5. 1⋅ a= a . r r 6. ( αβ ) a = α( βa) , α ³ β — ä³éñí³ ÷èñëà. r r r r 7. α ( a+ b) =α a+αb. r r r 8. ( α+β )a =α a+βa . 2.1.3. ˳í³éíèé âåêòîðíèé ïðîñò³ð Âåêòîðíèé ïðîñò³ð íàçèâàºòüñÿ ë³í³éíèì, ÿêùî â íüîìó âèçíà÷åí³ îïåðàö³¿ äîäàâàííÿ ³ ìíîæåííÿ íà ÷èñëî ç âëàñòèâîñòÿìè 1 – 8. 2.1.4. ˳í³éíà íåçàëåæí³ñòü âåêòîð³â r r r Íåõàé a1, a2,..., an — âåêòîðè ðîçì³ðíîñò³ n. Òîä³ âåêòîð r r r r b =λ 1a +λ 1 2 a + ... +λ 2 na n , äå λ i ( i = 1, n ) — ä³éñí³ ÷èñëà, íàçèâà- ºòüñÿ ë³í³éíîþ êîìá³íàö³ºþ âåêòîð³â. Îçíà÷åííÿ 2.1.3. Ñóêóïí³ñòü âåêòîð³â a r , a r ,..., a r 1 2 n (ïðè n ≥ 2) íàçèâàºòüñÿ ë³í³éíî çàëåæíîþ, ÿêùî õî- ÷à á îäèí ³ç öèõ âåêòîð³â º ë³í³éíîþ êîìá³íàö³ºþ ³íøèõ. Çîêðåìà, íà ïëîùèí³ áóäü-ÿê³ òðè âåêòîðè ë³í³éíî çàëåæí³, òîìó ùî îäèí ³ç íèõ ìîæíà çàïèñàòè ÿê ë³í³éíó êîìá³íàö³þ äâîõ ³íøèõ (ðèñ. 2.1). Ðèñ. 2.1 a r =λ 3 1a r +λ 1 2a r . 2 Îçíà÷åííÿ 2.1.4. ßêùî æîäíèé ç ñóêóïíîñò³ âåêòîð³â a r , a r , ..., a r íå º ë³í³éíîþ êîìá³íàö³ºþ ³íøèõ, òî ö³ âåêòîðè 1 2 íàçèâàþòüñÿ n ë³í³éíî íåçàëåæíèìè. r r r Äëÿ ë³í³éíî¿ íåçàëåæíîñò³ âåêòîð³â a , a , ..., a íåîáõ³äíî ³ äîñòàòíüî, ùîá ð³âí³ñòü 1 2 n r r r λ 1a +λ 2 a + ... +λ a = 0 1 2 n n âèêîíóâàëàñÿ ò³ëüêè ïðè λ 1 =λ 2 = ... =λ n = 0 . Öå òâåðäæåííÿ áåçïîñåðåäíüî âèïëèâຠç îçíà÷åííÿ ë³í³éíî¿ íåçàëåæíîñò³ âåêòîð³â. ² íàâïàêè, ÿêùî çàçíà÷åíà ð³âí³ñòü ìຠì³ñöå, êîëè íå âñ³ ÷èñëà λ 1 , λ 2 , …, λ n äîð³âíþþòü íóëþ, òî ñóêóïí³ñòü âåêòîð³â r r r a1, a 2,..., a n ë³í³éíî çàëåæíà. r r r Ïðèêëàä 2.1.1. Âåêòîðè a = (2,1,0), a = (0,1,1) ³ a = (4,5,3) 1 2 3 r r r ë³í³éíî çàëåæí³, òîìó ùî 2a + 3a − a = 0. 1 2 3 Ñïðàâä³ 2(2,1,0) + 3(0,1,1) – (4,5,3) = (4,2,0) + (0,3,3) + (–4, –5, –3) = = (4–4, 2+3–5, 3–3) = (0,0,0) = 0. r r r Ïðèêëàä 2.1.2. Âåêòîðè b 1 = (1,0,0) , b 2 = (0,12,0) ³ b 3 = (0,0,72) r r r ë³í³éíî íåçàëåæí³. ijéñíî, ³ç λ 1b1 +λ 2b2 +λ 3b 3 = 0 âèïëèâຠλ 1 (1,0,0) + λ 2 (0,12,0) + λ 3 (0,0,72) = (λ 1 , 12λ 2 , 72λ 3 ) ³ âåêòîð (λ 1 ,12λ 2 ,72λ 3 ) áóäå íóëüîâèì, ÿêùî λ 1 = λ 2 = λ 3 =0. Ïðèêëàä 2.1.3. Çíàéäåìî âñ³ çíà÷åííÿ λ 1 , λ 2 , λ 3 , ïðè ÿêèõ âèêîíóºòüñÿ ð³âí³ñòü r r r r r r λ 1a +λ 1 2a +λ 2 3a = 0 , äå a = (2,1,0), a = (0, − 2,1), a = (1,2, −1) . 3 1 2 3 ϳäñòàâëÿþ÷è â ð³âí³ñòü êîìïîíåíòè âåêòîð³â, îòðèìàºìî r r r λ a +λ a +λ a =λ (2,1,0) +λ (0, − 2,1) +λ (1,2, − 1) = 1 1 2 2 n n 1 2 3 =(2λ 1 ,λ 1 ,0) + (0,–2λ 2 ,λ 2 )+(λ 3 ,2λ 3 ,–λ 3 )= =(2λ 1 + λ 3 ,λ 1 –2λ 2 +2λ 3 ,λ 2 −λ 3 ) = (0,0,0). Ç óìîâè ð³âíîñò³ äâîõ âåêòîð³â âèïëèâຠ2λ 1 +λ 3 =0, λ 1 − 2λ 2 +2λ 3 =0, λ 2 −λ 3 =0. 34 35
Öÿ ñèñòåìà ð³âíÿíü ìຠò³ëüêè íóëüîâèé ðîçâ’ÿçîê: r r r λ 1 = λ 2 = λ 3 = 0. Îòæå, âåêòîðè a1, a2, a ë³í³éíî íåçàëåæí³. 3 2.1.5. Áàçèñ ë³í³éíîãî âåêòîðíîãî ïðîñòîðó Áàçèñîì ë³í³éíîãî âåêòîðíîãî ïðîñòîðó íàçèâàºòüñÿ áóäü-ÿêà óïîðÿäêîâàíà ñóêóïí³ñòü âåêòîð³â, ÿêà çàäîâîëüíÿº âèìîãè: 1) óñ³ âåêòîðè ö³º¿ ñóêóïíîñò³ ë³í³éíî íåçàëåæí³; 2) áóäü-ÿêèé âåêòîð öüîãî ïðîñòîðó º ë³í³éíà êîìá³íàö³ÿ äàíî¿ ñóêóïíîñò³ âåêòîð³â.  n-âèì³ðíîìó ë³í³éíîìó ïðîñòîð³ (àáî ë³í³éíîìó ïðîñòîð³ ðîçì³ðíîñò³ n) áóäü-ÿêà ñóêóïí³ñòü ³ç n ë³í³éíî íåçàëåæíèõ âåêòîð³â óòâîðþº áàçèñ ïðîñòîðó. Çâè÷àéíî â ÿêîñò³ áàçèñó îáèðàþòü íàéá³ëüø ïðîñòó ñóêóïí³ñòü âåêòîð³â. Íàïðèêëàä, r r r e1 = (1,0,...,0), e2 = (0,1,...,0),..., e n = (0,0,...,1) . Òåîðåìà 2.1.1.  n-âèì³ðíîìó ë³í³éíîìó ïðîñòîð³ ñèñòåìà âåêòîð³â e1, e2, ..., en ñêëàäຠáàçèñ öüîãî ïðîñòîðó. r r r r r r Äîâåäåííÿ. 1) Ïîêàæåìî, ùî âåêòîðè e1, e2, ..., en ë³í³éíî íåçàëåæí³. Äëÿ öüîãî ñë³ä äîâåñòè, ùî ñï³ââ³äíîøåííÿ r r r λ 1e1 +λ 2 e2 + ... +λ n en = 0 (2.1.1) ìຠì³ñöå ò³ëüêè ïðè λ 1 = λ 2 =…= λ n = 0. ²ç (2.1.1) ç óðàõóâàííÿì ÷èñëîâèõ çíà÷åíü êîìïîíåíò³â âåêòîð³â å 1 , å 2 ,…,å n , âèïëèâຠλ 1 ⋅1=0, λ 2 ⋅1=0,…, λ n ⋅1=0, òîáòî λ 1 = λ 2 =…=λ n =0. r 2) Áóäü-ÿêèé âåêòîð a = ( a1, a2,..., a n ) r r r º ë³í³éíîþ êîìá³íàö³ºþ âåêòîð³â e1, e2,..., e ç êîåô³ö³ºíòàìè à n 1 , à 2 ,…,à n . ijéñíî, r r r ae 1 1 + ... + ae n n = a1(1,0,...,0) + ... + an(0,0,...,1) = ( a1, a2,..., an) = a. (2.1.2) r r r Îòæå, ñèñòåìà e1, e2,..., en º áàçèñîì. Öåé áàçèñ íàçèâàþòü îðòîíîðìîâàíèì, à ð³âí³ñòü (2.1.2) — ðîçêëàäàííÿì âåêòîðà a r â ë³í³éíîìó ïðîñòîð³ çà îðòîíîðìîâàíèì áàçèñîì. Ó òðèâèì³ðíîìó ïðîñòîð³ äëÿ îðòîíîðìîâàíèõ r r âåêòîð³â áàçèñó çàñòîñîâóþòü òàêîæ ïîçíà÷åííÿ i = (1,0,0), j = (0,1,0) , r k = (0,0,1) . Ðîçêëàäàííÿ âåêòîðà a r â òðèâèì³ðíîìó ïðîñòîð³ çà îðòîíîðìîâàíèì áàçèñîì ìຠâèãëÿä: r r r r r r r a = a1i+ a2 j+ a3k = axi+ ay j+ azk , äå à 1 , à 2 , à 3 ³ à õ , à ó , à z º ð³çí³ ïîçíà÷åííÿ ïðîåêö³é âåêòîðà a r íà îñ³ êîîðäèíàò (Oõ, Oó, Oz). Ìíîæèíà âåêòîð³â äåÿêîãî ë³í³éíîãî ïðîñòîðó íàçèâàºòüñÿ ë³í³éíèì ï³äïðîñòîðîì (ë³í³éíèì ìíîãîâèäîì), ÿêùî îïåðàö³¿ äîäàâàííÿ, ð³çíèö³ ³ ìíîæåííÿ íà ÷èñëî äàþòü âåêòîð ³ç ö³º¿ æ ìíîæèíè. Òàê, äâîâèì³ðíèé ïðîñò³ð º ï³äïðîñòîðîì òðèâèì³ðíîãî ïðîñòîðó. 2.1.6. Ñêàëÿðíèé äîáóòîê äâîõ âåêòîð³â r Íà ïëîùèí³ r ñêàëÿðíèì äîáóòêîì äâîõ âåêòîð³â a = ( a1, a2) ³ b = ( b1, b2) íàçèâàºòüñÿ ÷èñëî, ÿêå äîð³âíþº äîáóòêó ¿õ äîâæèí íà êîñèíóñ êóòà ì³æ íèìè: r r r r r r r r a⋅ b = a b = a b a b. (2.1.3) ( , ) cos( ) ∧ Ç êóðñó ìàòåìàòèêè ñåðåäíüî¿ øêîëè â³äîìî, ùî íà ïëîùèí³ îðòè (îðòîíîðìîâàí³ âåêòîðè) i = (1, 0), j = (0,1) âçàºìíî r r ïåðïåíäèêóëÿðí³. Òîä³ ç (2.1.3) âèïëèâຠùå îäíå îçíà÷åííÿ ñêàëÿðíîãî äîáóòêó äâîõ âåêòîð³â a = ( a1, a r r 2) ³ b = ( b1, b 2) . r r r r r r r r ab ⋅ = ( ai+ a j)( bi+ b j) = aa 1 2 1 2 1 2 + bb 1 2; i= (1,0), j = (0,1) . Àíàëîã³÷íî ââîäèòüñÿ ñêàëÿðíèé äîáóòîê ³ â n-âèì³ðíîìó ïðîñòîð³ ðîçì³ðíîñò³ n>2. Äëÿ r n-âèì³ðíîãî ïðîñòîðó r ñêàëÿðíèé äîáóòîê äâîõ âåêòîð³â a = ( a1, a2,..., a n ) ³ b = ( b1, b2,..., b n ) âèçíà÷àºòüñÿ (çà àíàëî㳺þ ç îñòàííüîþ ð³âí³ñòþ) òàêèì ÷èíîì: r r r r ab ⋅ = ab , = ab+ ab+ ... + ab n n . (2.1.4) ( ) 1 1 2 2 36 37
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14.5. ÃÐÀÔ²ÊÈ ÄÅßÊÈÕ
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14.5.11. y = arctg x (ðèñ. 14.19
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14.7. ÍÀÁËÈÆÅÍÅ ÇÍÀ×Å
- Page 261:
Ê36 Êåðåêåøà Ï. Â. Ëå