1. Grammik 'Algebra
Orjogwniìthta - Jemelie¸dec Je¸rhma (mèroc 2o)
Tm ma Hlektrolìgwn Mhqanik¸n kai Mhqanik¸n Upologist¸n
Panepist mio JessalÐac
1 DekembrÐou 2014

2. M koc DianÔsmatoc
kxk2 Æ x21
Åx22
Å. . .Åx2
n
kxk Æ
q
xTx

3. Kajetìthta Dianusmˆtwn
kxk2Åkyk2 Æ kx¡yk2
Ï x kai y eÐnai kˆjeta metaxÔ touc ann xTy Æ 0
Ï O arijmìc xTy lègetai eswterikì ginìmeno

4. Orjog¸nia sÔnola dianusmˆtwn
'Ena sÔnolo mh-mhdenik¸n dianusmˆtwn {u1,u2, . . .um} ½ Rn eÐnai
orjog¸nio sÔnolo an ana dÔo ta dianusmatˆ tou eÐnai kˆjeta
metaxÔ touc.

5. Orjog¸nia sÔnola dianusmˆtwn
Ti
'Ena sÔnolo mh-mhdenik¸n dianusmˆtwn {u1,u2, . . .um} ½ Rn eÐnai
orjog¸nio sÔnolo an ana dÔo ta dianusmatˆ tou eÐnai kˆjeta
metaxÔ touc.
8ui6Æ 0 kai gia kˆje zeÔgoc ui,uj, i6Æ j èqoume uuj Æ 0.

6. Orjog¸nia sÔnola dianusmˆtwn
Ti
'Ena sÔnolo mh-mhdenik¸n dianusmˆtwn {u1,u2, . . .um} ½ Rn eÐnai
orjog¸nio sÔnolo an ana dÔo ta dianusmatˆ tou eÐnai kˆjeta
metaxÔ touc.
8ui6Æ 0 kai gia kˆje zeÔgoc ui,uj, i6Æ j èqoume uuj Æ 0.
Parˆdeigma:
u1 Æ
2
4
3
5, u2 Æ
1
2
1
2
4
3
5, u3 Æ
2
1
¡4
2
4
3
5
3
¡2
1

7. Orjog¸nia sÔnola dianusmˆtwn
Ti
'Ena sÔnolo mh-mhdenik¸n dianusmˆtwn {u1,u2, . . .um} ½ Rn eÐnai
orjog¸nio sÔnolo an ana dÔo ta dianusmatˆ tou eÐnai kˆjeta
metaxÔ touc.
8ui6Æ 0 kai gia kˆje zeÔgoc ui,uj, i6Æ j èqoume uuj Æ 0.
Parˆdeigma:
u1 Æ
2
4
3
5, u2 Æ
1
2
1
2
4
3
5, u3 Æ
2
1
¡4
2
4
3
5
3
¡2
1
LÔsh: Prèpei na elègxoume trÐa eswterikˆ ginìmena. OrÐste to
èna:
uT1
u3 Æ 1¢3Å2¢ (¡2)Å1¢1 Æ 3¡4Å1 Æ 0.

8. Kajetìthta & Grammik AnexarthsÐa
Eˆn èna sÔnolo dianusmˆtwn eÐnai
orjog¸nio (anˆ dÔo kˆjeta metaxÔ
touc) tìte autˆ eÐnai grammikˆ
anexˆrthta.

9. 'Ekfrash dianÔsmatoc wc proc orjog¸nia bˆsh
An {u1,u2, . . .um} ½ Rn eÐnai
orjog¸nio sÔnolo ekfrˆste to
opoiod pote stoiqeÐo tou q¸rou
pou autˆ parˆgoun san grammikì
sundoiasmì touc.

10. Orjog¸nioi Upìqwroi
DÔo upìqwroi V kai W tou Ðdiou
dianusmatikoÔ q¸rou (pq tou Rn)
eÐnai orjog¸nioi eˆn kˆje diˆnusma
v 2V eÐnai orjog¸nio se kˆje
diˆnusma w2W.

11. ParadeÐgmata
V Æ
8><
>:
c1
2
140
64
3
75
Åc2
2
¡170
64
3
75
,c1,c2 2 R
9>=
>;
WÆ
8><
>:
d
2
64
3
00
75
¡3
9>=
>;
,d 2 R

12. Orjogwniìthta upoq¸rwn kai bˆseic
DÔo upìqwroi eÐnai orjog¸nioi ann
ta sÔnola twn bˆse¸n touc eÐnai
orjog¸nia metaxÔ touc.

13. Orjogwniìthta upoq¸rwn kai bˆseic
DÔo upìqwroi eÐnai orjog¸nioi ann
ta sÔnola twn bˆse¸n touc eÐnai
orjog¸nia metaxÔ touc.
m
DÔo upìqwroi eÐnai orjog¸nioi ann
kˆje diˆnusma bˆshc tou enìc eÐnai
orjog¸nio se kˆje diˆnusma bˆshc
tou ˆllou.

14. Dojèntoc enìc upìqwrou V ½ Rn, o q¸roc ìlwn
twn orjogwnÐwn dianusmˆtwn ston V lègetai
orjog¸nio sumpl rwma tou V kai sumbolÐzetai me
V?.

15. Dojèntoc enìc upìqwrou V ½ Rn, o q¸roc ìlwn
twn orjogwnÐwn dianusmˆtwn ston V lègetai
orjog¸nio sumpl rwma tou V kai sumbolÐzetai me
V?.
WÆV?)V ÆW?,

16. Dojèntoc enìc upìqwrou V ½ Rn, o q¸roc ìlwn
twn orjogwnÐwn dianusmˆtwn ston V lègetai
orjog¸nio sumpl rwma tou V kai sumbolÐzetai me
V?.
WÆV?)V ÆW?,
¡
V?¢?
ÆV

17. Dojèntoc enìc upìqwrou V ½ Rn, o q¸roc ìlwn
twn orjogwnÐwn dianusmˆtwn ston V lègetai
orjog¸nio sumpl rwma tou V kai sumbolÐzetai me
V?.
WÆV?)V ÆW?,
¡
V?¢?
ÆV
Parˆdeigma:

18. Jemelei¸dec Je¸rhma
N (A) Æ
¡
R(AT)
¢?

19. Jemelei¸dec Je¸rhma
N (A) Æ
¡
R(AT)
¢?
R(AT) Æ (N (A))?

20. Jemelei¸dec Je¸rhma
N (A) Æ
¡
R(AT)
¢?
R(AT) Æ (N (A))?
N (AT) (R(A))?
Æ ¡
R(A) Æ
N (AT)
¢?

21. Jemelei¸dec Je¸rhma

22. Pìrisma
To Ax Æ b èqei lÔsh ann bTy Æ 0 opoted pote
ATy Æ 0

23. Pìrisma
To Ax Æ b èqei lÔsh ann bTy Æ 0 opoted pote
ATy Æ 0
m
To Ax Æ b èqei lÔsh ann to b eÐnai orjog¸nio se
kˆje diˆnusma pou eÐnai orjog¸nio stic st lec tou
A.

24. Pìrisma
To Ax Æ b èqei lÔsh ann bTy Æ 0 opoted pote
ATy Æ 0
m
To Ax Æ b èqei lÔsh ann to b eÐnai orjog¸nio se
kˆje diˆnusma pou eÐnai orjog¸nio stic st lec tou
A.
Parˆdeigma:
x1¡x2 Æ b1
x2¡x3 Æ b2
x3¡x1 Æ b3

25. H apeikìnish tou q¸rou gramm¸n
ston q¸ro sthl¸n eÐnai
antistrèyimh

26. H apeikìnish tou q¸rou gramm¸n
ston q¸ro sthl¸n eÐnai
antistrèyimh
m
Gia kˆje b ston q¸ro sthl¸n
upˆrqei monadikì xr ston q¸ro
gramm¸n

27. Probol se eujeÐa tou Rn
Na brejeÐ h probol p tou b epˆnw
sthn eujeÐa pou orÐzei to a
m
Na brejeÐ to plhsièstero sto b
shmeÐo p thc eujeÐac pou orÐzei to a

28. Probol se eujeÐa tou Rn

29. Probol se eujeÐa tou Rn

30. H probol p enìc dianÔmatoc b 2 Rn
se mia eujeÐa a 2 Rn pou pernˆei apo
to 0

31. H probol p enìc dianÔmatoc b 2 Rn
se mia eujeÐa a 2 Rn pou pernˆei apo
to 0
Ï eÐnai h p Æ aTb
aTaa

32. H probol p enìc dianÔmatoc b 2 Rn
se mia eujeÐa a 2 Rn pou pernˆei apo
to 0
Ï eÐnai h p Æ aTb
aTaa
Ï me antÐstoiqo pÐnaka probol c
P Æ aaT
aTa

33. H probol p enìc dianÔmatoc b 2 Rn
se mia eujeÐa a 2 Rn pou pernˆei apo
to 0
Ï eÐnai h p Æ aTb
aTaa
Ï me antÐstoiqo pÐnaka probol c
P Æ aaT
aTa
Ï pou eÐnai summetrikìc kai tˆxhc 1

34. Parˆdeigma
Ï Probol tou
2
123
64
3
75sto
2
111
64
3
75

35. Parˆdeigma
Ï Probol tou
2
123
64
3
75sto
2
111
64
3
75
Ï PÐnakac probol c sto
2
111
64
3
75

36. Parˆdeigma
Ï Probol tou
2
123
64
3
75sto
2
111
64
3
75
Ï PÐnakac probol c sto
2
111
64
3
75
kai
sto
·
cosµ
sinµ
¸

37. Efarmog

38. Efarmog
UpologÐste x tètoio ¸ste Ax Æ b

39. Efarmog
UpologÐste x tètoio ¸ste Ax Æ b kai x ÝR(A).

40. Efarmog
UpologÐste x tètoio ¸ste Ax Æ b kai x ÝR(A).

41. Elˆqista Tetrˆgwna
An Ax Æ b kai x ÝR(A) tìte

42. Elˆqista Tetrˆgwna
An Ax Æ b kai x ÝR(A) tìte mia
prosèggish thc lÔshc x eÐnai h
lÔsh y tou sust matoc Ay Æ p
ìpou p h probol tou b ston R(A).

43. Elˆqista Tetrˆgwna
Ax Æ b

44. Elˆqista Tetrˆgwna
Ax Æ b)ATAx ÆATb

45. Elˆqista Tetrˆgwna
Ax Æ b)ATAx ÆATb)
x Æ
¡
ATA
¢¡1ATb

46. Elˆqista Tetrˆgwna
Ax Æ b)ATAx ÆATb)
x Æ
¡
ATA
¢¡1ATb
Prˆgmati AT(Ax¡b) Æ 0

47. Elˆqista Tetrˆgwna
Ax Æ b)ATAx ÆATb)
x Æ
¡
ATA
¢¡1ATb
Prˆgmati AT(Ax¡b) Æ 0
Upìjesh: oi st lec tou A eÐnai
grammikˆ anexˆrthtec.

48. Parˆdeigma
2
64
1 4
1 5
0 6
3
75
x Æ
2
456
64
3
75