Ε. ΚΑΠΠΟΣ - ΓΡΑΜΜΙΚΗ ΑΛΓΕΒΡΑ

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,

, 2006

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Copyright by E Kappos, 2006

.

LaTEX

kerkis.

, , 10/2006.

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v , v, v

v.

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. , v .

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vi

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vii

1 1

2 3

2.1 . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 . . . . . . . . . . . . . . 7

2.3 . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 . . . . . . . . . . . . . . . . 16

2.4.1 . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.2 . . . . . . . . . . . . . . . . . . 20

2.4.3 R3 . . . . . . . . . . . . . . 25

2.5 . . . . . . . . . . 30

2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 33

3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 R2: . . . . . . . . . . . . . . . . . . 33

3.3 R3: . . . . . . . . . . . . 35

4 37

4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 . . . . . . . . . . . . . . . . . . . . . 40

4.2.1 . . . . . . . . . . . . . . . . . . . 43

4.3 ;

. . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 . . . . . . . . . . . . . . . . . . . 49

4.5 , 52

4.6 : . . . 54

4.6.1 . 56

4.6.2 . . . . . . . . . . . . . . . . . . . . 56

4.6.3 Gauss . . . . . . . . . . . . . . . . . . 59

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viii

4.7 . . . . . . . . . . 62

4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 65

5.1 . . . . . . . 65

5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3 . . . . . . . . . . . . . . . . . . 67

5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.5 . . . . . . . . . . 74

5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

I 77

: 79

.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

85

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1

, ,

.

(-

) , -

.

(.. -

)

.

-

:

:

-

.

:

s1, s2, . . . , s30 m1, m2, . . . , m45. sicij

mj . Si , ;

Xj, j = 1, . . . , 45 , :

c11X1+ c12X2+ + c1,45X45 = S1c21X1+ c22X2+ + c2,45X45 = S2

c30,1X1+ c30,2X2+ + c30,45X45 = S30

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2 1.

,

.

: , -

,

(inventory allocation and control.) -

,

.

(Linear Programming.)

:

(

Descartes), -

.

,

Rn.

:

.

,

, .

: -

( -

)

. -

:

, Googlec,

-

!

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2

2.1

v , . -

( ,

.)

,

. -

0. , , . -

, . -

, . ,

v1 + (v2 +v3) = (v1 +v2) +v3. , :

v+ w= (x1, y1, z1) + (x2, y2, z2) = (x1+ x2, y1+ y2, z1+ z2).

v -

v( .) ,

v= (x,y,z) = (x,y,z).

: -

n- (n = 2 , n = 4...),

a= (a1, a2, . . . , an),

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4 2.

a b n-, , ,

a + b= (a1+ b1, a2+ b2, . . . , an+ bn).

a= (a1, a2, . . . , an).

2.1. -

: R3 (

.) , -

( .) , ,

, . -

,

.

,

. 2.3.

2.1. F= R C. V F-- (.. F = R .. F = C)

v, w V v + wV v V, F, v V :

1. v, w V,v + w= w + v,2. v, w, u V,(v+ w) + u= v+ (w+ u),3. V 0, 0 + v= v

v V,

4. , F,v, w V,(v+ w) =v + w, ( + )v= v+ v, (v) = ()v,

5. 1 F, v V: 1v= v,6. 0 F, v V: 0v= 0.

2.2. 0 0.

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2.1 5

. , :

0= 0v= (1 + (1))v= 1v+ (1)v= v + (1)v,

(1)v, v, v v V.

1

Rn: , n- n- . - 0 = (0, 0, . . . , 0)(n.) n. - R2 () R3 (

), n 3 .

Cn: n- n- v = (z1, z2, . . . , z n)(zi C i), , C

v= (z1, z2, . . . , z n).

n = 1 C. C ,

!

z= 0 w

w= z = w/z.

, ( )

( ): -

z C z:

C= {z, C}.

2.1. C ,

R C, .

1 - , .

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6 2.

n = 2 , Cn , , -

2n(.. n = 2, C2 4.) -

,

Fourier

.

2.3.

-

!

: -

n(n >0).

p(x) =a0+ a1x+ a2x2 + . . .+ anx

n

(n+ 1)-v = (a0, a1, a2, . . . , an) Rn+1. , , p(x) + q(x) p(x) v.

Rn+1.

-

:

: -

,

: f, g , x f(x)+g(x) f(x). . : f x,f(x), .

, .

: ( )

( v = {a0, a1, . . . , an, . . .}) , - ( .)

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2.2 7

2.2

0 -

R3: v - , v

L(v) ={v; R}. - R3,v1, v2. :

:= {1v1+ 2v2; 1, 2 R}.

.

(), v2=v1 R, = 0. .

R3 :

1. 0 (1, 2 = 0.)2. w < v1, v2 >, w < v1, v2 >(

1, 2.)3. w1, w2 , w1+w2 ( v= 1v1+2v2,

w= 1v1+ 2v2 1+ 1, 2+ 2.)

4. v < v1, v2 >, R, v < v1, v2 >(:1, 2.)

< v1, v2> .

2.2. W V V V V.

, v, w W, R, v + wv W.

. R3

R3,

.

2.1. v1, v2, . . . , vk - V. ,< v1, v2, . . . , vk >, V.

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8 2.

.

v1

-. v2,

v1, v2

v1. : v1, v2 , v3, v1, v2, ( ) v3 < v1, v2>. :

2.3.

. v1, v2, . . . , vk , -

vk+1, v1, v2, . . . , vk, vk+1 vk+1 / . v1, v2, . . . , vk , k

< v1, v2, . . . , vk>.

-

, -.

:

2.4. v1, v2, . . . , vk 1, 2, . . . , k

F, , -

1v1+ 2v2+ . . .+ kvk =0.

v1, v2, . . . , vk . ,

v1, v2, . . . , vk , i.

! -

, -,

. , -

-, : (1, 2) = (0, 0)

1v1+ 2v2 = 0.

( , 2. v2=v1, = 1/2.)

,

,

:

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2.2 9

2.1.

(0, 1, 0), (0, 0, 1)(0, 1, 1) R3;,

(1) (0, 1, 0) + (1) (0, 0, 1) + (1) (0, 1, 1) = (0, 0, 0), .

2.2. -

(1, 1, 0), (0, 1, 1) (2, 1, 1)? x, y,z i :

x(1, 1, 0) +y(0, 1, 1) +z(2, 1, 1) = (0, 0, 0) - -

:x + 0 + 2z = 0x y z = 00 + y+ z = 0

2.2. .

; ;

2.5. V (..). V,dim V, .

V.

0 - (.. .) ..

..

,

, -

.

:

2.2. -

. ,

..V .{b1, b2, . . . , bn} ..V n.

v V x1, x2, . . . , xn v= x1b1+ x2b2+ . . .+ xnbn.

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10 2.

x1, . . . , xn v {b1, . . . , bn}.

:

! .

2.6. Rn, n

e1 = (1, 0, . . . , 0), e2= (0, 1, 0, . . . , 0), , en= (0, . . . , 0, 1).

(

!) ( . -

Rn) , .

2.3. R3

b1 = (0, 1, 1), b2= (1, 0, 1), b3= (0, 0, 2).

. R3

v = (3, 1, 1) . , ,

b1, b2, b3, (1, 2, 3)

=

(0, 0, 0) 1b1+ 2b2+ 3b3=0.

,

(2, 1, 1 2+ 23) = (0, 0, 0), 2 = 0, 1 = 0 3 = 0, -. b1, b2, b3 .

x1, x2, x3

v= x1b1+ x2b2+ x3b3.

(3, 1, 1) = (x2, x1, x1 x2+ 2x3),

x2= 3, x1= 1 x3 =11+3

2 = 0, 5. -

v {b1, b2, b3} (1, 3, 0, 5) (3, 1, 1). .

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2.3 11

2.1. v = (c1, c2, . . . , cn)

Rn

{e1, e2, . . . , en) c1, c2, . . . , cn.

2.4. -

. , -

( 2.2)

(;)

2.3

,

,

.

: ,

. ;

, S,

(s1, s2), , s1 s2 ,

,

, (s1 s2)s3, , s1 s2 S; ,

. , ,

V.2 , , , R2 R3. R

. R3

, v1 v2( .)

R2 -

! v1 = (c1, c2)

2 , ,

, :

(x1, x2, . . . , xn) (y1, y2, . . . , yn) = (x1y1, x2y2, . . . , xnyn).

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12 2.

z1 = c1+ ic2, . , v2= (d1, d2), :

v1 v2= (c1c2 d1d2, c1d2+ c2d1),

z1z2= (c1+ ic2)(d1+ id2) = (c1c2 d1d2) +i(c1d2+ c2d1). .

, , , (z1z2)z3=z1(z2z3) , ..(z1+ z2)z3=z1z3+ z2z3.

4, ( - R4!)

, ,

,

.

, (v1, v2)v1 v2. , v1 v2 v3 (;)

,

. :

2.7. V, v1, v2 ,v1 v2 R V :

1. v1 v2=v2 v1( ),2. v v 0 v v= 0 v= 0( ),3. (v1) v2 = (v1 v2),4. (v

1+ v

2)

w= v1

w + v2

w().

-

.

Rn :

v1, v2 Rn, v1 = (x1, x2, . . . , xn),v2= (y1, y2, . . . , yn)v1 v2=x1y1+ x2y2+ . . .+ xnyn.

. Rn -

, n.

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2.3 13

,

v v= x21+ x22+ . . .+ x2n = ||v||2.( -

n: , v =(x1, x2) -,||v|| =

x21+ x22, ,

.)

Rn. , R3

,

v1 v2 = 2x1y1+ 3x2y2+ x3y3, . -

( -

!) ,

v1 v2= 2x1y1+ 3x2y2+ x3y3 x1y2 x2y1. . -

: ,

, - ,

! , , -

,

. , , . 2.8. v1, v2 V ( ) v1 v2 = 0.

Rn,e1 = (1, 0, . . . , 0),e2= (0, 1, 0, . . . , 0), ... en= (0, . . . , 0, 1) : ei

ej = 0i

=j .

2.4. (1, 1, 3), (3, 0, 1) R3 :(1, 1, 3) (3, 0, 1) = (1)(3) + 0 + (3)(1) = 0.

-

;

(x,y,z) :

(1, 1, 3) (x,y,z) =x y+ 3z= 0, (3, 0, 1) (x,y,z) = 3x + 0 z= 0.

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14 2.

, z = 3x, y = x+ 3z = 10x.

, , :

L = {(1, 10, 3); R}.

(

.)

:

. ,

( ) . , ,

{e1, e2, . . . , en} Rn: v= (x1x2, . . . , xn) x1, x2, . . . , xn! (2.1)

2.9. {b1, b2, . . . , bn} V - ,

i =j , b1 bj = 0, , . ||bi|| = 1 i(. ||v|| =

v v).

2.3.

{b1, b2, . . . , bn

} V.

v V n v :

x1 = v b1, x2=v b2, . . . , xn = v bn.. {bi}ni+1 , v - ,v = x1b1 + x2b2 + . . . + xnbn. bi:

bi

v= bi

(x1b1+ x2b2+ . . .+ xnbn) =

n

j=1

xjbi

bj =xi,

{bi}.

, -

. ,

,

. , -

Gram-Schmidt. 10.

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2.3 15

: -

v1 v2 =x1y1+. . . xnyn . ,

v2=v1, ,

v1 v2=v1 v1=||v1||2. , -. , v1 v2 , .

: -

( -, .)

( ) v1 b1, ,

v2 < b1, b2 >( .)

2.3. .

v1 (x1, 0, . . . , 0), !

v2 = (y1, y2, . . . , yn)

.

, y1 v2 b1( .) ,

v1 v2= (x1, 0, . . . , 0) (y1, y2, . . . , yn) =x1y1, .

, ,

:

2.2. v1, v2 -

Rn

v1 v2 = ||v1| | | |v2|| cos , (0 180).2.5. (3, 1, 1)(1, 2, 0) .

:

||(3, 1, 1)|| =

32 + 12 + (1)2 =

11,

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16 2.

2.1: . () 0 , ()

()

||(1, 2, 0)|| =

(1)2 + (2)2 =

5.

,

cos =(3, 1, 1) (1, 2, 0)

11

5=

511

5

= cos1

511

= 132, 4

2.4

-

R2 () R3 (-

).

,

.

, ,

! (x, y x, y,z) : , :

(x,y,z), ; ,

,

R3, x, yz( , ;)

( R2

R3) ()

( 0). ,

.

. n-, Rn - e1 = (1, 0, . . . , 0), e2= (0, 1, 0, . . . , 0)...

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2.4 17

R3 ,

e1, e2, e3 i,j, k. v R3, v= xi + yj + zk,

x, y,z v. , -

, i,j, k . , (i,j, k). . (j, k, i)(k, i,j). , (j, i, k) -.

2.4.1

.

e1 = (1, 0), e2 = (0, 1). v = xe1+ye2 v = (x, y) R2 (. 2.1).3 .

v1= (x1, y1)v2 = (x2, y2)

v1

v2 = x1x2+ y1y2

||v|| =

x2 + y2.

e1 e2

( e1

e2.) (e1, e2) . - (b1, b2) b2 b190 ( .)

2.3. v = (x, y) - , v = (y, x) v (v, v) R2.

.

||v|| = 1( , ||v||.).

3

(x1, x2, . . . , xn) Rn, (x, y) .

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18 2.

.

(ABC) A = v1, B = v2 C = v3. AB , ,

|AB| = ||v2 v1|| =

(x2 x1)2 + (y2 y1)2.

, -

:

2.4. v1, v2 - , v1 = (x1, y1), v2 = (x2, y2), ,

x1y2 x2y1.

a b: E=ab sin , ( 90) .. -

, : v1, ,

(x1, 0) , v2 = (x2, y2), y2 x1y2( x2y1 = 0.)

A

B

C

x1

x2

y2

y1

x1+ x

2

C/

B/

0

2.2:

, .

v1, v2 , , A, B,C v1, v1 + v2, v2 , 0ABC.

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2.4 19

x1CB(x1 + x2) x1CB(x1 + x2) ( 0CC ABB .) :

E(x1CB(x1+ x2)) =1

2(y1+ (y1+ y2)) x2 = y1x2+

1

2x2y2

E(x1CB(x1+ x2) =

1

2

x1x2

y2+ y2+x1x2

y2

x2=x1y2+

1

2x2y2.

( Cx1 .) . , , .

:

, -

e1

.

.

v = (x, y) = xe1+ ye2 .

,

x= v e1= ||v||||e1|| cos = ||v|| cos , e1 . r= ||v||, x= r cos .r2 = x2 +y2 y2 = r2 x2 = r2(1 cos2 ) = r2 sin2 . -,y = r sin . v -.

0 < 2( 0 .)

x= r cos , y = r sin (0 0, - .

2.4. v =0 (r, ), - - (r, ), (0, )[0, 2).

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20 2.

2.5. ,

: (r, ) .

.

, , 0

r.

2.4.2

R3. e1 =(1, 0, 0), e2 = (0, 1, 0) e3 = (0, 0, 1)

i,j k e1 = i, e2 = j, e3 = k. v =(x,y,z)( ) v= xi + yj + zk. x, y,z v . {i,j, k} .

(

: ,

.)

xi, yjzk v -

/ ( .) -

v

||v|| =

x2 + y2 + z2,

v1 v2= (x1, y1, z1) (x2, y2, z2) =x1x2+ y1y2+ z1z2 R:||v|| = v v.

:

v1 v2= ||v1||||v2|| cos , ( [0, ]) . v :

v i= xi i + yj i + zk i= xi i= x,i j= 0, i k= 0. , ,

x= v i= ||v||||i|| cos 1 = ||v|| cos 1,

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2.4 21

1 vi. ,

y=v j= ||v|| cos 2 z=v k= ||v|| cos 3,2, 3 v j, k. - v

, :

||v||2 =x2 + y2 + z2 = ||v||2(cos2 1+ cos2 2+ cos2 3), :

cos2 1+ cos2 2+ cos

2 3 = 1

2.6.

: (1, 2, 3, r = ||v||). ,

. ,

(1, 2, r) (-) R3. : 2 1!

2.4. |2| cos1

1 cos2 1. 1 =.

2= ; .

, -

.

R3: -

,

: ,

, -

. ,

, -

. ,

i

Greenwich .

r= ||v|| . . :

2.10. R3 {i,j, k}, -v, :

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22 2.

1. kv(v

k=

||v

||cos ),

2. i v < i,j >, vzk, [0, 2).

3. r= ||v||, v. (,,r) - v

(x,y,z)

v= xi + yj + zk :

x = r sin cos y = r sin sin z = r cos

-

.

2.5. R3

( < k > k, ) (0, ) [0, 2) (0, ), (,,r).

k(-

0), -

. (x,y,z)

= cos1(v

k

||v||), = tan1

y

x, r= x2 + y2 + z2.

. , x2 +y2 = 0(- < i,j>.)

=

tan1 yx

x 0, y 0+ tan1 y

x x 0

tan1 yx

+ x 0, y 02+ tan1 y

x x >0, y

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2.4 23

,

, . ,

.

R3

r( ), ,

.

, , r ,

. , ,

, r = 0 x = y = 0 .

R3: -

< i,j >, z,

= x2 + y2, = tan1 yx

, z.

, -

(,,z)

x = cos y = sin z = z

= 0, , , - .

2.6. R3 < k > (0, )[0, 2)R, (,,z).

= z = =.

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24 2.

R3: .

, -

: (, ) (, +0). -

.

,

.

, -

. , .

,

.

2.11. T V W , v1, v2V 1, 2 R,

T(1v1+ 2v2) =1T(v1) +2T(v2).

R3

R3.

.

2.12. R3, -

(rotation) , -

( , ) , -

.

Rot : R3 R3. R3 - SO(3).

-

. ,

:

2.13. T :R3

R3

,

T(v1) T(v2) =v1 v2. : -

, -

, .

. O(R3). .

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2.4 25

:

, ,

(,,r) r, -(+0, +0, r), 0, 0, , r. , , 00 - , 0, . -

, . ,

(,,r) (, +0, r) (,,r) (+0, , r) . -, ,

r 0, - ,

,

! , -

, Roti, Rotj, Rotk. .

-

: i

( ). -

!

,

/ SO(3) .

2.4.3 R3

.

. -

.

,

.

-

.

2.14.v1 = (x1, y1, z1), v2 = (x2, y2, z2)

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26 2.

, v1

v2

v1 v2 = (y1z2 z1y2)i + (z1x2 x1z2)j + (x1y2 y1x2)k (2.1)

2.6. (2, 1, 4) (1, 1, 0)

(2, 1, 4) (1, 1, 0) = (4)i + (4)j + (2 1)k

:

1. :

v (v1+ v2) =v v1+ v v2

v1 (v2) = (v1) v2=(v1 v2),0 v= 0.

2.

v

v= 0

3. ,

v1 v2= v2 v1( (v1+ v2) (v1+ v2) = 0, .)

(1) (2)

(;)

.

2.5. v1 = (x1, y1, z1), v2 = (x2, y2, z2) , n

v1, v2 (v1, v2, n) R3. ,

v1 v2= (||v1||||v2|| sin ) n,

(0 ).

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2.4 27

||v1

||||v2

|| |sin

|

v1 v2.

, , v1 v2, ,

v1, v2 - .

, ,

( .)

, -

F r,F r.

. (u,v,w) v1= (x1, y1, z1),v2= (x2, y2, z2)

x1u + y1v+ z1w = 0

x2u + y2v+ z2w = 0

w ,4

u = wy1z2 z1y2x1y2 y1x2

v = w

z1x2

x1z2x1y2 y1x2

, w = x1y2 y1x2, u = y1z2 z1y2v = z1x2 x1z2.

(y1z2 z1y2)i + (z1x2 x1z2)j + (x1y2 y1x2)k

v1 v2.

v1 v2. ||v1||||v2|| | sin | . ,

, -

( ,

.)

-

:

4 x1y2y1x2 -. , ,

.

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28 2.

v1 v2 =

i j kx1 y1 z1x2 y2 z2

( .)

2.7.

(3, 1, 0) (2, 1, 1) =

i j k

3 1 02 1 1

= (1)i + (3)j + (3 + 2)k

:,

(v1 v2) v3=v1 (v2 v3)

.

2.5.

(v1 v2) v3=0, v1 (v2 v3) =0; 2.6.

1.(v1 v2) v3+ (v2 v3) v1+ (v3 v1) v2=0

2.

(v1 v2) v3= (v1 v3)v2 (v2 v3)v1,

:

v1 v2 v3 = v1 (v2 v3)

: (v1 v2) v3 ! ;

:

2.7. v1, v2, v3 ,

.

2.6. -

.

5/21/2018 . -

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2.4 29

:

:

v1 v2 v3=

x1 y1 z1x2 y2 z2x3 y3 z3

. ,

,

F Stokes.

5/21/2018 . -

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30 2.

2.5 -

: -

( R C.)

.

.

: n - n. -

. . ,

.

:

.

, .

R2 R3

((i,j)(i,j, k))

R3 :

(x1, x2, x3) (y1, y2, y3) =x1y1+ x2y2+ x3y3 R.

: -

,

. :

R3 -

.

: ,

, -

.

:

v1 v2 =

i j k

x1 y1 z1x2 y2 z2

.

5/21/2018 . -

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2.6 31

2.6

1. -

.

2. R4

v= (1, 3, 0, 4), w= (2, 0, 3, 1). 2v w v 3w.

3. C2

v= (1 +i, 1 + 2i), w= (2i, 1 i) v+ (2i)w(1 + 2i)v.

4.

{(1, 0), (0, 1)}, {(i, 0), (0, i)}, {(0, 1), (i, 0)} C2.

5.

(1, 2, 1), (2, 1, 0) (1, 0, 0) R3.

6. v V {b1, b2, . . . , bn} .

7. (2, 2, 4) b1 = (1, 2, 0),b2= (1, 1, 1)b3 = (0, 2, 3).

8. ;

() v1, v2, . . . , vk , (.. v1, v2, . . . , vk1) -.

() v1, v2, . . . , vk , (.. v1, v2, . . . , vk1) -.

() {b1, b2, . . . , bn}{c1, c2, . . . , cn} -V. bi - {c1, c2, . . . , cn}.

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32 2.

9. n Rn+1. -

p(x), q(x),p(x) q(x),

.

Rn+1 ( n) ; n = 1 ;

10. Gram-Schmidt: {b1, b2, . . . , bn} V. e1 = b1/

||b1

||.

e2, e3, . . . en , e1, V.

E2=b2(b2 e1)e1 e1. - e2=E2/||E2||. - : E3=b3 (b3 e1)e1 (b3 e2)e2. E2 e1 e2. e3=E3/||E3|| , e4, . . . , en V. - Ei

.

11. v (1, 2, 1) (0, 1, 1). - ,

R3.

12. e1= (1, 0, 0), e2= (0, 1, 0), e3 = (0, 0, 1)R3.

b1= (cos , sin , 0) b2 = ( sin , cos , 0)

.

b1 b2 b1, b2.

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3

3.1

( 16 ) -

,

.

(Rene Descartes, 15961650) -

.

,

-, .

,

,

.

.

3.2 R2:

:

L = {(x, y) R2; ax+ by= c}, a,b,c R.

b = 0,

y=x+ , (= a/b, =c/a.)

b = 0 . , (

5/21/2018 . -

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34 3.

)

n0. n0 = (a, b) ax+ by (x, y) n0:

n0 (x, y) =ax+ by= c. , : (x, y) (x1, y1) , n0:

(x x1, y y1) (a, b) = 0.c = 0 R2. c

= 0,

(affine).

(x, y) !,

! -

, ,

. , (x, y) :

ax+ by = cx+ dy =

3.1. ax +by = cx +dy =( ), ,

(c,d,) =(a,b,). , (c, d) =(a, b).

, ,

.

.

:

v0= (x0, y0) R >0 () R v0 :

C(v0, R) = {v R2;|v v0| =R}. , :

C(v0, R) = {(x, y) R2; (x x0)2 + (y y0)2 =R2}., :

x2 + y2 + dx+ ey = f, d= 2x0, e= 2y0, f=R2 x20 y20.

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3.3 R3: 35

3.1.

, x0, y0, R;

: (x, y) :

ax2 + by2 + cxy+ dx + ey = f .

,

. ( -

), , ,

...

3.3 R3:

: n0 = (a,b,c)= 0, (x,y,z)- n0 .

ax+ by+ cz= 0,

.

, :

ax+ by+ cz=.

R2 ,

, n0.

:

3.1. (1, 1, 2) (0, 3, 5).

(x,y,z)

(x 0, y (3), z 5) (1, 1, 2) = 0.

x + y+ 2z= 3 + 10 = 7.

.

.

5/21/2018 . -

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36 3.

, , .

. -

R3

. , -

: , ,

,

.

.

(

4. , , -

!

4.

:

R3 -

f(x,y,z) =ax2 + by2 + cz2 + 2dxy+ 2eyz+ 2f xz+ gx+ hy+ iz

. ,

. ,

Q b. :

Q=

a d fd b e

f e c

, b= gh

i

.

, x = xi + yj + zk :

f(x) =xTQx + bTx= x

Qx + b

x,

,

-

. Q ,QT =Q. (level sets)

, R, f1() = {x R3; f(x) =}.

[ ]

5/21/2018 . -

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4

4.1

.

:

, !

: , , Heisenberg,

,

Lie .

4.1. m n m n m n .

A=

a11 a12 a1na21 a22 a2n

am1,1 am1,2 am1,nam1 am2 amn

(4.1)

M(m, n).

4.1. ,

, -

5/21/2018 . -

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38 4.

.1 , -

aij R C. M(m, n; R)M(m, n; C).

, -

. ,

-

, :

n, , n, n ! , , - ,

n n

n

, A M(n, n; R)! A = (aij)

m,n

i=1,j=1 = (aij), . -

:

am1,n2 am1n2.

(column vector) -

a, A, A M(m, 1). ,

. , -

, (rowvector). ,

. ,

A M(m, n) n , m. a1, a2, . . . , an A( , m), :

A= (a1 a2 . . . an) .

4.1.

A= 2 3 18 1 112 3 3

:

a1 =

28

2

, a2 = 31

3

, a3 = 111

3

.

1 : ,

, , ,

.

5/21/2018 . -

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4.1 39

4.2. AT A

M(m, n) -

M(n, m) A A . ,

A=

a11 a12 a1na21 a22 a2n am1 am2 amn

AT

=

a11 a21 am1a12 a22

am2

a1n a2n anm

, -

:

(c1, c2, . . . , cn) =cT, c=

c1c2

cn

, A M(m, n) - m n:

A=

aT1aT2. . .aTm

,

aT1 = (a11, a12, . . . a1n), aT2 = (a21, a22, . . . a2n) ...

4.2. ,

A=

2 3 18 1 11

2 3 3

, :

a1=

23

1

, a2 = 81

11

, a3 = 23

3

.

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40 4.

4.2

,

, .

.

4.3. A B M(m, n)( ), , A+ B :

A+ B=

a11 a12 a1na21 a22 a2n


+

+

b11 b12 b1nb21 a22 b2n

bm1,1 bm1,2 bm1,nbm1 bm2 bmn

= (4.2)

=

a11+ b11 a12+ b12 a1n+ b1na21+ b21 a22+ b22 a2n+ b2n

am1,1+ bm1,1 am1,2+ bm1,2 am1,n+ bm1,n

am1+ bm1 am2+ bm2 amn+ bmn

, A+ B= (aij+ bij).

AB , .

4.4. A- A

A=

a11 a12 a1na21 a22 a2n


(4.3)

A B M(m, n)A + (1)B 1A1+ 2A2+ + kAk, Ai M(m, n) i = 1, 2, . . . , k.

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4.2 41

: ,

:

, A B aij bij . , , ,

(. 4.3 .)

:

.

m nx1, x2, . . . , xn

a11x1+ a12x2+

+ xna1n = b1

a21x1+ a22x2+ + xna2n = b2

am1x1+ am2x2+ + xnamn = bn(4.4)

, -

(x1, x2, . . . , xn) , bi. , A= (aij)

x=

x1

x2 xn

, b= b1

b2 bn

Ax= b

Ax , : A -

x m b.

A B . , A B, A B.

4.5.AB, A M(m, n)B M(n, k). A m

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42 4.

( ):

A=

a11 a12 a1na21 a22 a2n


=

aT1aT2 aTm

ai=

ai1 ai2 ain

, i= 1, . . . , m

B k :

B

b11 b12 b1kb21 a22 b2k

bn1,1 bn1,2 bn1,kbn1 bn2 bmk

=

b1 b2 bk

,

bj =

b1jb2j bnj

, j = 1, . . . , k .

, AB , (m, k),

AB=

a1 b1 a1 b2 a1 bka2 b1 a2 b2 a2 bk

am b1 am b2 am bk

(4.5)

, i j AB

(AB)ij =ai

bj =

n

l=1

ailblj .

4.3.

2 3 13 0 2

1 1 2 11 0 2 2

2 3 0 1

2 42 + 3 + 2 2 + 0 3 4 6 + 0 2 + 6 + 13 + 0 + 4 3 + 0 6 6 + 0 + 0 3 + 0 + 2

=

7 1 2 97 3 6 5

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4.2 43

: B A , B4, A2.

4.2.1

, , -

. , , -

12 -

,

A= (A1A2 A3),

, A1 , A2 A3 .

, -

(partition) A M(m, n):

A=

A11 A12 A1kA21 A22 A2k Al1,1 Al1,2 Al1,kAl1 Al2 Alk

,

Aij , (mi, nj). m:

m= m1+ m2+ . . .+ ml

n:

n= n1+ n2+ . . .+ nk.

A=

B CD E

,

B, D C, E B, C D, E .

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44 4.

: -

, . -

.

4.4.

2 21 3

1 45 3

0 00 0

2 21 1

1 11 2 2 12 00 00 0

3 12 1

4 22 7

11 329 9

0 00 0

10 41 0

-

. .

4.1. -

.

4.3 ;

: ,

() ,

a11, . . . , a1n, a21, . . . a2n, . . . , am1, . . . , amn.

;

.

; ,

.

5/21/2018 . -

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4.3 ;

45

2. V n -, n b1, . . . , bn , < b1, . . . , bn > V. - : B ={b1, . . . , bn}. v V

v= x1b1+ . . .+ xnbn (4.6)

x1, . . . , xn -v B.

v :

x1x2...

xn

. (4.7)

-

! , .

:

x1x2...

xn

B

. (4.8)

4.6 4.8 .

, ,

,

( ,

2.3.) V. E ={e1, . . . , en}

, :

1. , bi,i = 1, . . . , n, E= {e1, e2, . . . , en},

bi=a1ie1+ a2ie2+ . . .+ anien

( e1, . . . , en .)

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46 4.

2. E, ei, i = 1, . . . , n, B,

ei = 1ib1+ 2ib2+ . . .+ nibn

( b1, . . . , bn .)

v E= {e1, . . . , en}:v= x1(a11e1+ a21e2+ . . .+ an1en) +. . .+ xn(a1ne1+ a2ne2+ . . .+ annen)

v= (a11x1+ a12x2+ . . .+ a1nxn)e1+ (a21x1+ . . .+ an2xn)e2+ + (an1x1+ . . . annxn)en = y1e1+ y2e2+ + ynen (4.9)

, : -

Aaij( , m = n.) v B -

4.8. v E

y1

y2...

yn

E

=A

x1

x2...

xn

B

, (4.10)

, ,

vE= AvB (4.11)

, -

E B,

A:

vB= A1vE (4.12)

.

4.2. A, B

E. , .

,

.

.

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4.3 ;

47

4.5. R2: 10

,

01

,

11

,

11

,

E= {e1, e2} B= {b1, b2} . :

E B

b1 = e1+ e2b2 = e1+ e2

A

A=

1 11 1

.

4.12,

B A1,

A1 =1

2

1 11 1

.

2e1+ 0e2, ,

e

b

b

e

1

2

2

1

4.1: .

A1

20

=

11

B

.

: , ,

.

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48 4.

: T V W- 1v1+ 2v2

1T(v1) +2T(v2).

, v1, . . . , vn V

v= x1v1+ x2v2+ + xnvn, , v T

T(v) =x1T(v1) +x2T(v2) +

+ xnT(vn).

, T(vi) vi V W, w1, . . . , wm:

T(vi) =a1iw1+ a2iw2+ + amiwm.,

T(v) =x1(a11w1+a21w2+ +am1wm)+x2(a12w1+a22w2+ +am2wm)+ + xn(a1nw1+ a2nw2+ + amnwm). (4.13)

W,

T(v) = (x1a11+x2a12+ +xna1n)w1+(x1a21+x2a22+ +xna2n)w2+ (x1am1+ x2am2+ + xnamn)wm. (4.14)

A

A=

a11 a12 a1na21 a22 a2n


T v1, . . . , vn Vw1, . . . , wm W.

4.1. 1. x -

v v1, . . . , vn, T(v) w1, . . . , wm

y= Ax

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4.4 49

2. A V,

A=

a1 a2 an

,

ai=T(vi).

,

ai=a1iw1+ a2iw2+ + amiwm.

4.4

, ,

,

.

4.6. 1.

, m = n.

2. -

, aij = 0 i= j. A =(a1, a2, . . . , an), ai = aii, i =1, . . . , n .

3. n 1, , In, , , InA= A,BIn=B.

4.

, aij = 0 j > i. .

5. A i, j,aij =aji,

A=

a11 a12 a1na12 a22 a2n a1n a2n ann

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50 4.

6. A i, j,aij = aji,

A=

0 a12 a1na12 0 a2n

a1n a2n 0

7. A Toeplitz( ),

A=

a0 a1 a2 an1a1 a0 a1 an2a2 a1 a0 a1

an+2 a1 a0 a1an+1 an+2 a2 a1 a0

Toeplitz

, .

8. A Vandermonde,

A=

1 1 1 11 2 3 n21

22

23 2n

n11

n12

n13 n1n

: 1, 2, . . . , n, Vandermonde .

1i j, .

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4.4 51

4.4. , A = AT.

: Am n Bn k, AB. , n 1, M(n, n) M(n), .

4.1. M(n) - ,

.

(AB)C=A(BC)( ). M(n) In. ,

(n = 2, ): 1 10 0

0 10 1

=

0 20 0

0 10 1 1 1

0 0 = 0 0

0 0

4.7. A M(n) B M(n) AB=BA= In. B A B = A1. M(n, n) Gl(n).

4.2. Gl(n) .

: , -

AB , A, B Gl(n) B1A1. 4.8. -

,

AAT =ATA= I .

, ,

(

(4.5).

M(n) n 2, :

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52 4.

4.6.

A=

1 20 0

: B, AB .

M(n) , , : A(B+C) =AB+ AC (A+B)C= AC+BC. M(n)(-).

4.7.

A=

1 00 2

A1 =

1 00 1/2

4.2. A ,A = (a1, . . . , an)

ai= 0, A1

A1 =(1/a1, . . . , 1/an).

4.2.

: A - ,

, .

,

!

.

, .

4.5 ,

A VW, -. , A

5/21/2018 . -

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4.5 , 53

V:

A=

a1 a2 an

,

ai= T(vi) W.

4.9. < a1, a2, . . . , an > W A W (image) A Im(A). (rank) A rank(A).

A, .

4.10. (kernel) A,ker(A), V ,

ker(A) = {v V; Av= 0}.

ker(A)V: ker(A) V. A A,

A.

rank(A) max{m, n}.

4.3. A M(n) :

1. n: rank(A) =n

2. -: det A = 0.3. W: Im(A) =W

4. : ker(A) = {0}.

( .)

.

A .

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54 4.

4.11. A M(n). (adjoint matrix) A A:

AdjA=

11 12 1n21 22 2n n1 n2 nn

T

=

11 21 n112 22 n2 1n 2n nn

(4.15)

4.3. A M(n) , A,A1

A1 = 1

det AAdjA

.2.

4.8.

1 1 22 0 10 2 1

|A| = 8 + 2 + 2 = 4 = 0, A .

:

11= 2 12= 2 13= 421= 3 22= 1 23 = 231= 1 32= 3 33 = 2

( !) ,

A1 = 1

4 2 3 1

2 1 3

4 2 2 4.6 : -

. -

!

5/21/2018 . -

63/94

4.6 : 55

m n

a11x1+ a12x2+ + xna1n = b1a21x1+ a22x2+ + xna2n = b2

am1x1+ am2x2+ + xnamn = bn

(4.16)

, , ,

Ax= b.

4.4 ( ). A

A=

a1 a2 an

.

(4.17) b

ai:

b=n

i=1

xiai

,

. ba1, . . . , an m- R

m.

-

n b.: Rm, Rn!

:

4.4. A Im(A) =< a1, . . . , an > = Rm,

Rm, .

m= n, Im(A) =Rn a1, . . . , a

n. , ,

.

:

Rm, W Rm; :

4.5. Im(A) ==W=Rm, Rm, (4.17) b W.

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56 4.

4.6.1 -

- m = n, :

4.6.

a11x1+ a12x2+ + xna1n = b1a21x1+ a22x2+ + xna2n = b2

an1x1+ an2x2+ + xnann = bn(4.17)

:

1. det A = 0.

x= A1b

2. det A= 0, b A, :

b / .

3. det A= 0 b A:

b .

.

4.6.2

a11x1+ a12x2+ + xna1n = 0a21x1+ a22x2+ + xna2n = 0

am1x1+ am2x2+ + xnamn = 0

Ax= 0 (4.18)

: -

, 0. :

; .

5/21/2018 . -

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4.6 : 57

4.9.

2x + y = 06x+ 3y = 0

(4.19)

0= x

26

+ y

13

.

:

26

= 2 1

3

.

, (x, y) = (0, 0), -(x, y) = (1, 2), R . :

y = 2x. !

( , ),

, :

4.5. -

Rn,

A. m = n, - A .

dim ker A= n rank(A) . .

.

,

, R3.

4.10.

2x y+ z = 00 + 2y z = 0

4x + 2y+ 0 = 0(4.20)

:

det

2 1 10 2 14 2 0

= 4 8 + 4 = 0

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58 4.

: - . .

( ) :

z= 2y, x= y/2.

y =, z= 2,x = /2. A, (1/2, 1, 2):

ker A=< 1/21

2

>= { 1/212

; R}. 4.7( ). A k (k min{m, n}), k xi1, xi2 , . . . , xik k , xi1, . . . , xik ( ),

A

xi1

xi2 xik

= b({xj}j=i1,i2,...,ik),

A b - .

k , .

(. 4.6.1)

xi1

xi2 xik

= A1b({xj}j=i1,i2,...,ik)

, n k , Rn, A.

-

, , ,

,n

k .

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4.6 : 59

4.11 (). ,

0 z = 2y4x+ 0 = 2y , A=

0 14 0

z = 2y, x =y/2, y, .

4.6.3 Gauss

Gauss . -

,

.

.

,

Gauss

.

Gauss(Gauss Tableau).

Ax= b.

:

1. .

2. ( -

.)

3. A.

4. b.

5. .

, , / Gaussm:

N A b 1 a11 a12 a1n b1

ni=1 a1i+ b1

2 a21 a22 a2n b2n

i=1 a2i+ b2...

... ...

...

m am1 am2

amn bm ni=1 ami+ bm

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60 4.

m1() :

,

A. . .

N A b 1 a11 a12 a1n b1

ni=1 a1i+ b1

2 2 a21 a22 a2n b2n

i=1 a2i+ b2...

... ...

... ...

m m am1am2 amn bm ni=1 ami+ bmm+ 1 0 a(m+1)2 a(m+1)n bm+1 ni=1 a(m+1)i+ bm+1...

... ...

...

2m 1 0 a(2m1)2 a(2m1)n b2m1n

i=1 a(2m1)i+ b2m1

: -

...

.

, , -

, : -,

. -

(pivots).

. -

!

(

A, b) ( .

4.12. 3 3:N A b 1 2 2

1 1 4

2 1 3 2 1 13 1 1 2 2 4

:

N A b 1 2 2 1 1 42 1/2 1 3 2 1 13 1/2 1 1 2 2 44 0 4 2, 5 1, 5 35 0

2

1, 5

2, 5

6

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4.6 : 61

, :

N A b

1 2 2 1 1 42 1/2 1 3 2 1 13 1/2 1 1 2 2 44 0 4 2, 5 1, 5 35 1/2 0 2 1, 5 2, 5 66 0 0 2, 75 1, 75 4, 5

1,4 6:

(6) = z= 2, 25/ 2, 75 = 0, 637, (4) = y=0, 5 + 2, 5z4

= 0, 772,

(6) = x= 1 2y+ z2

= 0, 047.

LU A (LU-decomposition). , ,

-

.

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62 4.

4.7 -

: -

( : cA+ dB). AB A B. A AT .

.

:

. A

.

M(n, n; R): . , A: A= AT A= AT,AAT =I,Toeplitz Vendermonde

n 1 i. : A M(n, R) det A= 0,

A1 = 1

det AAdjT(A).

det A = 0, A : Av = 0. A ,

- -

.

: Ax = b A b,

b .

ker A A,

A.

Gauss , -

.

Gauss.

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4.8 63

4.8 1.

A=

3 0, 3 1, 40, 4 0 2, 7

1, 3 2, 5 4

, B= 0 1, 2 8, 30, 7 3, 3 5, 1

0 7 6, 2

.

A 2B,A2,3A+ B2,(A B)(B+ A).2.

A=

1 3 44 0 7

, B=

3 51 2

0 4

.

, -

, .

AB, BA, BT 2A, AAT 3BTB, BBT A, BBT + ATA.

3.

, ,

A1 = 1|A|AdjT(A).

A=

1 2 22 0 3

1 2 4

, B= 0 2 37 3 5

3 1 6

.

4. ,

Gauss.

5. .

()2x y+ 3z = 0

3x 2y+ z = 12x+ y 2z = 2

()

0 + 4y z = 103x + y+ 3z = 3

x + 0 4z = 0

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64 4.

()2x y+ 3z = 0

3x 2y+ z = 12x + y 2z = 2

6. .

()2x 3y+ z = 2

x 2y+ 3z = 1()

2x + y 2z = 1x + y z = 1

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5

5.1 -

.

A - Rn

Rn. A , , -

. v , x1, x2, . . . , xn , Av =

ni=1 xiai

v.

v

v,

Av= v (5.1)

; , , -

Rn!

, , (

) .

(

,

.) , , -

(5.1). n v ,(n+1) . n! , . .

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66 5.

- (5.1) :

, v (5.1).

5.2

(5.1) :

(A

In)v= 0 (5.2)

In Inv = v. , , ( 4.6.2).

- A I n.

det(A I) = 0 (5.3) .

5.1. (5.3) , n.

n , - C.

A n A.

5.1. 1 12 5

det(A I2) = 1 12 5 = ( + 1)( 5) + 2 = 2 4 3.

= 2 7.

: 1 2 7 12 5 2 7

=

3 7 12 3 7

(

72

32) + 2 = 0.

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5.3 67

5.2. 3 1 30 1 2

0 0 4

det(A I3) =

3 1 30 1 20 0 4

= ( + 3)( 1)( 4)

=

3, 1, 4, .

:

5.2.

: 1 =a11, 2 =a22, . . . , n=ann.

, -

n -n . n.

k , k.

5.3

, (AIn) n . - .

5.1. - v Rn (A

I)v= 0

. .

5.1. : v

, c R, cv - (

Rn). , ,

.

-

.

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68 5.

5.3. 3 0 20 3 0

0 0 2

= 3, 3, 2. = 3,

A I3=

0 0 20 0 00 0 1

, ker(A+3I), , !

ker(A+ 3I) =< e1, e2 >

(e1, e2 R3.)

= 3 . -

, .

5.1. A, 1, 2, . . . , n (i= j i= j), i Vi ()

. 1 Rn:

< V1, V2, . . . , V n > = Rn.

:

5.4.

10 10 2 1= 102 = 2( .)

1= 10, (A 1I)v= 0:

(A 1I)v=

0 10 8

uv

=

00

.

1 . :

.

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5.3 69

: v = 0 u! i, 1 = 10 . :

v1 = c

10

.

,2 = 2:

(A 2I)v= 8 1

0 0 uv

=

00

.

: 8u + v= 0 :

v2 = c

18

.

5.5. 3 12 2

:

det(A I) = ( 3)( 2) + 2 = 2 5 + 8 = 0

=5

21

2

25 32 =5

2 i

7

2 C.

, -

:

(A

1I)v=

12 i

72

1

2 1

27

2 u

v = 0

0 .

:

v=

1

2 i

7

2

u.

, :

v= c 112

i72 ,

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70 5.

c C! !

2u=

1

2+

7

2

v=

1

2+

7

2

1

2 i

7

2

u=

1

4+

7

4

u= 2u!

, A I -

( .)

-

M(n; R) M(n; C),

! -

( n n):5.3. C A M(n; R) () = 0, , , - .

Av= v:Av= v,

A . , :

5.1. v2 =52 i

72 .

-

. ,

-

. , ,

Jordan.

5.2. k > 1, k , -: v1, . . . , vm (m k) - w1, . . . , wkm. vi /Avi= vi.

w -

Aw= w + v (A I)w= v, (5.4)

v .

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5.4 71

:

1. .

k, (. .)

2. , vi -

: (5.4) v= vi .

v . ...

.

3. , , - .

-

k, .

5.4

: A M(n), - , , ,

, !

. ,

, 1, 2, . . . , n, i= j i= j, v1, . . . , vn R

n (

.)

, , -

A . A ( 4.3.)

U= {v1, . . . , vn}. ( 4.3

):

U= (v1, . . . , vn)

v, U:

v= y1v1+ y2v2+ . . .+ ynvn.

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72 5.

A :

Av= A(y1v1+ y2v2+ . . .+ ynvn) =

=y1Av1+ y2Av2+ . . .+ ynAvn=y11v1+ y22v2+ . . .+ ynnvn. (5.5)

, U, A ,

=

1 0 00 2 0 0 0 n

.

, A , !( .) -

, ,

A, M(n; C)!

A . - U :

vU=

y1y2...

yn

U

vE=

x1x2...

xn

E

,

, , , :

vE= UvU.

v= x1e1+ x2e2+ . . .+ xnen=y1v1+ y2v2+ . . .+ ynvn.

x = vEy = vU, x = Uy. :

Ax= A(Uy),

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5.4 73

5.5. Ax v

. , U,

y. , :

Ax= U(y)

( y .) x =Uy,

A(Uy) =U(y).

, ,

AU=U = U1AU=

A= UU1.

:

5.3. A M(n) , vi

U= (v1, v2, . . . , vn)

A= UU1 U1AU= (5.6) .

:

i vi, :

Avi=ivi.

n , :

A(v1 v2 . . . vn) = (1v1 2v2 . . . nvn).

AU=U,

! ,

.

5.2. .

,

, , -

.

. Jordan.

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74 5.

5.5 -

: , , - A I . det(AI) n, - A. A M(n; R) n, .

: i, AiI- i. k(k 1 ), - , ,

k.

(A iI)v= 0.

, -

. - k = 1( ), cvi, vi .

: A M(n) -, ,

U = (v1 vn)(vi) A- .

A= UU1 U1AU=

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5.6 75

5.6 1. :

A=

1 32 0

, B=

3 21 2

, C=

5 50 2

.

2. :

A= 1 4

2 4 , B=

3 3

1 1 .

3. :

A=

2 3 10 1 0

0 0 2

, B= 2 1 21 2 0

2 0 0

,

C=

1 1 22 0 2

1 1 3

, D=

2 0 20 2 0

2 0 2

.

: C = 2.

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I

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:

, .

M(n) , A, det A .

.

.1. A =

a bc d

.

A

det A= ad bc. det A =|A|. -

, A( ) , .

.1. v1, v2 R2

A= v1 v2 .

. ,

, .

, v1, v2 (1, 2) = (0, 0)

1v1+ 2v2 = 0.

(

, )

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80 . :

2 , . v2=v1, = 1/2. :

(c, d) =(a, b) = det A= (ab ba) = 0., , ad = bc,

a/b = c/d, ( b, d = 0;)

M(3): , : A

A=

a11 a12 a13a21 a22 a23

a31 a32 a33

2 2

det A= a11

a22 a23a32 a33 a12 a21 a33a23 a31+ a13 a21 a32a22 a31 . (.1)

:

: ,

a13( ) A . .

det A= a11a22a33+ a12a23a31+ a21a32a13 a31a22a13 a21a12a33 a32a23a11,(.2)

-

450 45o.

.1. .

, (.1),

M(n),n 4. .2. A M(n) n. (i, j), Aij(n 1) (n 1) A i j. (i, j) A (principal minor). (i, j) ij A(cofactor)

ij = (1)i+j|Aij|.

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81

.1. (1)i+j , .

.3. A M(n) det A= |A| =a1111+ a1212+ . . .+ a1n1n.

,

(

.1.)

.1. a1, a2, . . . , an Rn ( ) A =

a1 a2 . . . an

.

( .)

.2. 1. A M(n)

det A= ai1i1+ ai2i2+ . . .+ ainin

i = 1, 2, . . . , n. i -. j:

det A= a1j1j+ a2j2j+ . . .+ anjnj

2. i =k,ai1k1+ ai2k2+ . . .+ ainkn = 0,

i k, . .

-

, .2:

.2( ). 1. A,AT, :

det AT = det A.

2. A, . .

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82 . :

3. ( ) A , . , An ,

det(A) =n det A.

4. ( ) A ( ), (-

,

( .)

5. det A

ai ak, ai ai+ak. .

6. ( )

.

( ,

) :

.3. A, B M(n), det AB= det A det B.

- : m= n, - A M(m, n). , - A, min(m, n)! - .

.4. k min(m, n) k , 1 i1 < i2

Ε. ΚΑΠΠΟΣ - ΓΡΑΜΜΙΚΗ ΑΛΓΕΒΡΑ

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Transcript of Ε. ΚΑΠΠΟΣ - ΓΡΑΜΜΙΚΗ ΑΛΓΕΒΡΑ