2η διάλεξη Γραμμικής άλγεβρας

Grammikèc Exis¸seic Sust mata Grammik¸n Exis¸sewn
Grammik 'Algebra
Grammikˆ Sust mata, Gewmetrik Je¸rhsh
Tm ma Hlektrolìgwn Mhqanik¸n kai Mhqanik¸n Upologist¸n
Panepist mio JessalÐac
23 SeptembrÐou 2014

Qarakthristikˆ Grammik c 'Algebrac
I EÔkolh

I EÔkolh
I 'Omorfh

I EÔkolh
I 'Omorfh
I Qr simh

I EÔkolh
I 'Omorfh
I Qr simh
I Shmantik

ExÐswsh eujeÐac gramm c
b
m
1
} x
y

y y = mx + b
b
m
1
} x

y y = mx + b
b
m
1
} x
m
y mx = b

2 x y = mx + b
b
m
1
} x 1
m
y mx = b
m
x2 mx1 = b

2 x y = mx + b
b
m
1
} x 1
m
y mx = b
m
x2 mx1 = b
m
mx1 + x2 = b

2 x y = mx + b
b
m
1
} x 1
m
y mx = b
m
x2 mx1 = b
m
mx1 + x2 = b
+ (* a26= 0)
a1x1 + a2x2 = b0

Grammik exÐswsh wc proc tic ex c metablhtèc x1; :::; xn:
a1x1 + a2x2 + : : : + anxn = b
ìpou oi suntelestèc a1; :::; an kai endeqomènwc to b eÐnai gnwstˆ
ek twn protèrw.

a1x1 + a2x2 + : : : + anxn = b
ek twn protèrw.
LÔsh eÐnai mia lÐsta arijm¸n s1; :::; sn tètoiwn ¸ste
a1s1 + a2s2 + : : : + ansn = b

a1x1 + a2x2 + : : : + anxn = b
ek twn protèrw.
LÔsh eÐnai mia lÐsta arijm¸n s1; :::; sn tètoiwn ¸ste
a1s1 + a2s2 + : : : + ansn = b
Parˆdeigma 1: Gia thn exÐswsh thc gramm c a1x1 + a2x2 = b:
To zeÔgoc s1; s2 eÐnai lÔsh () to shmeÐo (s1; s2) brÐsketai
pˆnw sthn gramm .

Parˆdeigma 2 Upologismìc tou telikoÔ bajmoÔ sto mˆjhma:
I x1 o bajmìc thc telik c exètashc mou
I x2 o mèsoc ìroc twn bajm¸n twn exetˆsewn proìdou mou
I x3 o mèsoc ìroc twn bajm¸n twn test mou
I x4 o bajmìc summetoq c sto mˆjhma

Parˆdeigma 2 Upologismìc tou telikoÔ bajmoÔ sto mˆjhma:
I x1 o bajmìc thc telik c exètashc mou
I x2 o mèsoc ìroc twn bajm¸n twn exetˆsewn proìdou mou
I x3 o mèsoc ìroc twn bajm¸n twn test mou
I x4 o bajmìc summetoq c sto mˆjhma
Grammik exÐswsh
0:5x1 + 0:4x2 + 0:1x3 + 0:05x4 = 7

SÔsthma grammik¸n exis¸sewn eÐnai èna sÔnolo grammik¸n
exis¸sewn:

exis¸sewn:
a11x1 + a12x2 + : : : + a1nxn = b1
a21x1 + a22x2 + : : : + a2nxn = b2
...
am1x1 + am2x2 + : : : + amnxn = bm

exis¸sewn:
a11x1 + a12x2 + : : : + a1nxn = b1
a21x1 + a22x2 + : : : + a2nxn = b2
...
am1x1 + am2x2 + : : : + amnxn = bm
LÔsh tou sust matoc eÐnai mia lÐsta
s1; :::; sn 2 R
h opoÐa apoteleÐ lÔsh ìlwn twn m exis¸sewn tautìqrona.

exis¸sewn:
a11x1 + a12x2 + : : : + a1nxn = b1
a21x1 + a22x2 + : : : + a2nxn = b2
...
am1x1 + am2x2 + : : : + amnxn = bm
LÔsh tou sust matoc eÐnai mia lÐsta
s1; :::; sn 2 R
h opoÐa apoteleÐ lÔsh ìlwn twn m exis¸sewn tautìqrona.
Dhlad , ìlec oi m exis¸seic alhjeÔoun ìtan
x1 = s1; x2 = s2; :::; xn = sn.

Parˆdeigma
a11x1 + a12x2 = b1
a21x1 + a22x2 = b2

Parˆdeigma
a11x1 + a12x2 = b1
a21x1 + a22x2 = b2
LÔsh , h tom twn dÔo gramm¸n:
x1
x2

Parˆdeigma
a11x1 + a12x2 = b1
a21x1 + a22x2 = b2
LÔsh , h tom twn dÔo gramm¸n:
x1
x2 Parˆdeigma:
1x1 + 2x2 = 3
2x1 + 1x2 = 3
, (x1; x2) = (1; 1)

'Allec pijanìthtec:
x1
2 x
x1
2 x

x1
2 x
x1
2 x
1x1 + 2x2 = 3
1x1 + 2x2 = 4
asunèpeia

x1
2 x
x1
2 x
1x1 + 2x2 = 3
1x1 + 2x2 = 4
asunèpeia
1x1 + 2x2 = 3
2x1 + 4x2 = 6
aoristÐa

Upˆrqoun akrib¸c trÐa endeqìmena

I Den upˆrqei kammÐa lÔsh

I Upˆrqei mia monadik lÔsh

I Upˆrqei apeirÐa lÔsewn

Stìqoi:
I Melèthse to poiì endeqìmeno isqÔei.

Stìqoi:
I D¸se thn lÔsh an eÐnai monadik .

Stìqoi:
I D¸se thn lÔsh an eÐnai monadik .
I Brèc ènan trìpo na perigrˆyeic ìlec tic lÔseic ìtan autèc
eÐnai pollèc.

H krÐsimh plhroforÐa brÐsketai sta aij; bi.

'Ara topojèthse ìlouc autoÔc touc arijmoÔc se ènan pÐnaka
a11x1 + : : : + a1nxn = b1
a21x1 + : : : + a2nxn = b2
...
am1x1 + : : : + amnxn = bm

a11x1 + : : : + a1nxn = b1
a21x1 + : : : + a2nxn = b2
...
am1x1 + : : : + amnxn = bm
2
6664 a11 a12 : : : a1n
a21 a22 : : : a2n
...
...
. . .
...
am1 am2 : : : amn
3
7775
m n pÐnakac suntelest¸n

a11x1 + : : : + a1nxn = b1
a21x1 + : : : + a2nxn = b2
...
am1x1 + : : : + amnxn = bm
2
6664 a11 a12 : : : a1n
a21 a22 : : : a2n
...
...
. . .
...
am1 am2 : : : amn
3
7775
m n pÐnakac suntelest¸n
2
6664
a11 a12 : : : a1n b1
a21 a22 : : : a2n b2
...
...
. . .
...
...
am1 am2 : : : amn bm
3
7775
m (n + 1) epauxhmènoc
pÐnakac

Melet ste to sÔsthma
3x1 + 2x2 + 4x3 = 0
x1 x2 + 2x3 = 1
2x1 + 2x2 + 3x3 = 2
x1 + x3 = 3

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