢɫɬɟɦɚ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɜɟɤɬɨɪɨɜ

ɥɢɧɟɣɧɨ ɡɚɜɢɫɢɦɚ ɉɪɢɪɚɜɧɢɜɚɟɦ ɤ ɧɭɥɸ ɥɢɧɟɣɧɭɸ ɤɨɦɛɢɧɚɰɢɸ ɷɬɢɯ ɜɟɤɬɨ ɪɨɜ

ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɷɮɮɢɰɢɟɧɬɨɜ

69

ɗɬɚ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɛɟɫɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɧɟɧɭɥɟɜɵɯ ɪɟɲɟɧɢɣ ɧɚɩɪɢ ɦɟɪ B1 1,B2 2,B3 1 ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɫɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ u1 u2 u3 ɥɢ ɧɟɣɧɨ ɡɚɜɢɫɢɦɚ

Ɉɩɪɟɞɟɥɟɧɢɟ Ɇɚɤɫɢɦɚɥɶɧɚɹ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɚɹ ɫɢɫɬɟɦɚ ɜɟɤɬɨ ɪɨɜ ɩɪɨɫɬɪɚɧɫɬɜɚ ɧɚɡɵɜɚɟɬɫɹ ɛɚɡɢɫɨɦ ɷɬɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ

ɗɬɨ ɨɩɪɟɞɟɥɟɧɢɟ ɨɡɧɚɱɚɟɬ ɱɬɨ ɫɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ f1, f2, ..., fn ɥɢɧɟɣɧɨ ɧɟ ɡɚɜɢɫɢɦɚ ɧɨ ɞɥɹ ɥɸɛɨɝɨ ɜɟɤɬɨɪɚ x L ɫɢɫɬɟɦɚ f1, f2, ..., fn, x – ɥɢɧɟɣɧɨ ɡɚɜɢ

ɫɢɦɚ

Ɍɟɨɪɟɦɚ > 7] ȿɫɥɢ f1, f2,..., fn ɢ g1, g2,..., gm – ɞɜɚ ɪɚɡɥɢɱɧɵɯ ɛɚɡɢɫɚ ɩɪɨɫɬɪɚɧɫɬɜɚ L ɬɨ m n

Ɍ ɟ ɤɨɥɢɱɟɫɬɜɨ ɜɟɤɬɨɪɨɜ ɥɸɛɨɝɨ ɛɚɡɢɫɚ ɫɨɜɩɚɞɚɟɬ Ɉɩɪɟɞɟɥɟɧɢɟ Ʉɨɥɢɱɟɫɬɜɨ ɜɟɤɬɨɪɨɜ ɛɚɡɢɫɚ ɧɚɡɵɜɚɟɬɫɹ ɪɚɡɦɟɪɧɨ

ɫɬɶɸ ɥɢɧɟɣɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɉɪɢɦɟɪɊɚɫɫɦɨɬɪɢɦ ɥɢɧɟɣɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ {1} ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ

ɜɟɤɬɨɪɨɜ

1)Ʌɸɛɵɟ ɞɜɚ ɧɟɤɨɥɥɢɧɟɚɪɧɵɯ ɜɟɤɬɨɪɚ ɩɥɨɫɤɨɫɬɢ ɨɛɪɚɡɭɸɬ ɛɚɡɢɫ ɉɪɨ ɫɬɪɚɧɫɬɜɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɜɟɤɬɨɪɨɜ ɩɥɨɫɤɨɫɬɢ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ ɞɜɚ

2)Ʌɸɛɵɟ ɬɪɢ ɧɟɤɨɦɩɥɚɧɚɪɧɵɯ ɜɟɤɬɨɪɚ ɩɪɨɫɬɪɚɧɫɬɜɚ ɨɛɪɚɡɭɸɬ ɛɚɡɢɫ

ɗɬɨ ɩɪɨɫɬɪɚɧɫɬɜɨ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ ɬɪɢ ɉɪɢɦɟɪ Ɋɚɫɫɦɨɬɪɢɦ ɥɢɧɟɣɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ^ ` ɦɧɨɝɨɱɥɟɧɨɜ ɧɟ

ɜɵɲɟ ɜɬɨɪɨɣ ɫɬɟɩɟɧɢ ɉɨɤɚɠɟɦ ɱɬɨ ɫɢɫɬɟɦɚ ɦɧɨɝɨɱɥɟɧɨɜ f1 1 f2 x f3 x2 ɹɜɥɹɟɬɫɹ ɛɚɡɢɫɨɦ ɷɬɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ȼ ɩɪɢɦɟɪɟ 3.2.2 ɛɵɥɨ ɞɨɤɚɡɚɧɨ ɱɬɨ ɷɬɚ ɫɢɫɬɟɦɚ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɚ ɉɭɫɬɶ f ax2 bx c – ɩɪɨɢɡɜɨɥɶɧɵɣ ɜɟɤɬɨɪ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɉɨɤɚɠɟɦ ɱɬɨ ɫɢɫɬɟɦɚ f1, f2, f3, f ɥɢ ɧɟɣɧɨ ɡɚɜɢɫɢɦɚ ȼɟɤɬɨɪ f ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɜ ɜɢɞɟ ɥɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɢ ɜɟɤ ɬɨɪɨɜ f1, f2, f3 :

f ax2 bx c af3 bf2 cf1, ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɨ ɭɬɜɟɪɠɞɟɧɢɸ .2 ɫɢɫɬɟɦɚ f1, f2, f3, f ɥɢɧɟɣɧɨ ɡɚɜɢɫɢɦɚ Ɍ ɨ ɜɟɤɬɨɪɵ f1, f2, f3 ɨɛɪɚɡɭɸɬ ɛɚɡɢɫ ɉɪɨɫɬɪɚɧɫɬɜɨ ɦɧɨɝɨɱɥɟɧɨɜ ɫɬɟɩɟɧɢ ɧɟ ɜɵɲɟ ɱɟɦ ɞɜɚ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ ɬɪɢ

ɉɪɢɦɟɪɉɨɤɚɠɟɦ ɱɬɨ ɫɢɫɬɟɦɚ ɦɧɨɝɨɱɥɟɧɨɜ f1 x2 1 f2 x

f3 x2 x 1 ɹɜɥɹɟɬɫɹ ɛɚɡɢɫɨɦ ɥɢɧɟɣɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ {2} ɦɧɨɝɨɱɥɟɧɨɜ ɧɟ

ɜɵɲɟ ɜɬɨɪɨɣ ɫɬɟɩɟɧɢ Ɍ ɤ ɜ ɩɪɢɦɟɪɟ 3.2 ɛɵɥɨ ɞɨɤɚɡɚɧɨ ɱɬɨ ɥɸɛɨɣ ɛɚɡɢɫ ɷɬɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɫɨɫɬɨɢɬ ɢɡ ɬɪɟɯ ɜɟɤɬɨɪɨɜ ɧɚɦ ɞɨɫɬɚɬɨɱɧɨ ɩɨɤɚɡɚɬɶ ɱɬɨ ɞɚɧɧɵɟ ɜɟɤɬɨɪɵ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵ ɉɪɢɪɚɜɧɹɟɦ ɤ ɧɭɥɸ ɥɢɧɟɣɧɭɸ ɤɨɦɛɢ ɧɚɰɢɸ ɷɬɢɯ ɜɟɤɬɨɪɨɜ

B1 f1 B2 f2 B3 f3 B1(x2 1) B2x B3(x2 x 1) 0.

70