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Òåìà 2<br />
Îñíîâè àëãåáðè âåêòîð³â ³ ìàòðèöü<br />
2.1. ÂÅÊÒÎÐÈ<br />
2.1.1. Îñíîâí³ ïîíÿòòÿ<br />
Ó øê³ëüíîìó êóðñ³ ìàòåìàòèêè âåêòîð âèçíà÷åíèé ÿê<br />
uuur<br />
íàïðÿìëåíèé â³äð³çîê AB , ó ÿêîìó òî÷êà À ðîçãëÿäàºòüñÿ<br />
ÿê ïî÷àòîê, à òî÷êà  — ê³íåöü. Âåêòîð ìîæå ïîçíà÷àòèñÿ<br />
rrrr<br />
r r<br />
³ îäí³ºþ ë³òåðîþ, íàïðèêëàä abcx , , , , X,<br />
P . Äîâæèíîþ, àáî<br />
uuur<br />
ìîäóëåì, âåêòîðà AB íàçèâàþòü äîâæèíó â³äð³çêà AB ³<br />
uuur r<br />
ïîçíà÷àþòü AB , a .<br />
Ó ïðÿìîêóòí³é ñèñòåì³ êîîðäèíàò, â³äïîâ³äíî íà ïëîùèí³<br />
é ó ïðîñòîð³, âåêòîð uur<br />
X ìîæíà çàïèñàòè ó âèãëÿä³ óïîðÿäêîâàíî¿<br />
ïàðè ÷èñåë (õ 1 , õ 2 ) òà óïîðÿäêîâàíî¿ òð³éêè ÷èñåë<br />
uur<br />
uur<br />
(õ 1 , õ 2 , õ 3 ), òîáòî X = ( x<br />
1, x<br />
2)<br />
(íà ïëîùèí³) ³ X = ( x<br />
1, x<br />
2, x<br />
3)<br />
(ó ïðîñòîð³).<br />
Óçàãàëüíèìî òåïåð ïîíÿòòÿ âåêòîðà uur<br />
X çà äîïîìîãîþ<br />
òàêèõ îçíà÷åíü.<br />
Îçíà÷åííÿ 2.1.1. Äîâ³ëüíèé óïîðÿäêîâàíèé íàá³ð x 1 ,<br />
x 2 ,…, x n ³ç n ä³éñíèõ ÷èñåë íàçèâàºòüñÿ n-âèì³ðíèì âåêòîðîì<br />
uur<br />
X . ×èñëà x 1 , x 2 ,…, x n , ñêëàäîâ³ íàáîðó, íàçèâàþòüñÿ<br />
êîìïîíåíòàìè (êîîðäèíàòàìè) âåêòîðà uur<br />
X . ×èñëî n íàçèâàþòü<br />
ðîçì³ðí³ñòþ âåêòîðà.<br />
Îçíà÷åííÿ 2.1.2. Ñóêóïí³ñòü óñ³õ n-âèì³ðíèõ âåêòîð³â<br />
íàçèâàºòüñÿ n-âèì³ðíèì âåêòîðíèì ïðîñòîðîì.<br />
Êîìïîíåíòè n-âèì³ðíîãî âåêòîðà ìîæíà ðîçòàøîâóâàòè â<br />
ðÿäîê àáî â ñòîâïåöü. Ó ïåðøîìó âèïàäêó ãîâîðÿòü ïðî<br />
âåêòîð-ðÿäîê<br />
uur<br />
X = ( x1, x2,..., x n<br />
),<br />
â äðóãîìó — ïðî âåêòîð-ñòîâïåöü<br />
r<br />
X<br />
⎛x1<br />
⎞<br />
⎜ ⎟<br />
⎜<br />
x<br />
⎟.<br />
...<br />
⎜<br />
⎟<br />
⎝xn<br />
⎠<br />
2<br />
= ⎜ ⎟<br />
Äâà âåêòîðè ð³âí³ òîä³ é ò³ëüêè òîä³, êîëè ð³âí³ ¿õ êîìïîíåíòè.<br />
ßê âèäíî ç îçíà÷åííÿ, ãîâîðèòè ïðî ð³âí³ñòü àáî<br />
íåð³âí³ñòü âåêòîð³â ìîæíà ëèøå äëÿ âåêòîð³â îäí³º¿ ðîçì³ðíîñò³.<br />
Äîâæèíà (ìîäóëü) âåêòîðà îá÷èñëþºòüñÿ çà ôîðìóëîþ:<br />
n<br />
( )<br />
uur<br />
1/ 2<br />
X = x + x + ... + x = ∑ x .<br />
2 2 2 2<br />
1 2 n<br />
i=<br />
1<br />
1<br />
2.1.2. Îïåðàö³¿ íàä âåêòîðàìè<br />
r<br />
r<br />
Ñóìîþ äâîõ âåêòîð³â a=<br />
( a1, a2,..., a n<br />
) ³ b= ( b1, b2,..., b<br />
n)<br />
íàçèâàºòüñÿ<br />
òðåò³é âåêòîð c= ( c1, c2,..., c<br />
r<br />
n<br />
), êîìïîíåíòè ÿêîãî äîð³âíþþòü<br />
ñóì³ êîìïîíåíò äîäàíê³â âåêòîð³â: c1 = a1 + b1,<br />
c2 = a2 + b2,..., cn = an + b<br />
n.<br />
гçíèöåþ a r r<br />
³ b º âåêòîð<br />
r r r<br />
c = a− b= ( a1−b1, a2 −b2,..., an<br />
−bn)<br />
.<br />
Íåõàé ìàºìî âåêòîð a r ³ ä³éñíå ÷èñëî λ. Äîáóòêîì ÷èñëà<br />
λ íà âåêòîð a r º âåêòîð b r , ÿêèé âèçíà÷àºòüñÿ çà ïðàâèëîì<br />
λ a = ( λa1, λa2,..., λa n),<br />
ïðè÷îìó äîâæèíà éîãî äîð³âíþº<br />
r<br />
λ a r .<br />
Äëÿ áóäü-ÿêèõ âåêòîð³â ³ ÷èñåë ñïðàâåäëèâî:<br />
r r r r<br />
1. a+ b = b+<br />
a — çàêîí ïåðåì³ùåííÿ.<br />
r r r r r r<br />
2. ( a+ b) + c = a+ ( b+<br />
c ) — çàêîí ñïîëó÷åííÿ (ïîºäíàííÿ).<br />
r r r r<br />
3. a+ 0 = a(0 = (0,...,0)).<br />
4. Äëÿ áóäü-ÿêîãî a r r<br />
r r<br />
³ñíóº a ' òàêèé, ùî a+ a′ = 0.<br />
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