1 7

6

: : . 118 ( 4 ) . 119,

5, . 127,

6.8, . 143

: 3: \ \f 4

( ) [ ]T2 3 1 2 1 1 2x x x x x x xx = + + f

0 1 11 1 01 0 01 1 0

A

= .

5.7 (. 119), ()

4: \ \f 3

( ) 2 3 4T 1 2 41

0 1 1 1 y y y1 1 0 1 y y y1 0 0 0 y

y A y y+ + = = = +

f

.

, (5.11,

. 118). ,

( ) [ ]T1 0 1 1 1 =f , ( ) [ ]T2 1 1 0 1 = f , ( ) [ ]T3 1 0 0 0 =f

( )1 2 3y y yy 4= + +D f , ( )2 1 2y y yy 4= + D f , ( )3 1yy =D f ,

2 7

( ) ( ) ( ) [ ]T2 3 4 1 1 2 4 2 1 3 2 3 4 1 2 4 1y y y y y y y y y y y y y yy = + + + + + = + + + f .

. 119

( ) ( ) ( )x =f f x , : f . (5.12) : ( ) ( ) ( ) ( )x y x y x y = =: D :f f f . , ( ) ( ) ( )x x y 0 = :f f , y

( ) ( ) ( )x x 0 =f f .

, ( ) ( ) ( )( )y y = f h h f : , , D :h , : , , :f y . , (5.12)

( ) ( ) ( )( ) ( )( ) ( ) ( )( )( ) ( )( ).

x y x y x y x

x y x y

= = =

= = D :

D Df h f h f h h f

h f h f

y y y

, ( ) ( ) ( )( )( ) 0x y y D f h h f = , x ( ) ( ) ( )( )y y = f h h f .

, ( ) ( )dim dim = f f : f . : A f

, ( 4.9, . 90) ( )dim rank A=f ( )dim rank A =f . , rank rankA A= .

, ( )ker = f f : f . ( )( )ker = f f . ( )kerx x 0 =f f ( ) ( )0 x y x y= =: Df f .

3 7

6.1 Ax b= ,

b

A x 0 = x 0= . : : f . A Ax b= , b

( ) = f ( ) { }ker ker 0= = f f f . , A x 0 = x 0= . . x 0=

{ } ( )ker ,A x 0 0 b x = = = f f . ( )x b=f Ax b= b .

* * *

x ( )yf , y , ( )( ) ( )( )kerx f f f .

( )( )x y f , ( ) ( )0 , 0x y y x y = =D :f f ( ) ( )( )ker kerx 0 x = f f f f .

( ) ( )( ) ( ) ker = = f f f f .

, ( )ker = f f : f .

( ) ( ) ( )ker ker = = f f f f , ( ) =f f .

4 7

6.2 ,

. A ker A Im A

A : , kerx A ,

( ) kerAx 0 A Ax 0 Ax= = A . ker A A . , ( )Im Imx Ax A A Ax A ^ .

Im A

A * * *

6.3 x A ^ , { }1span , , ,x Ax A x= M

. A : , 10 1 1c c c x Ax A x= + + + " M

20 1 1c c cA Ax A x A x

= + + +" .

Cayley-Hamilton 11 0k kA A I O+ + + =" . , 10 1k kA I A = " ,

10 1 1c c cA x Ax A x

= + + + " M .

* * *

6.4 ,

1 2 k, , ,x x x A

{ }1 2 kspan , , ,x x x= M . A : 1 1 k kx x x= + + " M 1 1 k kAx Ax Ax= + + " i i iAx x= ( )i 1,2, , k=

( ) ( ) ( )1 1 1 2 2 2 k k kAx x x x= + + + " M .

* * *

5 7

6.5 ,A B ( )1A S BS= , ,

M

A

{ }:S Sx x= M M B . : ,

M A 1A S BS M M M M( )B S S M M S M B .

* * *

6.6

. A

A : M N A ,

x M y N x y+ +M N ( )A x y Ax Ay+ = + +M N . M N M N A M A N A M N .

* * *

6.7 ( ) ( )x, y x y, 3x y= + f . f

:

.

2: \ \f 2

1 13 1

A = 1 2 = 2 2 = ,

( ) 12 span1

= ( ) 12 span 3

= 1

. f

* * *

6 7

6.8 M A B

. M AB k A B+ . k, ^. M 1A , . A : . Bxx M ABx M M . ,

, k, ^ ( ) ( ) ( )k kA B x Ax Bx+ = + M , BxAx M . . M A , x M .

y My Ax= 1x A y= , , M 1A .

* * *

6.9 1 2, , , ,

: f

f( )i f ( ) . i 1,2, ,=

: , . x( )i f i 1, 2, ,= 1 1 2 2c c cx = + + +" 1 1 2 2c c cAx A A A = + + + " .

, f . , f , . i ( )i f

* * *

6.10 M A , .

M

A

: x Ax M M ( ) 0y Ax y =DM . , ( ) ( ) 0Ax y x A y A y = = D D M , ,

.

M

A

7 7

, M A ( ) =M M ( )A A = .

* * *

6.11 2: 2 f ,

( ) ( ) ( ) ( )2 2x x 5 2 4 x x 2 + + = + + + f , { }2span x, 1 2 x= + f . f

:

( ) ( )2x x 2 x 1= + f (*)

( ) ( )2 22 x 1 x 3 2 x 1+ = + + f , 6.9, f .

( ) ( ) ( )( ) ( )2 2x x 2 2 1 2x= + + + + f x , , 2 E= { }2span xE = . 2x ,

( ) ( )2 2x 0 x 2 2 x 1 x2= + + +f (*) f

1 1 01 3 2

0 0 1

A

=

##

# #

.

1 11 3

|f .

* * *