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Öÿ ñèñòåìà ð³âíÿíü ìຠò³ëüêè íóëüîâèé ðîçâ’ÿçîê:<br />
r r r<br />
λ 1 = λ 2 = λ 3 = 0. Îòæå, âåêòîðè a1, a2,<br />
a ë³í³éíî íåçàëåæí³.<br />
3<br />
2.1.5. Áàçèñ ë³í³éíîãî âåêòîðíîãî ïðîñòîðó<br />
Áàçèñîì ë³í³éíîãî âåêòîðíîãî ïðîñòîðó íàçèâàºòüñÿ<br />
áóäü-ÿêà óïîðÿäêîâàíà ñóêóïí³ñòü âåêòîð³â, ÿêà çàäîâîëüíÿº<br />
âèìîãè:<br />
1) óñ³ âåêòîðè ö³º¿ ñóêóïíîñò³ ë³í³éíî íåçàëåæí³;<br />
2) áóäü-ÿêèé âåêòîð öüîãî ïðîñòîðó º ë³í³éíà êîìá³íàö³ÿ<br />
äàíî¿ ñóêóïíîñò³ âåêòîð³â.<br />
 n-âèì³ðíîìó ë³í³éíîìó ïðîñòîð³ (àáî ë³í³éíîìó ïðîñòîð³<br />
ðîçì³ðíîñò³ n) áóäü-ÿêà ñóêóïí³ñòü ³ç n ë³í³éíî íåçàëåæíèõ<br />
âåêòîð³â óòâîðþº áàçèñ ïðîñòîðó. Çâè÷àéíî â ÿêîñò³<br />
áàçèñó îáèðàþòü íàéá³ëüø ïðîñòó ñóêóïí³ñòü âåêòîð³â. Íàïðèêëàä,<br />
r r r<br />
e1 = (1,0,...,0), e2<br />
= (0,1,...,0),..., e n<br />
= (0,0,...,1) .<br />
Òåîðåìà 2.1.1.  n-âèì³ðíîìó ë³í³éíîìó ïðîñòîð³ ñèñòåìà<br />
âåêòîð³â e1, e2, ..., en<br />
ñêëàäຠáàçèñ öüîãî ïðîñòîðó.<br />
r r r<br />
r r r<br />
Äîâåäåííÿ. 1) Ïîêàæåìî, ùî âåêòîðè e1, e2, ..., en<br />
ë³í³éíî<br />
íåçàëåæí³. Äëÿ öüîãî ñë³ä äîâåñòè, ùî ñï³ââ³äíîøåííÿ<br />
r r r<br />
λ<br />
1e1 +λ<br />
2 e2<br />
+ ... +λ<br />
n<br />
en<br />
= 0<br />
(2.1.1)<br />
ìຠì³ñöå ò³ëüêè ïðè λ 1 = λ 2 =…= λ n = 0. ²ç (2.1.1) ç óðàõóâàííÿì<br />
÷èñëîâèõ çíà÷åíü êîìïîíåíò³â âåêòîð³â å 1 , å 2 ,…,å n ,<br />
âèïëèâàº<br />
λ 1 ⋅1=0, λ 2 ⋅1=0,…, λ n ⋅1=0, òîáòî λ 1 = λ 2 =…=λ n =0.<br />
r<br />
2) Áóäü-ÿêèé âåêòîð a = ( a1, a2,..., a n<br />
)<br />
r r r<br />
º ë³í³éíîþ êîìá³íàö³ºþ<br />
âåêòîð³â e1, e2,..., e ç êîåô³ö³ºíòàìè à<br />
n<br />
1 , à 2 ,…,à n . ijéñíî,<br />
r r r<br />
ae 1 1<br />
+ ... + ae<br />
n n<br />
= a1(1,0,...,0) + ... + an(0,0,...,1) = ( a1, a2,..., an)<br />
= a. (2.1.2)<br />
r r r<br />
Îòæå, ñèñòåìà e1, e2,..., en<br />
º áàçèñîì. Öåé áàçèñ íàçèâàþòü<br />
îðòîíîðìîâàíèì, à ð³âí³ñòü (2.1.2) — ðîçêëàäàííÿì âåêòîðà<br />
a r â ë³í³éíîìó ïðîñòîð³ çà îðòîíîðìîâàíèì áàçèñîì.<br />
Ó òðèâèì³ðíîìó ïðîñòîð³ äëÿ îðòîíîðìîâàíèõ<br />
r r<br />
âåêòîð³â<br />
áàçèñó çàñòîñîâóþòü òàêîæ ïîçíà÷åííÿ i = (1,0,0), j = (0,1,0) ,<br />
r<br />
k = (0,0,1) . Ðîçêëàäàííÿ âåêòîðà a r â òðèâèì³ðíîìó ïðîñòîð³<br />
çà îðòîíîðìîâàíèì áàçèñîì ìຠâèãëÿä:<br />
r r r r r r r<br />
a = a1i+ a2 j+ a3k = axi+ ay j+<br />
azk<br />
,<br />
äå à 1 , à 2 , à 3 ³ à õ , à ó , à z º ð³çí³ ïîçíà÷åííÿ ïðîåêö³é âåêòîðà<br />
a r<br />
íà îñ³ êîîðäèíàò (Oõ, Oó, Oz).<br />
Ìíîæèíà âåêòîð³â äåÿêîãî ë³í³éíîãî ïðîñòîðó íàçèâàºòüñÿ<br />
ë³í³éíèì ï³äïðîñòîðîì (ë³í³éíèì ìíîãîâèäîì), ÿêùî<br />
îïåðàö³¿ äîäàâàííÿ, ð³çíèö³ ³ ìíîæåííÿ íà ÷èñëî äàþòü<br />
âåêòîð ³ç ö³º¿ æ ìíîæèíè. Òàê, äâîâèì³ðíèé ïðîñò³ð º ï³äïðîñòîðîì<br />
òðèâèì³ðíîãî ïðîñòîðó.<br />
2.1.6. Ñêàëÿðíèé äîáóòîê äâîõ âåêòîð³â<br />
r Íà ïëîùèí³ r ñêàëÿðíèì äîáóòêîì äâîõ âåêòîð³â<br />
a = ( a1, a2)<br />
³ b = ( b1, b2)<br />
íàçèâàºòüñÿ ÷èñëî, ÿêå äîð³âíþº äîáóòêó<br />
¿õ äîâæèí íà êîñèíóñ êóòà ì³æ íèìè:<br />
r r r r r r r r<br />
a⋅ b = a b = a b a b. (2.1.3)<br />
( , ) cos( )<br />
∧<br />
Ç êóðñó ìàòåìàòèêè ñåðåäíüî¿ øêîëè â³äîìî, ùî íà ïëîùèí³<br />
îðòè (îðòîíîðìîâàí³ âåêòîðè) i = (1, 0), j = (0,1) âçàºìíî<br />
r r<br />
ïåðïåíäèêóëÿðí³. Òîä³ ç (2.1.3) âèïëèâຠùå îäíå îçíà÷åííÿ<br />
ñêàëÿðíîãî äîáóòêó äâîõ âåêòîð³â a = ( a1, a<br />
r<br />
r<br />
2)<br />
³ b = ( b1, b<br />
2)<br />
.<br />
r r r r r r r r<br />
ab ⋅ = ( ai+ a j)( bi+ b j)<br />
= aa<br />
1 2 1 2 1 2<br />
+ bb<br />
1 2; i= (1,0), j = (0,1) .<br />
Àíàëîã³÷íî ââîäèòüñÿ ñêàëÿðíèé äîáóòîê ³ â n-âèì³ðíîìó<br />
ïðîñòîð³ ðîçì³ðíîñò³ n>2.<br />
Äëÿ r n-âèì³ðíîãî ïðîñòîðó r ñêàëÿðíèé äîáóòîê äâîõ âåêòîð³â<br />
a = ( a1, a2,..., a n<br />
) ³ b = ( b1, b2,..., b n<br />
) âèçíà÷àºòüñÿ (çà àíàëî㳺þ<br />
ç îñòàííüîþ ð³âí³ñòþ) òàêèì ÷èíîì:<br />
r r r r<br />
ab ⋅ = ab , = ab+ ab+ ... + ab<br />
n n . (2.1.4)<br />
( ) 1 1 2 2<br />
36 37