Аналитическая геометрия и линейная алгебра. Учебное пособие
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ɋɢɫɬɟɦɚ ɢɦɟɟɬ ɬɨɥɶɤɨ ɧɭɥɟɜɨɟ ɪɟɲɟɧɢɟ ɋɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ ɥɢɧɟɣɧɨ ɧɟɡɚ ɜɢɫɢɦɚ ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɨɛɪɚɡɭɟɬ ɛɚɡɢɫ
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ɨɛɪɚɡɭɟɬ ɛɚɡɢɫ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɜɟɤɬɨɪɨɜ {3} ȼ ɩɪɢɦɟɪɟ ɛɵɥɨ ɞɨɤɚɡɚɧɨ ɱɬɨ ɞɚɧɧɚɹ ɫɢɫɬɟɦɚ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɚ ɉɭɫɬɶ
v1¬ v v2® v3
ɹɜɥɹɟɬɫɹ ɩɪɨɢɡɜɨɥɶɧɵɦ ɜɟɤɬɨɪɨɦ ɞɚɧɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɉɨɤɚɠɟɦ ɱɬɨ ɫɢ ɫɬɟɦɚ u1,u2,u3,v ɥɢɧɟɣɧɨ ɡɚɜɢɫɢɦɚ ȼɟɤɬɨɪ v ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɥɢ ɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɢ ɜɟɤɬɨɪɨɜ u1,u2,u3 :
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ɋɨɝɥɚɫɧɨ ɭɬɜɟɪɠɞɟɧɢɸ 3.2 ɫɢɫɬɟɦɚ u1,u2,u3,v ɥɢɧɟɣɧɨ ɡɚɜɢɫɢɦɚ Ɍ ɨ ɜɟɤɬɨɪɵ u1,u2,u3 ɨɛɪɚɡɭɸɬ ɛɚɡɢɫ Ɋɚɫɫɦɚɬɪɢɜɚɟɦɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɢɦɟɟɬ ɪɚɡ ɦɟɪɧɨɫɬɶ ɬɪɢ
71
ɉɪɢɦɟɪɉɨɤɚɠɟɦ ɱɬɨ ɫɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ
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ɹɜɥɹɟɬɫɹ ɛɚɡɢɫɨɦ Ɍ ɤ ɛɚɡɢɫ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɫɨɫɬɨɢɬ ɢɡ ɬɪɟɯ ɜɟɤɬɨɪɨɜ ɬɨ ɞɨɫɬɚɬɨɱɧɨ ɞɨɤɚɡɚɬɶ ɱɬɨ ɞɚɧɧɵɟ ɜɟɤɬɨɪɚ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵ ɉɪɢɪɚɜɧɢɜɚɟɦ ɤ ɧɭɥɸ ɥɢɧɟɣɧɭɸ ɤɨɦɛɢɧɚɰɢɸ
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Ɉɬɧɨɫɢɬɟɥɶɧɨ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ:
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Ⱦɚɧɧɚɹ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɬɨɥɶɤɨ ɧɭɥɟɜɨɟ ɪɟɲɟɧɢɟ ɋɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ u1,u2,u3 ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɚ ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɨɛɪɚɡɭɟɬ ɛɚɡɢɫ
Ɍɟɨɪɟɦɚ [5] ȿɫɥɢɫɢɫɬɟɦɚɜɟɤɬɨɪɨɜ f1, f2,..., fn ɹɜɥɹɟɬɫɹɛɚɡɢɫɨɦɥɢ
ɧɟɣɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ L ɬɨ x L ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɥɢɧɟɣɧɨɣ ɤɨɦ ɛɢɧɚɰɢɢ ɛɚɡɢɫɧɵɯ ɜɟɤɬɨɪɨɜ
x1, x2,..., xn R : x x1 f1 x2 f2 ... xn fn.
ɑɢɫɥɚ x1, x2,..., xn ɧɚɡɵɜɚɸɬɫɹ ɤɨɨɪɞɢɧɚɬɚɦɢ ɪɚɡɥɨɠɟɧɢɹ ɜɟɤɬɨɪɚ x ɩɨ ɡɚ
ɞɚɧɧɨɦɭ ɛɚɡɢɫɭ Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɞɚɧɧɨɣ ɬɟɨɪɟɦɵ ɨɱɟɜɢɞɧɵɦ ɨɛɪɚɡɨɦ ɜɵɬɟɤɚɟɬ ɢɡ ɭɬɜɟɪ
ɠɞɟɧɢɹ ɍɬɜɟɪɠɞɟɧɢɟ ɉɪɢ ɭɦɧɨɠɟɧɢɢ ɜɟɤɬɨɪɚ ɧɚ ɜɟɳɟɫɬɜɟɧɧɨɟ ɱɢɫɥɨ
ɜɫɟ ɟɝɨ ɤɨɨɪɞɢɧɚɬɵ ɭɦɧɨɠɚɸɬɫɹ ɧɚ ɷɬɨ ɱɢɫɥɨ
Mx M(x1 f1 x2 f2 ... xn fn ) (Mx1) f1 (Mx2) f2 ... (Mxn ) fn.
ɉɪɢ ɫɥɨɠɟɧɢɢ ɞɜɭɯ ɜɟɤɬɨɪɨɜ ɢɯ ɤɨɨɪɞɢɧɚɬɵ ɫɤɥɚɞɵɜɚɸɬɫɹ
x y (x1 f1 x2 f2 ... xn fn ) ( y1 f1 y2 f2 ... yn fn )(x1 y1) f1 (x2 y2) f2 ... (xn yn ) fn.
ɍɬɜɟɪɠɞɟɧɢɟ ɫɥɟɞɭɟɬ ɢɡ ɫɜɨɣɫɬɜ 1, 2, ɥɢɧɟɣɧɵɯ ɨɩɟɪɚɰɢɣ ɢɡ ɨɩɪɟ ɞɟɥɟɧɢɹ ɥɢɧɟɣɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ.
72
ɉɪɢɦɟɪ ɇɚɣɞɟɦ ɤɨɨɪɞɢɧɚɬɵ ɜɟɤɬɨɪɚ
x11¬2 ®
ɜ ɛɚɡɢɫɟ ɢɡ ɩɪɢɦɟɪɚ 3.2.8 |
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ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɹ ʋ
ɉɪɢɜɟɫɬɢ ɩɪɢɦɟɪ ɛɚɡɢɫɚ ɧɚ ɦɧɨɠɟɫɬɜɟ ɦɚɬɪɢɰ ɪɚɡɦɟɪɧɨɫɬɢ 2q2 Ⱦɨ ɤɚɡɚɬɶ ɱɬɨ ɷɬɨ ɛɚɡɢɫ
ɉɪɢɜɟɫɬɢ ɩɪɢɦɟɪ ɛɚɡɢɫɚ ɧɚ ɦɧɨɠɟɫɬɜɟ ɦɚɬɪɢɰ ɪɚɡɦɟɪɧɨɫɬɢ 2q3 Ⱦɨɤɚ ɡɚɬɶ ɱɬɨ ɷɬɨ ɛɚɡɢɫ
ɉɪɢɜɟɫɬɢ ɩɪɢɦɟɪ ɛɚɡɢɫɚ ɧɚ ɦɧɨɠɟɫɬɜɟ ɦɚɬɪɢɰ ɪɚɡɦɟɪɧɨɫɬɢ 3q2 Ⱦɨɤɚ ɡɚɬɶ ɱɬɨ ɷɬɨ ɛɚɡɢɫ
ɉɪɢɜɟɫɬɢ ɩɪɢɦɟɪ ɛɚɡɢɫɚ ɧɚ ɦɧɨɠɟɫɬɜɟ ɦɧɨɝɨɱɥɟɧɨɜ ɫɬɟɩɟɧɢ ɧɟ ɜɵɲɟ ɱɟɦ ɬɪɢ Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɷɬɨ ɛɚɡɢɫ
Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɦɧɨɝɨɱɥɟɧɨɜ ɫɬɟɩɟɧɢ ɞɜɚ ɧɟ ɹɜɥɹɟɬɫɹ ɥɢɧɟɣ ɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ
Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɦɧɨɝɨɱɥɟɧɨɜ ɫɬɟɩɟɧɢ ɧɟ ɜɵɲɟ ɱɟɦ ɞɜɚ ɫ ɩɨ ɥɨɠɢɬɟɥɶɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɧɟ ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ
Ⱦɨɤɚɡɚɬɶ ɱɬɨɦɧɨɠɟɫɬɜɨɦɧɨɝɨɱɥɟɧɨɜ ɜɢɞɚ f (x) ax3 b ɹɜɥɹɟɬɫɹ ɥɢ ɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɇɚɣɬɢ ɛɚɡɢɫ ɷɬɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ
Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɦɧɨɝɨɱɥɟɧɨɜ ɜɢɞɚ f (x) ax3 bx2 c ɹɜɥɹ ɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɇɚɣɬɢ ɛɚɡɢɫ ɷɬɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ
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ɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ
Ⱦɨɤɚɡɚɬɶ ɱɬɨɦɧɨɠɟɫɬɜɨɦɧɨɝɨɱɥɟɧɨɜ ɜɢɞɚ f (x) ax3 1 ɧɟɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ
Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɦɧɨɝɨɱɥɟɧɨɜ ɜɢɞɚ f (x) ax2 x b ɧɟ ɹɜ ɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ
Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɬɨɱɟɤ ɩɥɨɫɤɨɫɬɢ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ (x, y), x p0 ɧɟ ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ
Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɬɨɱɟɤ ɩɥɨɫɤɨɫɬɢ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ (x, y), y b0 ɧɟ ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ
Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɬɨɱɟɤ ɩɥɨɫɤɨɫɬɢ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ (1, y) ɹɜɥɹ ɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɇɚɣɬɢ ɛɚɡɢɫ ɷɬɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ
Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɬɨɱɟɤ ɩɥɨɫɤɨɫɬɢ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ (x,3) ɹɜɥɹ ɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɇɚɣɬɢ ɛɚɡɢɫ ɷɬɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ
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ɹɜɥɹɟɬɫɹ |
ɥɢ ɫɢɫɬɟɦɚ a1 (2, 3,1), a2 (3, 1,5), |
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(1, 4,3) ɛɚɡɢɫɨɦ ɧɚ ɦɧɨɠɟɫɬɜɟ ɫɬɪɨɤ ɜɢɞɚ a,b,c . |
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ɛɚɡɢɫɨɦ ɧɚ ɦɧɨɠɟɫɬɜɟ ɫɬɪɨɤ ɜɢɞɚ a,b,c . |
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23. ɉɪɨɜɟɪɢɬɶ |
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ɥɢ ɫɢɫɬɟɦɚ a1 (4, 5,2,6), a2 (2, 2,1,3), |
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(6, 3,3,9), a4 (4, 1,5,6) ɛɚɡɢɫɨɦ ɧɚ ɦɧɨɠɟɫɬɜɟ ɫɬɪɨɤ ɜɢɞɚ a,b,c,d . |
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ɉɪɨɜɟɪɢɬɶ ɹɜɥɹɟɬɫɹ ɥɢ ɫɢɫɬɟɦɚ a1 (5,0,3),a2 (3,1,2),a3 (8,2,3) ɛɚɡɢɫɨɦ ɧɚ ɦɧɨɠɟɫɬɜɟ ɫɬɪɨɤ ɜɢɞɚ a,b,c .
ɉɪɨɜɟɪɢɬɶ ɹɜɥɹɟɬɫɹ ɥɢ ɫɢɫɬɟɦɚ a1 (1,4,3),a2 (2,3,2),a3 ( 1,1,3) ɛɚɡɢɫɨɦ ɧɚ ɦɧɨɠɟɫɬɜɟ ɫɬɪɨɤ ɜɢɞɚ a,b,c .
ɉɪɨɜɟɪɢɬɶ ɹɜɥɹɟɬɫɹ ɥɢ ɫɢɫɬɟɦɚ a1 (5,4,4),a2 (3,3, 1),a3 (8,7,3) ɛɚɡɢɫɨɦ ɧɚ ɦɧɨɠɟɫɬɜɟ ɫɬɪɨɤ ɜɢɞɚ a,b,c .
ɉɪɨɜɟɪɢɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɫɬɪɨɤ ɜɢɞɚ a,b,0,c ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɉɪɢɜɟɫɬɢ ɩɪɢɦɟɪ ɛɚɡɢɫɚ ɷɬɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ
ɉɪɨɜɟɪɢɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɫɬɪɨɤ ɜɢɞɚ 0,a,0,b ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɉɪɢɜɟɫɬɢ ɩɪɢɦɟɪ ɛɚɡɢɫɚ ɷɬɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ
Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɫɬɪɨɤ ɜɢɞɚ a,b,c,1 ɧɟ ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ
Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɫɬɪɨɤ ɜɢɞɚ 1,a,0,b ɧɟ ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ
ɩɪɨɫɬɪɚɧɫɬɜɨɦ
3.3. dzȇșȗȏȝȢ ȏ ȕȖȌȗȇȝȏȏ Ȕȇȋ Ȕȏȓȏ
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ɧɨɦɟɪ ɫɬɪɨɤɢ ɜɬɨɪɨɣ ɢɧɞɟɤɫ j – ɧɨɦɟɪ ɫɬɨɥɛɰɚ ɧɚ ɩɟɪɟɫɟɱɟɧɢɢ ɤɨɬɨɪɵɯ ɫɬɨɢɬ ɞɚɧɧɵɣ ɷɥɟɦɟɧɬ Ɇɚɬɪɢɰɭ ɜɢɞɚ 3.3 ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ A {aij}im, ,nj 1 Ⱦɜɟ ɦɚɬɪɢɰɵ A {aij}im, ,nj 1 ɢ B {bij}im, ,nj 1 ɧɚɡɵɜɚɸɬɫɹ ɪɚɜɧɵɦɢ A B ɟɫɥɢ ɢɯ ɪɚɡɦɟɪɧɨɫɬɢ ɫɨɜɩɚɞɚɸɬ ɢ aij bij i, j .
Ɇɚɬɪɢɰɚ AT {a ji}in,,jm 1 ɧɚɡɵɜɚɟɬɫɹ ɬɪɚɧɫɩɨɧɢɪɨɜɚɧɧɨɣ ɦɚɬɪɢɰɟɣ ɞɥɹ ɦɚɬɪɢɰɵ A.
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ɉɪɨɢɡɜɟɞɟɧɢɟɦ ɦɚɬɪɢɰɵ A {aij}im, ,nj 1 ɧɚ ɱɢɫɥɨ M ɧɚɡɵɜɚɟɬɫɹ ɦɚɬɪɢɰɚ |
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ɬɨɣ ɠɟ ɪɚɡɦɟɪɧɨɫɬɢ C {cij}im, ,nj 1 ɬɚɤɚɹ ɱɬɨ cij Maij . |
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76
ǸȉȕȐȘșȉȇ ȕȖȌȗȇȝȏȐ Ȕȇȋ ȓȇșȗȏȝȇȓȏ
ɉɭɫɬɶ A, B,C – ɦɚɬɪɢɰɵ ɨɞɢɧɚɤɨɜɨɣ ɪɚɡɦɟɪɧɨɫɬɢ
1)A, B A B B A;
2)A, B,C (A B) C A (B C);
3)ɇɚ ɦɧɨɠɟɫɬɜɟ ɦɚɬɪɢɰ ɫɭɳɟɫɬɜɭɟɬ ɧɭɥɟɜɨɣ ɷɥɟɦɟɧɬ
0
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4)A Aa: A Aa ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɣ ɷɥɟɦɟɧɬ
5)A 1¸ A A;
6)A,M,N R M(NA) (MN)A;
7)A,M,N R (MN)A MA NA;
8)A, B, M R M(A B) MA MB.
Ɍ ɨ ɦɧɨɠɟɫɬɜɨ ɦɚɬɪɢɰ ɮɢɤɫɢɪɨɜɚɧɧɨɣ ɪɚɡɦɟɪɧɨɫɬɢ ɫ ɜɜɟɞɟɧɧɵɦɢ ɨɩɟ ɪɚɰɢɹɦɢ ɨɛɪɚɡɭɟɬ ɥɢɧɟɣɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ
Ɉɩɪɟɞɟɥɟɧɢɟ ɉɭɫɬɶ A {aij}im, ,kj 1 ɢ B {bij}ik,n, j 1 ɬ ɟ ɤɨɥɢɱɟɫɬɜɨ ɫɬɨɥɛɰɨɜ ɩɟɪɜɨɣ ɦɚɬɪɢɰɵ ɫɨɜɩɚɞɚɟɬ ɫ ɤɨɥɢɱɟɫɬɜɨɦ ɫɬɪɨɤ ɜɬɨɪɨɣ ɦɚɬɪɢɰɵ ɉɪɨɢɡɜɟɞɟɧɢɟɦ ɦɚɬɪɢɰ A ɢ B ɧɚɡɵɜɚɟɬɫɹ ɦɚɬɪɢɰɚ C {cij}im, ,nj 1 ɫ ɷɥɟɦɟɧɬɚɦɢ
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Ʉɚɤ ɜɢɞɧɨɢɡ ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɩɪɢɜɟɞɟɧɧɵɯ ɩɪɢɦɟɪɨɜ ɨɩɟɪɚɰɢɹ ɩɪɨɢɡɜɟɞɟ ɧɢɹ ɦɚɬɪɢɰ ɧɟ ɨɛɥɚɞɚɟɬ ɫɜɨɣɫɬɜɨɦ ɩɟɪɟɫɬɚɧɨɜɨɱɧɨɫɬɢ ȿɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɩɪɨ ɢɡɜɟɞɟɧɢɟ ɦɚɬɪɢɰ AB ɬɨ ɦɨɠɟɬ ɧɟ ɫɭɳɟɫɬɜɨɜɚɬɶ ɩɪɨɢɡɜɟɞɟɧɢɟ BA ȿɫɥɢ ɨɛɚ ɷɬɢ ɩɪɨɢɡɜɟɞɟɧɢɹ ɫɭɳɟɫɬɜɭɸɬ ɬɨ ɨɧɢ ɦɨɝɭɬ ɢɦɟɬɶ ɪɚɡɧɭɸ ɪɚɡɦɟɪɧɨɫɬɶ ɩɪɢ ɦɟɪ 3.3.3 (3, Ɉɛɚ ɩɪɨɢɡɜɟɞɟɧɢɹ ɨɩɪɟɞɟɥɟɧɵ ɢ ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɭɸ ɪɚɡɦɟɪ ɧɨɫɬɶ ɬɨɥɶɤɨ ɜ ɫɥɭɱɚɟ ɟɫɥɢ ɨɛɟ ɦɚɬɪɢɰɵ A ɢ B ɤɜɚɞɪɚɬɧɵɟ ɨɞɢɧɚɤɨɜɨɣ ɪɚɡ ɦɟɪɧɨɫɬɢ ɉɪɨɢɡɜɟɞɟɧɢɹ AB ɢ BA ɬɚɤɠɟ ɛɭɞɭɬ ɤɜɚɞɪɚɬɧɵɦɢ ɦɚɬɪɢɰɚɦɢ ɬɨɝɨ ɠɟ ɩɨɪɹɞɤɚ ɇɨ ɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ AB v BA.
ɉɪɢɦɟɪ
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Ɉɛɚ ɩɪɨɢɡɜɟɞɟɧɢɹ AB ɢ BA ɨɩɪɟɞɟɥɟɧɵ ɢ ɫɨɜɩɚɞɚɸɬ ɜ ɫɥɭɱɚɟ ɤɨɝɞɚ A ɤɜɚɞɪɚɬɧɚɹ ɦɚɬɪɢɰɚ ɚɦɚɬɪɢɰɚ B ɤɜɚɞɪɚɬɧɚɹ ɦɚɬɪɢɰɚ ɬɨɣ ɠɟɪɚɡɦɟɪɧɨɫɬɢ ɫɩɟ ɰɢɚɥɶɧɨɝɨ ɜɢɞɚ ɧɚ ɞɢɚɝɨɧɚɥɢ ɫɬɨɢɬ ɨɞɧɨ ɱɢɫɥɨ ɚ ɜɫɟ ɨɫɬɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ ɪɚɜɧɵ ɧɭɥɸ
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Ɍɨɝɞɚ AB BA bA ɉɪɨɜɟɪɢɦ ɷɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɞɥɹ ɫɥɭɱɚɹ ɦɚɬɪɢɰ 2q2 . |
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|||||||
16 |
A |
¨ |
7 |
1 |
5¸ |
|
B |
¨ |
5 |
6 |
5¸ |
|
C |
¨ |
4¸ |
|
D 1 5 0 |
|
|
¨ |
|
|
¸ |
|
|
¨ |
|
|
¸ |
|
|
¨ |
¸ |
|
|
|
|
¨ |
2 |
8 |
¸ |
|
|
¨ |
9 |
6 |
¸ |
|
|
¨ |
¸ |
|
|
|
|
© |
6¹ |
|
|
© |
0¹ |
|
|
© |
2¹ |
|
|
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80