Διανυσματικοί χώροι και υπόχωροι

1. Jemelei¸deic Q¸roi
Grammik 'Algebra
DianusmatikoÐ q¸roi kai upìqwroi
Tm ma Hlektrolìgwn Mhqanik¸n kai Mhqanik¸n Upologist¸n
Panepist mio JessalÐac
30 OktwbrÐou 2014

2. Jemelei¸deic Q¸roi
Tèssera shmantikˆ sÔnola
Ï Mhdenìqwroc N (A)
Ï Q¸roc Sthl¸n R(A)
Ï Q¸roc Gramm¸n R
¡
AT¢
Ï Aristerìc Mhdenìqwroc N
¡
AT¢

3. Jemelei¸deic Q¸roi
Mhdenìqwroc N (A) enìc PÐnaka A2 Rm£n eÐnai to
sÔnolo ìlwn twn dianusmˆtwn x gia ta opoÐa isqÔei
ìti Ax Æ 0.

4. Jemelei¸deic Q¸roi
Mhdenìqwroc N (A) enìc PÐnaka A2 Rm£n eÐnai to
sÔnolo ìlwn twn dianusmˆtwn x gia ta opoÐa isqÔei
ìti Ax Æ 0.
N (A) Æ
©
x 2 Rn : Ax Æ 0
ª

5. Jemelei¸deic Q¸roi
Q¸roc Sthl¸n R(A) enìc pÐnaka A2 Rm£n eÐnai to
sÔnolo ìlwn twn grammik¸n sunduasm¸n twn
sthl¸n tou A.

6. Jemelei¸deic Q¸roi
Q¸roc Sthl¸n R(A) enìc pÐnaka A2 Rm£n eÐnai to
sÔnolo ìlwn twn grammik¸n sunduasm¸n twn
sthl¸n tou A.
R(A) Æ
(
x 2 Rm : x Æ
Xn
kÆ1
)
ckA¤,k,8ck 2 R

7. Jemelei¸deic Q¸roi
Q¸roc Gramm¸n R
¡
AT¢
enìc pÐnaka A2 Rm£n eÐnai
to sÔnolo ìlwn twn grammik¸n sunduasm¸n twn
gramm¸n tou A.

8. Jemelei¸deic Q¸roi
Q¸roc Gramm¸n R
¡
AT¢
enìc pÐnaka A2 Rm£n eÐnai
to sÔnolo ìlwn twn grammik¸n sunduasm¸n twn
gramm¸n tou A.
R
¡
AT¢
Æ
(
x 2 Rn : x Æ
mX
kÆ1
ckAk,¤,8ck 2 R
)

9. Jemelei¸deic Q¸roi
Aristerìc Mhdenìqwroc N
¡
AT¢
enìc pÐnaka A
eÐnai to sÔnolo ìlwn twn dianusmˆtwn x gia ta
opoÐa isqÔei ìti xTAÆ 0.

10. Jemelei¸deic Q¸roi
Aristerìc Mhdenìqwroc N
¡
AT¢
enìc pÐnaka A
eÐnai to sÔnolo ìlwn twn dianusmˆtwn x gia ta
opoÐa isqÔei ìti xTAÆ 0.
N (AT) Æ
©
x 2 Rm : xTAÆ 0
ª

11. Jemelei¸deic Q¸roi
Aristerìc Mhdenìqwroc N
¡
AT¢
enìc pÐnaka A
eÐnai to sÔnolo ìlwn twn dianusmˆtwn x gia ta
opoÐa isqÔei ìti xTAÆ 0.
N (AT) Æ
©
x 2 Rm : xTAÆ 0
ª
N (AT) Æ
©
x 2 Rm :ATx Æ 0
ª

12. Jemelei¸deic Q¸roi
Jewr mata
'Estw ìti h apaloif metatrèpei to sÔsthma Ax Æ b
sto sÔsthma Ux Æ c.
Ï N (A) ÆN (U).

13. Jemelei¸deic Q¸roi
Jewr mata
'Estw ìti h apaloif metatrèpei to sÔsthma Ax Æ b
sto sÔsthma Ux Æ c.
Ï N (A) ÆN (U).
Ï x lÔsh tou Ax Æ b,b 2R(A).

14. Jemelei¸deic Q¸roi
Orismìc
Dianusmatikìc q¸roc eÐnai èna sÔnolo antikeimènwn (pou
sun jwc onomˆzoume dianÔsmata) gia ta opoÐa èqoume orÐsei
tic prˆxeic
Ï ˆjroisma dÔo dianusmˆtwn kai
Ï pollaplasiasmìc enìc dianÔsmatoc me arijmì.

15. Jemelei¸deic Q¸roi
Orismìc
Dianusmatikìc q¸roc eÐnai èna sÔnolo antikeimènwn (pou
sun jwc onomˆzoume dianÔsmata) gia ta opoÐa èqoume orÐsei
tic prˆxeic
Ï ˆjroisma dÔo dianusmˆtwn kai
Ï pollaplasiasmìc enìc dianÔsmatoc me arijmì.
ParadeÐgmata: To sÔnolo twn pragmatik¸n arijm¸n, to
sÔnolo twn dianusmˆtwn sto epÐpedo (ston trisdiˆstato
q¸ro), to sÔnolo twn pragmatik¸n sunart sewn, . . .

16. Jemelei¸deic Q¸roi
Orismìc
Dianusmatikìc upoq¸roc Y enìc dianusmatikoÔ q¸rou V
eÐnai èna uposÔnolo tou V tètoio ¸ste
Ï to ˆjroisma dÔo opoiond pote dianusmˆtwn tou Y na
an kei kai autì sto Y kai
Ï o pollaplasiasmìc opoioud pote dianÔsmatoc Y me ènan
arijmì na an kei kai autì sto Y.

17. Jemelei¸deic Q¸roi
Orismìc
Dianusmatikìc upoq¸roc Y enìc dianusmatikoÔ q¸rou V
eÐnai èna uposÔnolo tou V tètoio ¸ste
Ï to ˆjroisma dÔo opoiond pote dianusmˆtwn tou Y na
an kei kai autì sto Y kai
Ï o pollaplasiasmìc opoioud pote dianÔsmatoc Y me ènan
arijmì na an kei kai autì sto Y.
ParadeÐgmata: To sÔnolo twn dianusmˆtwn tou Rn, to
sÔnolo twn summetrik¸n n£n pinˆkwn, to sÔnolo twn
suneq¸n pragmatik¸n sunart sewn, . . .

18. Jemelei¸deic Q¸roi
Enalaktikìc Orismìc
Dianusmatikìc upoq¸roc Y enìc dianusmatikoÔ q¸rou V
eÐnai èna uposÔnolo tou V tètoio ¸ste kˆje grammikìc
sundoiasmìc twn stoiqeÐwn tou Y an kei sto Y.
Dianusmatikìc upoq¸roc Y enìc dianusmatikoÔ q¸rou V
eÐnai èna uposÔnolo tou V tètoio ¸ste 8x,y 2 Y kai
8®,¯ 2 R, ®xůy 2 Y.

19. Jemelei¸deic Q¸roi
'Askhsh
Poiˆ apo ta parakˆtw sÔnola eÐnai dianusmatikoÐ upìqwroi
1. Oi n£n ˆnw trigwnikoÐ pÐnakec.

20. Jemelei¸deic Q¸roi
'Askhsh
Poiˆ apo ta parakˆtw sÔnola eÐnai dianusmatikoÐ upìqwroi
1. Oi n£n ˆnw trigwnikoÐ pÐnakec.
2. Oi n£n antistrèyimoi pÐnakec.

21. Jemelei¸deic Q¸roi
'Askhsh
Poiˆ apo ta parakˆtw sÔnola eÐnai dianusmatikoÐ upìqwroi
1. Oi n£n ˆnw trigwnikoÐ pÐnakec.
2. Oi n£n antistrèyimoi pÐnakec.
3. Oi lÔseic tou sust matoc Ax Æ b.

22. Jemelei¸deic Q¸roi
'Askhsh
Poiˆ apo ta parakˆtw sÔnola eÐnai dianusmatikoÐ upìqwroi
1. Oi n£n ˆnw trigwnikoÐ pÐnakec.
2. Oi n£n antistrèyimoi pÐnakec.
3. Oi lÔseic tou sust matoc Ax Æ b.
4. Oi lÔseic tou omogenoÔc sust matoc Ax Æ 0.

23. Jemelei¸deic Q¸roi
'Askhsh
Poiˆ apo ta parakˆtw sÔnola eÐnai dianusmatikoÐ upìqwroi
1. Oi n£n ˆnw trigwnikoÐ pÐnakec.
2. Oi n£n antistrèyimoi pÐnakec.
3. Oi lÔseic tou sust matoc Ax Æ b.
4. Oi lÔseic tou omogenoÔc sust matoc Ax Æ 0.
5. To sÔnolo twn dianusmˆtwn (x,y,z) pou an koun sto
epÐpedo z Æ 2.

24. Jemelei¸deic Q¸roi
Je¸rhma
'Estw o m£n pÐnakac A. Tìte
1. O mhdenìqwrìc tou A eÐnai dianusmatikìc upìqwroc tou
Rn.

25. Jemelei¸deic Q¸roi
Je¸rhma
'Estw o m£n pÐnakac A. Tìte
1. O mhdenìqwrìc tou A eÐnai dianusmatikìc upìqwroc tou
Rn.
2. O aristerìc mhdenìqwrìc tou A eÐnai dianusmatikìc
upìqwroc tou Rm.

26. Jemelei¸deic Q¸roi
Je¸rhma
'Estw o m£n pÐnakac A. Tìte
1. O mhdenìqwrìc tou A eÐnai dianusmatikìc upìqwroc tou
Rn.
2. O aristerìc mhdenìqwrìc tou A eÐnai dianusmatikìc
upìqwroc tou Rm.
3. O q¸roc gramm¸n tou A eÐnai dianusmatikìc upìqwroc
tou Rn.

27. Jemelei¸deic Q¸roi
Je¸rhma
'Estw o m£n pÐnakac A. Tìte
1. O mhdenìqwrìc tou A eÐnai dianusmatikìc upìqwroc tou
Rn.
2. O aristerìc mhdenìqwrìc tou A eÐnai dianusmatikìc
upìqwroc tou Rm.
3. O q¸roc gramm¸n tou A eÐnai dianusmatikìc upìqwroc
tou Rn.
4. O q¸roc sthl¸n tou A eÐnai dianusmatikìc upìqwroc
tou Rm.