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ÒÅÌÀ 3<br />
ÑÈÑÒÅÌÈ Ë²Í²ÉÍÈÕ ÀËÃÅÁÐÀ¯×ÍÈÕ<br />
вÂÍßÍÜ<br />
3.1. ÏÎ×ÀÒÊÎ<strong>²</strong> ÇÍÀÍÍß ÏÐÎ ÑÈÑÒÅÌÈ<br />
ÀËÃÅÁÐÀ¯×ÍÈÕ Ð²ÂÍßÍÜ<br />
Ñèñòåìè ë³í³éíèõ àëãåáðà¿÷íèõ ð³âíÿíü (ÑËÀÐ) çàéìàþòü<br />
äîñòîéíå ì³ñöå ÿê ó ñàì³é ìàòåìàòèö³, òàê ³ â ¿¿ ïðèêëàäíèõ<br />
ïèòàííÿõ, çîêðåìà ïðè ðîçâ’ÿçàíí³ çàäà÷ ç åêîíîì³êè.<br />
3.1.1. Çîáðàæåííÿ ÑËÀÐ<br />
Íåõàé çàäàíà ñèñòåìà m ë³í³éíèõ ð³âíÿíü â³äíîñíî n<br />
íåâ³äîìèõ õ 1 , õ 2 ,…, õ n . ¯¿ ìîæíà çàïèñàòè ó âèãëÿä³:<br />
⎧a11x1 + a12x2 + ... + a1nxn<br />
= b1<br />
⎪a21x1 + a22 x2 + ... + a2nxn<br />
= b2<br />
⎨<br />
⎪.........................................<br />
⎪<br />
⎩a x + a x + ... + a x = b<br />
m1 1 m2 2<br />
mn n m<br />
. (3.1.1)<br />
Òóò a ij (i = 1,2,…,m; j=1,2,…,n) ³ b i (i = 1,2,…,m) — â³äîì³<br />
÷èñëà, à õ 1 , õ 2 ,…, õ n — íåâ³äîì³.<br />
Ñèñòåìó ð³âíÿíü (3.1.1) ìîæíà çàïèñàòè ó âåêòîðí³é<br />
ôîðì³. Äëÿ öüîãî ââåäåìî â ðîçãëÿä m-âèì³ðí³ âåêòîðè (â<br />
öüîìó ðîçä³ë³ äëÿ çðó÷íîñò³ âåêòîðè áóäåìî ïîçíà÷àòè áåç<br />
ðèñî÷êè çâåðõó)<br />
⎛a11 ⎞ ⎛a12 ⎞ ⎛a1n<br />
⎞ ⎛b1<br />
⎞<br />
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟<br />
a21 a22 a2n<br />
b2<br />
a1 = ⎜ ⎟, a2<br />
= ⎜ ⎟,..., an<br />
= ⎜ ⎟,<br />
b = ⎜ ⎟<br />
⎜... ⎟ ⎜... ⎟ ⎜...<br />
⎟ ⎜...<br />
⎟.<br />
⎜ a ⎟ ⎜ a ⎟ ⎜a<br />
⎟ ⎜b<br />
⎟<br />
⎝ m1⎠ ⎝ m2⎠ ⎝ mn ⎠ ⎝ m⎠<br />
Òîä³ ñèñòåìà (3.1.1) ïðèéìຠâèãëÿä:<br />
à 1 õ 1 + à 2 õ 2 +…+a n x n = b. (3.1.2)<br />
Âèõîäÿ÷è ç ð³âíÿííÿ (3.1.2), ïèòàííÿ ïðî ðîçâ’ÿçí³ñòü<br />
ñèñòåìè (3.1.1) ìîæíà çâåñòè äî ïèòàííÿ ïðî âñòàíîâëåííÿ<br />
ë³í³éíî¿ çàëåæíîñò³ âåêòîð³â à 1 , à 2 ,…,à n ³ b.<br />
Çâåäåìî êîåô³ö³ºíòè ïðè íåâ³äîìèõ â ñèñòåì³ (3.1.1) äî<br />
ìàòðèö³<br />
A<br />
⎛a a ... a<br />
⎜<br />
⎜<br />
a a ... a<br />
... ... ... ...<br />
⎜<br />
⎝a a ... a<br />
11 12 1n<br />
21 22 2n<br />
= ⎜ ⎟<br />
m1 m2<br />
mn<br />
Íåâ³äîì³ õ 1 , õ 2 ,…, õ n çîáðàçèìî ó âèãëÿä³ ìàòðèö³-ñòîâïöÿ<br />
x<br />
⎛x1<br />
⎞<br />
⎜ ⎟<br />
⎜<br />
x<br />
⎟<br />
... .<br />
⎜<br />
x ⎟<br />
⎝ n ⎠<br />
2<br />
= ⎜ ⎟<br />
Âèêîðèñòîâóþ÷è îçíà÷åííÿ äîáóòêó ìàòðèöü, ñèñòåìó<br />
(3.1.1) ìîæíà çàïèñàòè ó âèãëÿä³<br />
Àõ = b. (3.1.3)<br />
Öÿ ôîðìà çàïèñó ñèñòåìè (3.1.1) íàçèâàºòüñÿ ìàòðè÷íîþ.<br />
3.1.2. Îñíîâí³ ïîíÿòòÿ ³ îçíà÷åííÿ<br />
×èñëà a ij , b i íàçèâàþòüñÿ â³äïîâ³äíî êîåô³ö³ºíòàìè ïðè<br />
íåâ³äîìèõ ³ â³ëüíèìè ÷ëåíàìè.<br />
Ðîçâ’ÿçêîì ÑËÀÐ (3.1.1) íàçèâàºòüñÿ òàêà óïîðÿäêîâàíà<br />
ñóêóïí³ñòü ÷èñåë α 1 , α 2 ,…, α n , ùî ïðè â³äïîâ³äí³é çàì³í³ íåâ³äîìèõ<br />
íà ö³ ÷èñëà êîæíå ð³âíÿííÿ ñèñòåìè ñòຠòîòîæí³ñòþ.<br />
Ðîçâ’ÿçàòè ÑËÀÐ (3.1.1) îçíà÷ຠâèçíà÷èòè âñ³ ¿¿ ðîçâ’ÿçêè<br />
àáî äîâåñòè ¿¿ íåðîçâ’ÿçí³ñòü (íå ³ñíóþòü ðîçâ’ÿçêè<br />
ÑËÀÐ (3.1.1)).<br />
⎞<br />
⎟<br />
⎟<br />
.<br />
⎟<br />
⎠<br />
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