ЛЕКЦІЇ ² ВПРАВИ

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 âèïëèâàº
λ 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 âèïëèâàº
2λ 1 +λ 3 =0,
λ 1 − 2λ 2 +2λ 3 =0,
λ 2 −λ 3 =0.
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