Математика для менеджеров. Часть I. Учебное пособие

ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ ɩɪɨɜɟɪɢɬɶ ɱɬɨ ɝɥɚɜɧɵɣ ɨɩɪɟɞɟɥɢɬɟɥɶ ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɵ ɪɚɜɟɧ ɧɭɥɸ

2.5. Ɍɟɨɪɟɦɚ Ʉɪɨɧɟɤɟɪɚ - Ʉɚɩɟɥɥɢ ɋɨɜɦɟɫɬɧɚɹ ɧɟɫɨɜɦɟɫɬɧɚɹ ɨɩɪɟɞɟɥɟɧɧɚɹ ɢ ɧɟɨɩɪɟɞɟɥɟɧɧɚɹ ɫɢɫɬɟɦɚ

Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ P ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɫ Q ɧɟɢɡ ɜɟɫɬɧɵɦɢ

əɫɧɨ ɱɬɨr(A) r(A) ɬɚɤ ɤɚɤ ɤɚɠɞɵɣ ɦɢɧɨɪ ɦɚɬɪɢɰɵ Ⱥ

ɛɭɞɟɬ ɢ ɦɢɧɨɪɨɦ ɦɚɬɪɢɰɵ A ɧɨ ɧɟ ɧɚɨɛɨɪɨɬ

ɋɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɧɚɡɵɜɚɟɬɫɹ ɫɨɜɦɟɫɬɧɨɣ ɟɫɥɢ ɨɧɚ ɢɦɟɟɬ ɯɨɬɹ ɛɵ ɨɞɧɨ ɪɟɲɟɧɢɟ ɢ ɧɟɫɨɜɦɟɫɬɧɨɣ ɟɫɥɢ ɨɧɚ ɧɟ ɢɦɟ ɟɬ ɪɟɲɟɧɢɣ

ɋɨɜɦɟɫɬɧɚɹ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɧɚɡɵɜɚɟɬɫɹ ɨɩɪɟɞɟɥɟɧ ɧɨɣ ɟɫɥɢ ɨɧɚ ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ ɢ ɧɟɨɩɪɟɞɟɥɟɧɧɨɣ, ɟɫɥɢ ɨɧɚ ɢɦɟɟɬ ɛɨɥɟɟ ɨɞɧɨɝɨ ɪɟɲɟɧɢɹ

Ɍɟɨɪɟɦɚ Ʉɪɨɧɟɤɟɪɚ-Ʉɚɩɟɥɥɢ ɤɪɢɬɟɪɢɣ ɫɨɜɦɟɫɬɧɨɫɬɢ ɫɢ ɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɫɢɫɬɟɦɚ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɛɵɥɚ ɫɨɜɦɟɫɬɧɨɣ ɧɟɨɛɯɨɞɢɦɨ ɢ ɞɨɫɬɚɬɨɱɧɨ ɱɬɨɛɵ ɪɚɧɝ ɦɚɬɪɢɰɵ ɫɢɫɬɟɦɵ ɛɵɥ ɪɚɜɟɧ ɪɚɧɝɭ ɟɟ ɪɚɫɲɢɪɟɧɧɨɣ ɦɚɬɪɢ

ɰɵ ɬ ɟ r(A) r(A).

Ɂɚɦɟɱɚɧɢɟ ȿɫɥɢ ɪɚɧɝ ɦɚɬɪɢɰɵ ɫɨɜɦɟɫɬɧɨɣ ɫɢɫɬɟɦɵ ɪɚ ɜɟɧ ɱɢɫɥɭ ɧɟɢɡɜɟɫɬɧɵɯ ɬɨ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟ ɧɢɟ ɟɫɥɢ ɠɟ ɪɚɧɝ ɦɟɧɶɲɟ ɱɢɫɥɚ ɧɟɢɡɜɟɫɬɧɵɯ ɬɨ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɦɧɨɠɟɫɬɜɨ ɪɟɲɟɧɢɣ

ɉɭɫɬɶr n ɬɨɝɞɚ r – ɩɟɪɟɦɟɧɧɵɯ x1,x2 ,...xr ɧɚɡɵɜɚ

ɸɬɫɹ ɨɫɧɨɜɧɵɦɢ ɢɥɢ ɛɚɡɢɫɧɵɦɢ ɟɫɥɢ ɨɩɪɟɞɟɥɢɬɟɥɶ ɦɚɬɪɢɰɵ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɪɢ ɧɢɯ ɬ ɟ ɛɚɡɢɫɧɵɣ ɦɢɧɨɪ ɨɬɥɢɱɟɧ ɨɬ ɧɭɥɹ Ɉɫɬɚɥɶɧɵɟ n r ɧɚɡɵɜɚɸɬɫɹ ɧɟɨɫɧɨɜɧɵɦɢ ɢɥɢ ɫɜɨɛɨɞ ɧɵɦɢ

32

Ɉɛɳɢɦ ɪɟɲɟɧɢɟɦ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɧɚɡɵɜɚɟɬɫɹ ɫɨɜɨ ɤɭɩɧɨɫɬɶ ɜɵɪɚɠɟɧɢɣ ɪɚɡɪɟɲɟɧɧɵɯ ɱɟɪɟɡ ɫɜɨɛɨɞɧɵɟ ɧɟɢɡɜɟɫɬ ɧɵɟ

Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɜ ɤɨɬɨɪɨɣ ɜɫɟ n r ɧɟɨɫɧɨɜɧɵɯ ɩɟɪɟ ɦɟɧɧɵɯ ɪɚɜɧɵ ɧɭɥɸ ɧɚɡɵɜɚɟɬɫɹ ɛɚɡɢɫɧɵɦ.

ɑɚɫɬɧɵɦ ɪɟɲɟɧɢɟɦ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɧɚɡɵɜɚɟɬɫɹ ɪɟ ɲɟɧɢɟ ɩɨɥɭɱɚɸɳɢɟɫɹ ɢɡ ɨɛɳɟɝɨ ɩɪɢ ɤɨɧɤɪɟɬɧɵɯ ɡɧɚɱɟɧɢɹɯ ɫɜɨɛɨɞɧɵɯ ɩɟɪɟɦɟɧɧɵɯ ɢ ɧɟɢɡɜɟɫɬɧɵɯ

ɉɪɢɦɟɪ. Ɇɟɬɨɞɨɦ Ƚɚɭɫɫɚ ɪɟɲɢɬɶ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ

Ɋɟɲɟɧɢɟ ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɦɚɬɪɢɱɧɨɣ ɮɨɪɦɨɣ ɡɚɩɢɫɢ ɫɢ

ɫɬɟɦɵ

ɉɨɥɭɱɢɥɢ ɱɬɨ ɪɚɧɝ ɦɚɬɪɢɰɵ ɫɢɫɬɟɦɵ ɪɚɜɟɧ ɪɚɧɝɭ ɟɟ ɪɚɫɲɢɪɟɧɧɨɣ ɦɚɬɪɢɰɵ ɬ ɟ r(A) r(A) r . ɋɢɫɬɟɦɚ ɫɨɜɦɟɫɬɧɚ

ɫɨɝɥɚɫɧɨ ɬɟɨɪɟɦɟ Ʉɪɨɧɟɤɟɪɚ–Ʉɚɩɟɥɥɢ ɂɦɟɟɦ ɱɬɨ ɪɚɧɝ ɦɚɬɪɢɰɵ ɫɢɫɬɟɦɵ ɪɚɜɟɧ ɱɢɫɥɭ ɧɟɢɡ

ɜɟɫɬɧɵɯ Ɂɧɚɱɢɬ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ ɬɨ ɟɫɬɶ ɨɩɪɟɞɟɥɟɧɚ

33

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɦɵ ɩɪɢɲɥɢ ɤ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɣ

ɂɡ ɩɨɫɥɟɞɧɟɝɨ ɭɪɚɜɧɟɧɢɹ ɧɚɯɨɞɢɦ x4 ɉɨɞɫɬɚɜɥɹɹ x4ɜ

ɉɪɢɦɟɪ ɇɚɣɬɢ ɨɛɳɟɟ ɛɚɡɢɫɧɨɟ ɢ ɨɞɧɨ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ

ɫɢɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ Ɋɟɲɟɧɢɟ Ɂɚɩɢɲɟɦ ɪɚɫɲɢɪɟɧɧɭɸ ɦɚɬɪɢɰɭ ɫɢɫɬɟɦɵ ɢ ɫ

ɩɨɦɨɳɶɸ ɷɥɟɦɟɧɬɚɪɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɪɢɜɟɞɟɦ ɟɟ ɤ ɫɬɭɩɟɧ ɱɚɬɨɦɭ ɜɢɞɭ

Ʉɨ ɜɬɨɪɨɣ ɫɬɪɨɤɟ ɩɪɢɛɚɜɥɹɟɦ ɩɟɪɜɭɸ ɫɬɪɨɤɭ Ʉ ɬɪɟ ɬɶɟɣ ɫɬɪɨɤɟ ɩɪɢɛɚɜɥɹɟɦ ɩɟɪɜɭɸ ɫɬɪɨɤɭ ɭɦɧɨɠɟɧɧɭɸ ɧɚ Ʉ ɱɟɬɜɟɪɬɨɣ ɫɬɪɨɤɟ ɩɪɢɛɚɜɥɹɟɦ ɩɟɪɜɭɸ ɫɬɪɨɤɭ ɭɦɧɨɠɟɧɧɭɸ ɧɚ

34

Ʉ ɬɪɟɬɶɟɣ ɫɬɪɨɤɟ ɩɪɢɛɚɜɥɹɟɦ ɜɬɨɪɭɸ ɫɬɪɨɤɭ ɭɦɧɨ ɠɟɧɧɭɸ ɧɚ – Ʉ ɱɟɬɜɟɪɬɨɣ ɫɬɪɨɤɟ ɩɪɢɛɚɜɥɹɟɦ ɜɬɨɪɭɸ ɫɬɪɨɤɭ ɭɦɧɨɠɟɧɧɭɸ ɧɚ –7.

Ɍɪɟɬɶɹ ɢ ɱɟɬɜɟɪɬɚɹ ɫɬɪɨɤɢ ɨɞɢɧɚɤɨɜɵ ɨɞɧɭ ɢɡ ɧɢɯ ɭɞɚɥɹɟɦ

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ:

ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ.

ɉɪɢɦɟɪ ɇɚ ɩɪɟɞɩɪɢɹɬɢɢ ɢɦɟɟɬɫɹ ɱɟɬɵɪɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɫɩɨɫɨɛɚ ɢɡɝɨɬɨɜɥɟɧɢɹ ɢɡɞɟɥɢɣ Ⱥ ɢ Ȼ ɢɡ ɧɟɤɨɬɨɪɨɝɨ ɫɵɪɶɹ ȼ ɬɚɛɥɢɰɟ ɭɤɚɡɚɧɨ ɤɨɥɢɱɟɫɬɜɨ ɢɡɞɟɥɢɣ ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɛɵɬɶ ɩɪɨ

35

ɢɡɜɟɞɟɧɨ ɢɡ ɟɞɢɧɢɰɵ ɫɵɪɶɹ ɤɚɠɞɵɦ ɢɡ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɫɩɨɫɨ ɛɨɜ

Ɂɚɩɢɫɚɬɶ ɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɨɪɦɟ ɭɫɥɨɜɢɹ ɜɵɛɨɪɚ ɬɟɯɧɨɥɨ ɝɢɣ ɩɪɢ ɩɪɨɢɡɜɨɞɫɬɜɟ ɢɡ ɟɞ ɫɵɪɶɹ ɢɡɞɟɥɢɣ Ⱥ ɢ ɢɡɞɟ ɥɢɣ Ȼ

ɂɡɞɟɥɢɟ

Ɋɟɲɟɧɢɟ Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ x1, x2, x3, x4 ɤɨɥɢɱɟɫɬɜɨ ɫɵɪɶɹ ɤɨ ɬɨɪɨɟ ɫɥɟɞɭɟɬ ɩɟɪɟɪɚɛɨɬɚɬɶ ɩɨ ɤɚɠɞɨɣ ɬɟɯɧɨɥɨɝɢɢ ɱɬɨɛɵ ɜɵ ɩɨɥɧɢɬɶ ɩɥɚɧɨɜɨɟ ɡɚɞɚɧɢɟ ɉɨɥɭɱɢɦ ɫɢɫɬɟɦɭ ɬɪɟɯ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɫ ɱɟɬɵɪɶɦɹ ɧɟɢɡɜɟɫɬɧɵɦɢ

ɂɦɟɟɦ U Ⱥ U Ⱥ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɱɢɫɥɨ ɝɥɚɜɧɵɯ ɧɟɢɡ ɜɟɫɬɧɵɯ ɪɚɜɧɨ ɬɪɟɦ ɨɞɧɨ ɧɟɢɡɜɟɫɬɧɨɟ x4 - ɫɜɨɛɨɞɧɨɟ ɂɫɯɨɞɧɚɹ ɫɢɫɬɟɦɚ ɪɚɜɧɨɫɢɥɶɧɚ ɫɥɟɞɭɸɳɟɣ ɫɢɫɬɟɦɟ

x1 x2 x3 94 x4,x2 5x3 386 2x4,

26x3 2080 9x4.

ɋ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɛɟɫɱɢɫɥɟɧɧɨɟ ɦɧɨɠɟɫɬɜɨ ɪɟɲɟɧɢɣ ɬ ɟ ɧɟɨɩɪɟɞɟɥɟɧɧɚ ɋ ɭɱɟɬɨɦ ɪɟɚɥɶɧɨɝɨ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɫɨɞɟɪɠɚɧɢɹ ɧɟɢɡɜɟɫɬɧɵɟ ɜɟɥɢɱɢɧɵ ɧɟ ɦɨɝɭɬ ɛɵɬɶ ɨɬɪɢɰɚɬɟɥɶɧɵɦɢ ɉɨɥɭɱɚɟɦ ɜɟɤɬɨɪ ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ

36

ɌȿɆȺ ɗɅȿɆȿɇɌɕ ȼȿɄɌɈɊɇɈȽɈ ȺɇȺɅɂɁȺȼɟɤɬɨɪɵ ɧɚ ɩɥɨɫɤɨɫɬɢ ɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ

ȼɟɤɬɨɪɨɦ ɪɚɡɦɟɪɧɨɫɬɢ n ɧɚɡɵɜɚɟɬɫɹ ɭɩɨɪɹɞɨɱɟɧɧɵɣ ɧɚɛɨɪ ɢɡ n ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ

Ȼɭɞɟɦ ɡɚɩɢɫɵɜɚɬɶ ɜɟɤɬɨɪ ɜ ɜɢɞɟ X (x1,x2 ,...,xn ) ɝɞɟ xi (i 1,2,...,n) - ɤɨɨɪɞɢɧɚɬɵ ɤɨɦɩɨɧɟɧɬɵ) ɜɟɤɬɨɪɚ

ȿɫɥɢ ɧɚɩɪɢɦɟɪ ɧɟɤɨɬɨɪɵɣ ɚɜɬɨɦɨɛɢɥɶɧɵɣ ɡɚɜɨɞ ɞɨɥ ɠɟɧ ɜɵɩɭɫɬɢɬɶ ɜ ɫɦɟɧɭ ɥɟɝɤɨɜɵɯ ɚɜɬɨɦɨɛɢɥɟɣ ɝɪɭɡɨɜɵɯɚɜɬɨɛɭɫɨɜ ɤɨɦɩɥɟɤɬɨɜ ɡɚɩɱɚɫɬɟɣ ɞɥɹ ɥɟɝɤɨɜɵɯ ɚɜɬɨɦɨɛɢ ɥɟɣ ɢ ɤɨɦɩɥɟɤɬɨɜ ɞɥɹ ɝɪɭɡɨɜɵɯ ɚɜɬɨɦɨɛɢɥɟɣ ɢ ɚɜɬɨɛɭɫɨɜ ɬɨ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɭɸ ɩɪɨɝɪɚɦɦɭ ɷɬɨɝɨ ɡɚɜɨɞɚ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ ɜɟɤɬɨɪɚ ɢɦɟɸɳɟɝɨ ɩɹɬɶ ɤɨɦɩɨɧɟɧɬ

Ɋɚɡɦɟɪɧɨɫɬɶ - ɷɬɨ ɱɢɫɥɨɦ ɟɝɨ ɤɨɨɪɞɢɧɚɬ n).

Ⱦɥɹ ɜɟɤɬɨɪɨɜ ɧɚ ɩɥɨɫɤɨɫɬɢ n=2, X (x1,x2 ) ɚ ɜ ɩɪɨ

ɫɬɪɚɧɫɬɜɟ Q=3, X (x1,x2 ,x3 ) ,

ȼɟɤɬɨɪɵ ɪɚɜɧɵ ɟɫɥɢ ɨɧɢ ɨɞɧɨɣ ɪɚɡɦɟɪɧɨɫɬɢ ɢ ɢɦɟɸɬ ɪɚɜɧɵɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɤɨɨɪɞɢɧɚɬɵ

ɇɭɥɶ-ɜɟɤɬɨɪ 0 ɧɟ ɫɥɟɞɭɟɬ ɩɭɬɚɬɶ ɫ ɱɢɫɥɨɦ

ɧɭɥɶ

Ⱦɟɣɫɬɜɢɹ ɧɚɞ n-ɦɟɪɧɵɦɢ ɜɟɤɬɨɪɚɦɢ

X Y (x1 y1, x2 y2,..., xn yn ) ɬ ɟ ɩɪɢ ɫɥɨɠɟɧɢɢ ɜɟɤɬɨ ɪɨɜ ɢɯ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɤɨɨɪɞɢɧɚɬɵ ɫɤɥɚɞɵɜɚɸɬɫɹ

(2, –4) + (–2, 4) = (0, 0); (3,0,1) + (0,1,4)+(–1, –7,0) = (2, –6,5).

ɉɪɨɢɡɜɟɞɟɧɢɟɦ ɜɟɤɬɨɪɚ X (x1,x2 ,...,xn ) ɧɚ ɱɢɫɥɨ

ɧɚɡɵɜɚɟɬɫɹ ɜɟɤɬɨɪ X ( x1, x2,..., xn ), ɬ ɟ ɩɪɢ ɭɦɧɨɠɟɧɢɢ

ɜɟɤɬɨɪɚ ɧɚ ɱɢɫɥɨ ɤɚɠɞɚɹ ɟɝɨ ɤɨɨɪɞɢɧɚɬɚ ɭɦɧɨɠɚɟɬɫɹ ɧɚ ɷɬɨ ɱɢɫ ɥɨ

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ɋɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɟɤɬɨɪɨɜ ȼɟɤɬɨɪɧɨɟ ɢ ɥɢɧɟɣɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ

ɋɤɚɥɹɪɧɵɦ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɞɜɭɯ ɜɟɤɬɨɪɨɜ

X (x1,x2 ,...,xn ), ɢ Y (y1, y2 ,..., yn ) ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨ ɪɚɜ ɧɨɟ ɫɭɦɦɟ ɩɪɨɢɡɜɟɞɟɧɢɣ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɤɨɨɪɞɢɧɚɬ ɜɟɤɬɨ

ɪɨɜ X Y x1 y1 x2 y2 ... xn yn .

ɉɪɢɦɟɪ ɉɭɫɬɶ A (2,4)ɢ B ( 1,7).

Ɍɨɝɞɚ A B 2 ( 1) 4 7 26

ɋɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɨɛɥɚɞɚɟɬ ɫɥɟɞɭɸɳɢɦɢ ɫɜɨɣ

Ɇɧɨɠɟɫɬɜɨ ɜɫɟɯ ɜɟɤɬɨɪɨɜ ɪɚɡɦɟɪɧɨɫɬɢ n ɧɚɡɵɜɚɟɬɫɹ ɚɪɢɮɦɟɬɢɱɟɫɤɢɦ n-ɦɟɪɧɵɦ ɜɟɤɬɨɪɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɢ ɨɛɨ ɡɧɚɱɚɟɬɫɹ Rn.

ɗɤɨɧɨɦɢɱɟɫɤɢɟ ɜɟɥɢɱɢɧɵ ɹɜɥɹɸɬɫɹ ɦɧɨɝɨɮɚɤɬɨɪɧɵɦɢɦɧɨɝɨɦɟɪɧɵɦɢ ɢ n-ɦɟɪɧɵɟ ɜɟɤɬɨɪɵ ɫɥɭɠɚɬ ɭɞɨɛɧɨɣ ɮɨɪɦɨɣ ɢɯ ɩɪɟɞɫɬɚɜɥɟɧɢɹ

ɇɚɩɪɢɦɟɪ ɧɟɤɨɬɨɪɵɣ ɧɚɛɨɪ ɬɨɜɚɪɨɜ ɪɚɡɥɢɱɧɵɯ ɫɨɪɬɨɜ ɦɨɠɧɨ ɨɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɜɟɤɬɨɪɨɦ T (t1,t2 ,...,tn ) ɚ ɫɨɨɬɜɟɬ

ɫɬɜɭɸɳɢɟ ɰɟɧɵ – ɜɟɤɬɨɪɨɦ P (p1, p2 ,..., pn ).

Ɇɧɨɠɟɫɬɜɨ U ɨɛɪɚɡɭɟɬ ɥɢɧɟɣɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɟɫɥɢ

ɞɥɹ ɥɸɛɵɯ ɞɜɭɯ ɟɝɨ ɷɥɟɦɟɧɬɨɜ X ɽ U ɢ Y ɽ U ɨɩɪɟɞɟɥɟɧɵ ɨɩɟ ɪɚɰɢɹ ɫɥɨɠɟɧɢɹ X Y U ɢ ɨɩɟɪɚɰɢɹ ɭɦɧɨɠɟɧɢɹ ɥɸɛɨɝɨ ɷɥɟɦɟɧɬɚ ɧɚ ɱɢɫɥɨ X U ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɫɜɨɣɫɬɜɚɦ

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1)X Y Y X ,

2)X 0 X ,

3)(X Y ) X Y ,

4)(X Y ) Z X (Y Z ),

5)X ( X ) 0,

6)( )X X X ,

7)( X ) ( )X ,

8)1 X X ,

ɝɞɟ Z U , 0, – ɧɭɥɟɜɨɣ ɷɥɟɦɟɧɬ (0 U ) ɚ ɤɨɷɮɮɢɰɢɟɧɬɵ Į

ȕ Ȝ – ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɱɢɫɥɚ Ⱦɜɚ ɜɟɤɬɨɪɚ ɧɚɡɵɜɚɸɬɫɹ ɨɪɬɨɝɨɧɚɥɶɧɵɦɢ ɟɫɥɢ ɢɯ ɫɤɚ

Ʌɢɧɟɣɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɫ ɜɜɟɞɟɧɧɵɦ ɫɤɚɥɹɪɧɵɦ ɩɪɨɢɡ ɜɟɞɟɧɢɟɦ ɧɚɡɵɜɚɟɬɫɹ ɟɜɤɥɢɞɨɜɵɦ n-ɦɟɪɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ.

ɉɪɢɦɟɪɵ

1.Ɇɧɨɠɟɫɬɜɨ ɬɪɟɯɦɟɪɧɵɯ ɜɟɤɬɨɪɨɜ 53.

2.Ɇɧɨɠɟɫɬɜɨ ɞɜɭɦɟɪɧɵɯ ɜɟɤɬɨɪɨɜ 52.

3.Ɇɧɨɠɟɫɬɜɨ 51 = R – ɦɧɨɠɟɫɬɜɨ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ

3.3. Ʌɢɧɟɣɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɢ ɧɟɡɚɜɢɫɢɦɨɫɬɶ ɜɟɤɬɨɪɨɜ

ɉɭɫɬɶ A1,A2,...,An – ɜɟɤɬɨɪɵ ɢɡ ɧɟɤɨɬɨɪɨɝɨ ɥɢɧɟɣɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ

Ʌɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɟɣ ɜɟɤɬɨɪɨɜ A1, A2,..., An ɧɚɡɵɜɚɟɬ ɫɹ ɜɵɪɚɠɟɧɢɟ ɜɢɞɚ 1 A1 2 A2 ... n An ɝɞɟ 1, 2,..., n –

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ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɱɢɫɥɚ ɧɚɡɵɜɚɟɦɵɟ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɥɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɢ

Ʌɢɧɟɣɧɚɹ ɤɨɦɛɢɧɚɰɢɹ ɞɚɟɬ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɥɨɠɟɧɢɹ ɜɟɤ ɬɨɪɨɜ ɭɦɧɨɠɟɧɧɵɯ ɧɚ ɱɢɫɥɨ( 1 A1, 2 A2,..., n An ) ɬɚɤɠɟ ɜɟɤ

ɬɨɪ

ɉɪɢɦɟɪɵ 1) 2 (2,5,1) – 4 (1,3,0) + (0,0,1) = (0,-2,3); 2) 3 (5,4) – 5 (-1,2) +2 (-10,-1) = (0,0).

ɉɨɫɥɟɞɧɢɣ ɩɪɢɦɟɪ ɩɨɤɚɡɵɜɚɟɬ ɱɬɨ ɜ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɦɨɠɧɨ ɜ ɪɟɡɭɥɶɬɚɬɟ ɥɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɢ ɜɟɤɬɨɪɨɜ A1, A2 ,..., An

ɩɨɥɭɱɢɬɶ ɧɭɥɟɜɨɣ ɜɟɤɬɨɪ 0 ɩɪɢ ɧɟɧɭɥɟɜɵɯ ɤɨɷɮɮɢɰɢɟɧɬɚɯ ɩɪɢ ɜɫɟɯ ɧɭɥɟɜɵɯ ɤɨɷɮɮɢɰɢɟɧɬɚɯ i 0 ɦɵ ɜɫɟɝɞɚ ɩɨɥɭɱɢɦ 0 ).

ɋɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ ɧɚɡɵɜɚɟɬɫɹ ɥɢɧɟɣɧɨ ɡɚɜɢɫɢɦɨɣ ɟɫɥɢ ɢɡ ɷɬɢɯ ɜɟɤɬɨɪɨɜ ɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ ɧɭɥɟɜɭɸ ɥɢɧɟɣɧɭɸ ɤɨɦɛɢɧɚ ɰɢɸ ɤɨɝɞɚ ɯɨɬɹ ɛɵ ɨɞɢɧ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɟɟ ɨɬɥɢɱɟɧ ɨɬ ɧɭɥɹ Ɍɚɤ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɪɢɦɟɪɟ ɜɟɤɬɨɪɵ -1,2), (-10,- ɥɢɧɟɣ ɧɨ ɡɚɜɢɫɢɦɵ

ɋɢɫɬɟɦɚ ɜɟɤɬɨɪɨɜ ɧɚɡɵɜɚɟɬɫɹ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɨɣ ɟɫ ɥɢ ɢɡ ɷɬɢɯ ɜɟɤɬɨɪɨɜ ɧɟɜɨɡɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ ɧɭɥɟɜɭɸ ɥɢɧɟɣɧɭɸ ɤɨɦɛɢɧɚɰɢɸ ɜ ɤɨɬɨɪɨɣ ɯɨɬɹ ɛɵ ɨɞɢɧ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɛɵɥ ɛɵ

ɨɬɥɢɱɟɧ ɨɬ Ɍ ɟ ɜɟɤɬɨɪɵ A1, A2,..., An ɛɭɞɭɬ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢ ɦɵ ɟɫɥɢ ɪɚɜɟɧɫɬɜɨ 1 A1 2 A2 ... n An 0 ɜɨɡɦɨɠɧɨ ɥɢɲɶ ɩɪɢ ɜɫɟɯ i 0 (i 1.2,...,n). Ɉɱɟɜɢɞɧɨ ɧɢ ɨɞɢɧ ɢɡ ɷɬɢɯ ɜɟɤɬɨ ɪɨɜ ɧɟɥɶɡɹ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɨɫɬɚɥɶɧɵɟ

ɉɪɢɦɟɪ Ȼɭɞɭɬ ɥɢ ɜɟɤɬɨɪɵ A (2,4) ɢ B (5,1) ɥɢɧɟɣɧɨ ɡɚ

ȼ ɪɚɜɧɵɯ ɜɟɤɬɨɪɚɯ ɞɨɥɠɧɵ ɛɵɬɶ ɪɚɜɧɵ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɤɨɨɪɞɢɧɚɬɵ

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