.

( )

, 2003

i

, !, "# "-

$ !%$. & ' $ $ # #

, !%# *"-

. ! '! # # . !

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! $ ! . ! "# /

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% " '! %$ + ! .

/ '$ ,

+, # 1/ $ '$.

4!# %# 1/ %+ % *"$ % ! -

'!. " ' !" %/ #: +

5+ . 4 %'+ '+ #' -

%!+ '. % 1/ -

/ 5/ ! + % '

# ! !, $ ! $ # + +

+ 5+.

& '# " # #

! / 5/, %% '$ # -

+ + #!, !$, ! -

'+ + &! !, + "-

..

* $ $ !$ 1984, ! -

# ! *"$ ! ! ;. <

$ '", /, + % ! !%#! -

" !, ! ;. & / $-

! # ! *"$ &-

! ! ;/!.

ii

4 $ #!= + %% ! / " -

# !# +, %!/!

! # + # $ ' -

. > %1 %$! # / -

% ! , % , "# -

.

4 $ ! # . & $,

/-

, ! !+, ! / , -

"$ # ! $, + + !#'.

%/ $ ! / '$!

# 1/ , ', + #

. & # %! % %+ -

, ! +! ! , #

'# 1/ $ .

# !, %% # $ 1-

$" / '$! # % . & #%

# ' , !, $ % %+-

# . +%

$ / '$! +

+, , #, # ! %# %%/

$ .

2001

.

1.

: &/, , &$ 1

1.1. &/ 1

1.2. 4

1.3. &$ 6

1.4. 11

2. A$ 13

2.1. # ! / '$! 13

2.2. 1 , , % 17

2.3. " !"/ " $ '$ 29

2.4. 33

3. # 35

3.1. # 35

3.2. ! + 39

3.3. 42

4. 45

4.1. 45

4.2. 1 52

4.3. 55

5. # 57

5.1. 4 57

5.2. D"+ , , 59

5.3. 68

6. &! 69

6.1. 4 ! 70

iii

iv EFEA4*E

6.2. 4 !+' ! 79

6.3. #"% ! Gauss 83

6.4. 86

7. 4! 89

7.1. 4+, /1 ! 89

7.2. H+ !$ # 102

7.3. 106

8. A$ E+ + 109

8.1. # ! '$! + + 109

8.2. 4"

116

8.3. 121

9. %#, %%/ > 125

9.1. %# %%/ $ 125

9.2. > 132

9.3. 140

D 143

E! 145

1. : !", #$, !

&

/ !+ " ! !+ "

"/ # "$% # /'*"$ !-

#, " "!"/ # / " ! # +-

, % ! $. 4 '$, ! /

# , " . $

! ! + + # %/, $ $

"$ R $ %$ "$ C !" 1 +"

+.

1.1. !"

! + # #, / + +

/' *" , # ! !+!. E%$ " ! $

!+ " ! # # #, '

/.

2 1. F4;F;;: &H45, E;4&E&, &*

#! + / !$ "$ %# N0,

N0 := {0, 1, 2, . . . },

/ "$ Z,

Z := {0,1,2, . . . },

/ $ "$ Q,

Q := {pq

: p, q #, q = 0}.

/ $ "$ ! R, / %$

"$ C.

:

1.1. &H45 3

& ! I #, I = {1, 2, . . . , n}, '/ ! !/

X1 X2 Xn :=n

i=1Xi :=

iIXi,

X1 X2 Xn :=n

i=1Xi :=

iIXi.

4 1. F4;F;;: &H45, E;4&E&, &*

1.2. #$

* # *"$, # +,

! +. I ! " %/

# %+ $ .

1.2. E;4&E& 5

+ f : X Y , + " y Y /

1f ({y}), %% + ! !+! {y}, #' $

# ', !! f1(y). ; !+ +

+f1 : Y X,

y f1(y), f.

! 1.1

6 1. F4;F;;: &H45, E;4&E&, &*

", %

[0, 1] [0, 1],x x,

+,

[0, 1] [0, 1],x x2.

I/ + f : X Y # !/ M ! X. +-

g : M Y,x f(x),

# f M ! f |M . 4 f

g %#! + % / , ! %/

! % #.

1.3. &* 7

* % (G, ) # ( ! % ! / "-/ Niels Henrik Abel), " a, b G '/

a b = b a ("+ +).

% (G, ) 1 ! , + !/ G (G, ), ab a b.

I %/ $ # # %+ %. &!#, " %/ +

a ! a, + a ! a, + -

+ '! , + ! ' $ # !%#

', + !%# ' + ! ea = a '/ ae = a

" ' a %.

% 1.1 # G . $ %:

1. & a G % a, % aa = e.2. ' % e G % ae = a a G.3. (% % e G.4. ' a G % % a G, - a1.

&.

1.

8 1. F4;F;;: &H45, E;4&E&, &*

3.

1.3. &* 9

y = ey = (a1a)y = a1(ay) = a1b = y.

'/ $ " !"#! + 1$ (1.1) #'!

/ " a, b G " %1! + (G, ) %.

10 1. F4;F;;: &H45, E;4&E&, &*

: K K K,

(a, b) a b, 1 %+ (1$ $):

(&1) (K,+) %.

( !%# ' !! 0, + '! a K 1 + a, "

/ !#' ! a.)(&2) (K, ), K := K \ {0}, %.

( !%# ' !! 1, + '! a K 1 a1.)(&3) a, b, c K / '/!

a(b + c) = (ab) + (ac)

(a + b)c = (ac) + (bc)

( +).

&$! +, + %$ #, '/ # + ! -

/ +!, + '/ .

!/! !+ / # "#,

!/ + + !%# '!+ + +".

1.4. &;&E& 11

3. ' a, b K % a(b) = (a)b = (ab).&.

1. ; ' #'!

0 a = (0 + 0) a = 0 a + 0 a,

5 1.2 !! + 0 a = 0. a 0 = 0 %/ (# + + % %'"#, % % .)

2. !"#! + a = 0 b = 0, + / (&2) !! +" '/ ab = 0, .3. 0}. %1 + / (Q, ), (Q+, ), (R, ), (R+, ) !"

12 1. F4;F;;: &H45, E;4&E&, &*

+ # %, $ / (Z, ), (N, ), Z := Z\{0},% %.

1.6. %1 + (Q,+, ), (R,+, ) (C,+, ) !" 1 +"- + $, $ (Z,+, ) % $.1.7.

2. &

& !+ " ! # -

, # !

% , " %/ # # %+ $

'$, " ! # "$% # / '$!,

# 1 , % , "$ #-

! !'$! + / '$! ! " $ '$.

I ! # # %+ $ '$ " %$-

! % $ '$.

4 '$ ! "+ + + *" ,

' ! % # + + .

2.1. ' ( " (

& ! + " ! # ! / '$! ! !-

'$! + / '$!, " %/ # % $ '$

" ! # # %+ $ '$.

" 2.1

14 2. F**;4 AF4

(A2) x, y X , K '/!(a) ( + ) x = ( x) + ( x)

(b) (x + y) = ( x) + ( y)

(c) () x = ( x)

(d) 1 x = x.

&!$! + '$ ' + %

(linear space, vector space). E/ ' ! $ ' !

%! !+!, ! 0, + !%# ' ! $ K, +

%+ % ! !X, "$ ! !+! + +" +

K + X. #' %1 + + 1

" /'! ! %'#

!+ !+ !

1#! $.

1 + ! , + !' !X (X,+, ). E x, K, x X, !" ! x.

/! # "#, !/ + -

+ + '! ! K # ' ! X !%# '!+ +

+" X, %% $ # #

"# ' ! / '$!, x + y (x) + y.

& # '$ '%+ $

$ "$ R $ %$ "$ C. +

!+ % # / / / !"/, !#

$ " '"/ + / / '$! (%%

#! $ K = R) %/ / '$! (K = C). M

!#' " / '$!, "

/ / /

'$! ! '/ %/ / '$! "

! $ " #.

! 2.1 ' + / '$! #

! / '$!. E1 /, +, # % ! # ' +

%!/ '$!.

2.1. E4 4H F**;4H AF4H 15

! 2.1

1. %, %/

% + ! ! "! %! " %!.

& + 5 " %/ + + + / "/ #

% ! %, + + "+ % ! ( -

+) %#. E " %/ + (1)x " ! x.

% 2.1 # (X,+, )

% . $ %:1. $ ,

,

(i)x X 0 x = 0 R 0 = 0

16 2. F**;4 AF4

, ,

(ii) ( R, x X, x = 0) = ( = 0 x = 0).2. $ 1 x x, x,

x X (1)x = x.

&.

1. (i)

2.2. F**; ERF&, D&, >&& 17

S! , + '$ #' " # $ K, + +

$ "$ / + ! ' ! K.

% 2.2 # (X,+, )

% Y % X. $ (Y,+, ), X % Y, -

% .

&. 1 (A2) '/ $ Y, / '/ X. 4 -

"+ + + '/! Y, / '/! X.

/ Y +, #' # ' y Y. E# %+ % !,0 = 0 y, + ! (HA2) Y. ! $ # % !'$. 4 %1 /

. &! ! # / !+-

' %/ %, %$! !# %

, + + '$ R2.

! 2.2

1.

18 2. F**;4 AF4

! #, " ! 1 1 %! -

, " ! % !" !#, " ! #

+ / '$!.

" 2.2

2.2. F**; ERF&, D&, >&& 19

! ! '# 1/j %"/

xj = mi=1i=j

ij

xi,

! xj + !%!+ ! %-

! .

$, !"#! +, j {1, . . . ,m}, xj + !%!+ ! %! , " #'!

xj = mi=1i=j

ixi,

+ j := 1 ! '#

1x1 + + mxm = 0,

/ , + ! + j = 0, x1, . . . , xm 1#-.

!%! %/ x1, . . . , xm 1#, !+ %

+ "# + ! + !-

%!+ !. & R2, % ' , %/ x1, x2, x3

x1 = (1, 0), x2 = (2, 0) x3 = (0, 1) 1#, x3 + %

+ !%!+ x1 x2.

! 2.3

1. %/ x, y, 2x + y !% / / '$! X

1#, /

2x + y + (1)(2x + y) = 0.

2. % + !#' " 1! %/ ! Rn, #'!

+ !$ ! %#, / %.

j {1, . . . , n}, !! ej % ! ! Rn ! ! !$ "# j % + !+ %#, ej = (0, . . . , 0, 1, 0, . . . , 0)

% "# j.

20 2. F**;4 AF4

%/ e1, . . . , en Rn 1 . '#

1e1 + + nen = 0

%/

(1, . . . , n) = 0 Rn,

! % # 1 = = n = 0. =! !$ + %/ ej =+, !

# / ' + *" # -

Kronecker Kronecker. %/ !'+ ! i j ij

/ % i = j %# % , %%

ij =

{1, i = j,

0, i = j.

* !+ !+ / =! % ! ej Rn ej = (1j, . . . , nj).

3. # !+ "+ n, !! Pn / !/

"/ / n,

Pn := {p : R R / p !$! "/ / n}.

+ "$% "$ ! + # !$! "/

n #' $ n , %# " -

! ! += + $.

I %/ $ + !$! pi, pi(t) := ti, i = 0, . . . ,m, m n, 1 Pn. , ' + ! + m n #'!pi Pn, i = 0, . . . ,m. $ !"#! + # + !%!+ q pi,q = 1p1+ +mpm, %#, q = 0, + !! + 1 = = m = 0. %/ !+, !"#! + + i % !

%+, + q " #', / "$% "$ ,

/ m , / q = 0, %% q(t) = 0 " t R, + q #' .

2.2. F**; ERF&, D&, >&& 21

" 2.4

< x1, . . . , xm >:= {x X : x =m

i=1

ixi, i R},

# !+' ! X.

I %/ $ # + ' '+ 1.

$ 2.2 # X

% x1, . . . , xm X. $

:

i. $ x1, . . . , xm

.

ii. / x < x1, . . . , xm >

x1, . . . , xm.

&. H"#! ' + '/ i. " %1! + '/

ii. % '/ ii., + x < x1, . . . , xm > "

%/ ! ' +! + !%!+ x1, . . . , xm, + "

#'!

(2.2) x =m

i=1

ixi =m

i=1

ixi

(1, . . . , m) = (1, . . . , m). + (2.2) + #m

i=1

(i i)xi = 0,

+ 1 x1, . . . , xm %/ !#

i i = 0, i = 0, . . . ,m, %% (1, . . . , m) = (1, . . . , m), . E# ii.'/.

$ !"#! + '/ ii. " %1! + '/

i. ; ' # #'!

0 x1 + + 0 xm = 0. !"#! + + x1, . . . , xm 1#, +

" ! ' (1, . . . , m) = 0 Rm # $1x1 + + mxm = 0,

22 2. F**;4 AF4

%% %+ % ! %/ +! +

!%!+ x1, . . . , xm, / ii. . '/

i.

% #'! 1 %/ !

# + '$. ; " # / %! ! '$-

!.

" 2.5

(B2) (B2) x1, . . . , xm 1 .

I %$! $ # % $ '$. I '-

! !+ ! % 2.3.

! 2.4

1. / {e1, . . . , en} ! Rn. #'! % % + e1, . . . , en 1 . E, " a = (a1, . . . , an) Rn ,+ %$ #,

a = a1e1 + + anen.

&Rn ! '! # . ! ! # %$ #' %

! Rn.

2. / {p0, . . . , pn} ! Pn. , 1- p0, . . . , pn #'! % %1, # " !$! p "/

/ n, p Pn,p(t) =

ni=0

aiti

p =n

i=0

aipi.

2.2. F**; ERF&, D&, >&& 23

>! !#' #! '/ # /

+ '$!.

$ 2.3 # X

% , X = {0}, x1, . . . , xm X. $ :

i. {x1, . . . , xm} X.ii. $ x1, . . . , xm X, , ,

X, < x1, . . . , xm >= X j {1, . . . ,m} %

< x1, . . . , xj1, xj+1, . . . , xm > = X.

iii. $ x1, . . . , xm

-

% X

,

{y1, . . . , yn} X {x1, . . . ,xm}, y1, . . . , yn

.iv. $ x1, . . . , xm X, x X

x1, . . . , xm.

&. I %1! + ii. # + i., iii. # + ii., iv.

# + iii., i. # + iv. !+ %/ %! !

"#!, !"$ ! ' / + % + !

%"/ % .

H"#! ' + '/ i. " %1! + '/ ii.

H"#! + + ii. % '/ + %"/

. S! , % '/ ii., " ! ' j {1, . . . ,m} # $

< x1, . . . , xj1, xj+1, . . . , xm >= X.

$ xj ' ! X " + !%!+ x1, . . . ,

xj1, xj+1, . . . , xm, %%

xj = mi=1i=j

ixi.

!+ + + %/ x1, . . . , xm 1#,

{x1, . . . , xm} % ! X, i. ! !"# +'/, # %" .

24 2. F**;4 AF4

H"#! $ + '/ ii. " %1! + '/ iii. $-

" %1! + x1, . . . , xm 1 . m = 1, !+

#, /, + ! + X = {0}, " #'! x1 = 0.

2.2. F**; ERF&, D&, >&& 25

E #, / + 2.2, " x X #

/ + + !%!+ %! x1, . . . , xm, / !

1 . , / x1, . . . , xm ! X, " x X - + !%!+ !. &! , " x X # $ + + !%!+ %! x1, . . . , xm.

#, / + 2.2, i. # + iv., !+

+

$ /.

!# ! !#! I, " %/ $ + +

# / %! ! ! # '$ / #1! -

# ! / !.

$ 2.1 # X

% , X = {0}, x1, . . . , xm X % X. $ % {i1, . . . , in} {1, . . . ,m} {xi1, . . . , xin} X.

&. x1, . . . , xm X 1 , + {x1, . . . , xm} ! X.

x1, . . . , xm X 1#, + ! ' # 1 !-$ + !%!+ !. A + -

+ / %'"/ + xm + !%!+ x1, . . . ,

xm1, !+ !' ". + x1, . . . , xm1" ! ! X, < x1, . . . , xm1 >= X. E ! $ %

%%. !+ " / m 1 #, !"# +

X %+ + '$, + % " %"/

1 %/ xi1, . . . , xin +" " !

X, + / {xi1, . . . , xin} " ! X.

# ! "#! ! !#' ! -

+ / '$!, # # ! #"! +

'$!. % " ' ! '$!,

+ " # %1! $ + + + -

/ '$! #'! % " '.

26 2. F**;4 AF4

I '! . $ %/ " %$!

#, !' " %! / "!+ -

# + + + / '$! #'! % "

'.

&/ $

+ #, 1$ +

"$ # ' # + !%!+ '

'+ ' !# ! , #-

! + 5 ! !", %/ !

'$! . & %/

+ # " %/ + / -

! + ' '

1 ! . !+ ! # !+ -

" , !# ! + +

#'! % " '.

% 2.3 # X

% {x1, . . . , xm} X. # y X,

(2.4) y = 1x1 + + mxm.

& j = 0 (2.4), j {1, . . . ,m}, - xj y, {x1, . . . , xj1, y, xj+1, . . . , xm}, X.

&. A + / %'"/ + j = 1, !+

!' ". >'+ + + 1 = 0, "#! %1! + / {y, x2, . . . , xm} ! X. ;' (2.4) % $

(2.5) x1 =1

1y +

mi=2

(i1

)xi.

2.2. F**; ERF&, D&, >&& 27

A$ (2.5) (2.6) !

(2.7) x =11

y +m

i=2

(i 1 i1

)xi,

!! + %/ y, x2, . . . , xm ! X, < y, x2, . . . ,

xm >= X.

# $ %1! 1 y, x2, . . . , xm.

28 2. F**;4 AF4

1#. !+ + !+" , , + m 1 < n, !$ m n.

/ $ < y1, . . . , ym1, xm, . . . , xn >= X, ! '! " 1,

. . . , n # $

(2.10) ym = 1y1 + + m1ym1 + mxm + + nxn.

& (2.10) % '/ m = = n = 0, / y1, . . . , ym 1 . * " xm, . . . , xn / /'! m =0. &/ 5 2.3 / $ ! xm ym

{y1, . . . , ym, xm+1, . . . , xn} ! X, + !

$ +%1.

E $ # %1! + %/ + / '$!

#'! % " '.

$ 2.2 # {x1, . . . , xn} {y1, . . . , ym}

% X.$ % m = n.

&. / / {x1, . . . , xn} ! X y1, . . . , ym X 1 , / I$ 2.1 " '/ m n. $' " '/ n m, + m = n.

&/ I$ 2.1 + 2.2, # +, #

+ '$ X ! ' {x1, . . . , xn}, + + ! X #'!n ', X % ! '! n + 1 1 %/. *

! # / $ ! # % -

.

" 2.6

2.3. IF4&* ; EHIH IF4&* F**; AF 29

# (dimension) ! / '$! X. & ! % -

+ / '$! %# # !+ "+, # + '$

% , % '$ # .

M % % 2.4, n N, / {e1, . . . , en} ! Rn. E# % ! Rn n, dimRn = n.

!# ! I 2.1 + #.

+ 2.2 # X

% . & y1, . . . , ym X

, % X %

y1, . . . , ym.

/ +%1 ! #! # , .

2.7.

% 2.4 # X

% n. $ %:

1. & y1, . . . , yn X

, {y1, . . . , yn} X.

2. # Y % X. $:

i. dim Y dimX.ii. & dimY = dimX, Y = X.

!%! & % ! '$! X, ii. %/ # !

5 2.4 % '/. >% ! '! !+' Y ! X #

$ dimY = dimX = .

2.3. -/$ (/" /$ 0

& ! + " ! # ! !

, " %/ # '# 1/ % -

' $ '$.

" 2.7

30 2. F**;4 AF4

!%! " %/ $ '$ % # !'# #

Y Z. Y Z % + '$, . 2.3.

M % ! " $ '$, #'! 1

#:

$ 2.4#X

% , Y, Z %

X. $ %

dim (Y + Z) = dimY + dimZ dim (Y Z).

&.

2.3. IF4&* ; EHIH IF4&* F**; AF 31

*# $ %1! + x1, . . . , xn, y1, . . . , y, z1, . . . , zm

1 .

32 2. F**;4 AF4

$ 2.5#X

% Y, Z % X. $

:

i. X = Y Z.ii. ' x X % y Y z Z x = y + z.

&. H"#! ' + '/ i. " %1! + '/

ii. H"#! %% + Y Z = {0}.

2.4. &;&E& 33

!"/ " Y Z. X R2 # Y

+ Z % .

$ 2.7 # Y %

% X .

$ % % Z X X = Y Z.&.

34 2. F**;4 AF4

2.7.

3. ' #$

36 3. F**;E& E;4&E&

S , # %$ + +L : R2 R2, L(x) :=x1x2

x x2 = 0 L(x) := 0 x2 = 0, x = (x1, x2), %/ '#,%% L(x) = L(x) R, +' $, /, % ' , x = (1, 2) y = (2, 1) #'! L(x + y) = (3, 3) L(x) + L(y) = (4.5, 3).

%$! $ # % + !

$ !" 4+ 3.1, "/ + L : C C,L(x) := a ib, +! a b + + # ! x,'. , x, y C, x = a + ib y = c + id, #'!

L(x + y) = L(a + c + i(b + d)) = a + c i(b + d) = (a ib) + (c id) = L(x) + L(y),

$, .'., L(ix) = L(b + ia) = b ia iL(x) = i(a ib) = b + ia, %%L(ix) = iL(x), x = 0.

& 5 ! !" %! # # %+ $ -

, + ! / !+ +%1

" %" ! . ! % %+ + %

+ + % ! % !

%! / . & !#' " %/ # '# !

!%# !# % , + , # + -

+ # .

% 3.1 # X,Y

% L : X Y

. $ %:

i. L0 = 0 x, y X L(x y) = Lx Ly.ii. # x1, . . . , xn X. & x1, . . . , xn

, Lx1, . . . , Lxn

. & Lx1, . . . , Lxn

,

x1, . . . , xn

.

iii. & X Y % X Y, %, L(X) 1L (Y )

% Y X, %.

iv. dimL(X) dimX.&.

i. ; ', $,

L0 = L(0 0) = 0 L0 = 0.

3.1. E4 & F**;& E;4&E& 37

(& # ! + %/ # 0, $

+ "+ %# %/ %+ % !.)

E, x, y X #'!L(x y) = L(x + (1)y) = Lx + (1)Ly = Lx Ly.

ii.

38 3. F**;E& E;4&E&

. & L (

), , y1, . . . , yn

.

&. ; ' " %1! + ! ' / # +.

3.2. HF& ; E;4 39

.$L(X) Y , / " x X x = 1x1+ +nxn, + Lx = 1Lx1+ +nLxn = 1y1+ +nyn.E1 !, y Y , + y = 1y1 + + nyn, + y = 1Lx1 + + nLxn y = L(1x1 + + nxn), %% y L(X). Y L(X), + ! #'!L(X) = Y .

40 3. F**;E& E;4&E&

! "! # + "+ %/ %/ x1, x2 !

! L, x1, x2 KerL. + " #'! L(x1+x2) = Lx1+Lx2 = 0+0 =0, !$ x1 + x2 ' ! KerL, + ! %/

/.

& +!" # " %/ + + #

#, + ! #' + %+ % !.

$ 3.2 # X Y

% L : X Y

. $ L , KerL = {0}.

&. H"#! ' + L # #. x KerL, +! Lx = 0, , % L0 = 0, !! + x = 0.

3.2. HF& ; E;4 41

/ 1 Lx = 0, $ % ! ! %

%#.

+ 3.1 # X Y

% , dimX < , L : X Y

. $ %

(3.2) dimX = dimKerL + dim mL.

&. M # %, / 5 3.1, '/ dim mL dimX. . x1, . . . , x+m ' ! X, ++ / !! + X =< x1, . . . , x+m > .

$! +%1, # %1! + x1, . . . , x+m

1 .

42 3. F**;E& E;4&E&

+! ! + ' + + L(1x1+ +x) = 0.

3.3. &;&E& 43

(ii)

{L2 : Pn P2n,

x x x,

(iii)

{L3 : Pn Pn,

x x,+! x x $ %/ ! x, -

'.

3.4.

4.

4 / + ! -

. & !+ " %/ #

, " ! "+ # !

!, " ! / , " %/ #

+ #'! + n %/ ! Rn / !, " -

! 1 1/ .

4.1. #

& ! + " ! , " ! '$%

'/ $ , " ! "+ +

# !, " "/ / , " %/ $,

" , / #1!, + n %/ ! Rn -

/ !.

46 4. ;E&

4.1. E; ;E& 47

...

ai...

aj...

...

aj...

ai...

.

&$! + '$% ' $ iii. iv.

/ +

/! # '$

i. ii., + / / %$!, . 4.1.

" 4.2 I/ # A Rm,n. 4 + '$ ! + # !, < a1, . . . , am >, # %

! A "

!! A(A). ' A&(A), A&(A):=< a1, . . . , an >, #

% ! A. % ! A(A) # A

, D(A), % ! A&(A) # A ,

D&(A).

& !#' " %/ + '$% ' $ % -

! '$ $ + .

% 4.1 # A B m n . & B A % %

, %

A B , A(A) = A(B).

&. &/ # 4+ 4.1, %-

1! '!+ '/ $ i. ii. E,

$, %1! # # '+, -

# !/ % + #.

48 4. ;E&

# x A(B). * % + %/ + x A(B) +

x A(A), !+

$ +%1 '/ i.

4.1. E; ;E& 49

& !#' " %/ + " m n %/ #

+ . %% " %"/ !+ #-

$ %, + , / '

#"% ! $ ! , #"% ! Gauss,

#"% " #"! # ; .

$ 4.2 / m n

m n.

&. / ! !/ "#! A(1) := A ,

A(1) =(a(1)ij

)i=1,...,mj=1,...,n

.

A %+, %% + ' ! %#, +

% ' ! . > , %% ' %

. & $ %! ' $ % -

! A, # + ! , , # $,#! $ , %% "# (1, ), # %+ '

. H"#, ' + +, + a(1)1 = 0, $ #!, ! '/ $, + '

+ + $ % . !+ 1: 4!

' ! mi , i = 2, . . . ,m ,

(4.2) mi :=a(1)i

a(1)1

, i = 2, . . . ,m .

;+ ! $ ! mi , /

# + i ! "/ i !

/ !+ +. !+ i = 2, . . . ,m . * !

%% $ %/ %/ ! A(1)

A(2)

50 4. ;E&

A(2) :=

a(1)11 . . . a

(1)1 a

(1)1,+1 . . . a

(1)1n

0 . . . 0 a(2)2,+1 . . . a

(2)2n

......

......

0 . . . 0 a(2)m,+1 . . . a

(2)mn

,

! ! $ 1 , + ! A(1), %#

a(2)ij := a

(1)ij mi a(1)1j , i = 2, . . . ,m, j = , . . . , n .

; !+ + ! ' ai, i = 2, . . . ,m, - ! A, !+ #"% # #"% .

'/ $ %/ . E1 ! (m 1) (n 1) !- A(2) ! A(2) '

A(2)ij := a

(2)ij , i = 2, . . . ,m, j = 2, . . . , n .

& %/ ! $ % ! $ ,

! + A(2). ; !+ + %/ #

A(3)

. . .

0 . . .

0 0 . . . ...

......

...

0 0 . . .

,

+! %/ $ # ! A(3) % %/ $ # !

A(2), %/ ! % $ %/

.

4.1. E; ;E& 51

%% ' % $ +. + #'

1: ; r, 1 r min(m,n) 1 , #'! # A(r), V-%/ ! A,

A(r) :=

a(1)11 a

(1)12 . . . . a

(1)1n

0 a(2)22 . . . . a

(2)2n

. 0 .

.

. a(r1)r1 r1 .

. 0 a(r)rr . . . a

(r)rn

.. . .

...

0 0 0 a(r)mr . . . a

(r)mn

.

I/ !

A(r) =(a(r)ij

)i=r,...,mj=r,...,n

.

A(r) %+, %% $ !+ . >-

, # $ % ! A(r). * #

$ ! A(r) #! "# (r, ) # %+ '.

52 4. ;E&

(4.4)

a(r+1)ij = 0 , r + 1 i m, 1 j ,

a(r+1)ij = a

(r)ij mi a(r)rj , r + 1 i m, + 1 j n .

E%$ $ r+ . * + / r = min(m,n) 1 ,

V%/ ! '/ A(r+1) +.

A$ 5 4.1 + 4.2 / %$! #

+ #'! + %%# %/ x1, . . . , xn ! Rn -

/ !, + %! ! '$! ! !.

"#! $ # +, %+

#! , . 4.2, 4.3, 4.4. E%+, /

%! + + Rn

Rm, . 4.5.

$ 4.1 # x1, . . . , xn Rn. $ {x1, . . . , xn} Rn, n n

x1, . . . , xn,

x1...

xn

,

x1...

xn

,

xij = 0 i > j, %

.

4.2. ) #

& ! + " ! 1 1/ , "$

+ "/ .

4.2. FRE& *E ;E& 53

+ / "/ 1

A + B := (aij + bij) i=1,...,mj=1,...,n

, A := (aij) i=1,...,mj=1,...,n

,

+! A = (aij) i=1,...,mj=1,...,n

, B = (bij) i=1,...,mj=1,...,n

R.* !# 1 Rm,n + '$. E+ + %-

" ' !, '$ Rm,n Rmn %,

+

A =

a11 a12 . . . a1n

a21 a22 . . . a2n...

......

am1 am2 . . . amn

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

. %+ ' ! Rm,n m n % ', " ! A = (aij) A = (aij). /

{Iij Rm,n : i = 1, . . . ,m, j = 1, . . . , n}, +! Iij #' + ' !% , + + % "# (i, j), %% i j , ! Rm,n.

4 AT := (bij) Rn,m bij := aji # ! A =(aij) Rm,n. # %$ + ! ! '/!

(A + B)T = AT + BT , (A)T = AT , (AT )T = A.

!

%!/ '$! % + %!

' ! '$!. 4 + + 1/ #-

.

I/ %/ A = (aij) Rm,n B = (bij) Rn,, +! %/#' % " $ " $ ! $!. 4 m C ' cij, cij = ai1b1j +ai2b2j + +ainbnj, A B ! AB. E/ ' ! $ + +

#' #! ! '+ !

#! ! BA ( %

% ) , + ! ,

54 4. ;E&

%+ ! AB. S #'!(1 0

0 0

) (0 1

0 0

)=

(0 1

0 0

), $

(0 1

0 0

) (1 0

0 0

)=

(0 0

0 0

).

! +, + + ! %, + +

%/ %$ %+ .

% 4.2 # A,A Rm,n, B,B Rn,, C, C R,r . $ %:

i. "

A(B + B) = AB + AB

(A + A)B = AB + AB.

ii. A(B) = (A)B = (AB).

iii. A(BC) = (AB)C ( ).

iv. (AB)T = BTAT .

&. i. ii. .

iii.

4.3. &;&E& 55

iv.

56 4. ;E&

4.5. % ! '$! L(R5), +! + L :

R5 R4 1: Lx := (2x2 + 4x4, x3 + x5 x4, x2 x1, 3x2 x3 + x4) x = (x1, x2, x3, x4, x5).

[(: $ #'! Lx = x1(0, 0,1, 0) + x2(2, 0, 1, 3) + x3(0, 1, 0,1)+x4(4,1, 0, 1). E#, '$ L(R5) < (0, 0,1, 0), (2, 0, 1, 3), (0, 1, 0,1),(4,1, 0, 1) > . , !$, %! ! '$!,

! %/ (0, 0,1, 0), (2, 0, 1, 3), (0, 1, 0,1), (4,1, 0, 1).]4.6.

5. ' #$

& !+ '# 1/ $ -

. % " %/ + " + 1/ $ '$

# % # . & !+ -

+ , + "+,

.

5.1. ; # % #$0

I/ %/ / '$! # % %/ !'/,

"#, !. M " %/ ! +, "

+ 1/ !$ '$ + # , , -

, " , % , ' # +.

E " %/ + /" %/ $

+ ' .

58 5. F**;E& E;4&E& ; ;E&

+ 1/ X Y, !! LA, # $ LAxj =

yj, j = 1, . . . , n, %%

LAxj =m

i=1

aijyi, j = 1, . . . , n.

M + " # + ! / '$! # % X

Y

"! , + " + L : X Y' # dim Y dimX , " dimY dimX -' + L : X Y.

%# , % + '/ %-

.

5.2. DI*4& *& F**;& E;4&E&, &4*4FS&*4, 55 D&E& 59

R3 R2 (1 1 12 0 1

).

60 5. F**;E& E;4&E& ; ;E&

5.2. DI*4& *& F**;& E;4&E&, &4*4FS&*4, 55 D&E& 61

L1 : X Rn

xi ei, i = 1, . . . , n,A : Rn Rm

x Ax,L2 : Rm Y

ei yi, i = 1, . . . ,m,

+! {e1, . . . , en} {e1, . . . , em} # Rn Rm, '.&$! + + + L1 + x1, . . . , xn, !

, / + 3.1, $ # + X '

'/! + L2. $ #'!

(L2 A L1)xj = (L2 A)(L1xj) = (L2 A)ej = L2(Aej)= L2(a1je

1 + + amjem) = a1jL2e1 + + amjL2em

= a1jy1 + + amjym = Lxj, j = 1, . . . , n,

# L2 A L1 = L.E% L1(X) = Rn L2 +, !!

{v1, . . . , v} ! A(Rm), + / {L2v1, . . . , L2v} " ! L1(A(Rn)), ! L(X). ; !+ + /' !

+ +!" %+ +:

62 5. F**;E& E;4&E& ; ;E&

&/ /, "+ A

"+ $ ! A, D&(A), . 4+ 4.2, /

A, "+(A).

! 5.2

5.2. DI*4& *& F**;& E;4&E&, &4*4FS&*4, 55 D&E& 63

% 5.1#X Y

% , dimX =

dimY < . & L : X Y

, :i. < L , .

ii. < L , .

iii. < L , .

&. +%1 !# ! /! %

dimX = dim mL + dimKerL,

. I$ 3.1, + 3.2, / L #

$ +, ! #' + %+ % !.

&/ / #, + L : X Y 1/ %/ $ '$ X Y, dimX = dimY = n, -

+, + "+ ! L, %% "+ $ !/ !

, n.

! 5.3

1. * + L : R6 R2 % +,/ % ! %! / ! %! $ %-

#.

2. E +L : R3 R3, L(x1, x2, x3) := (x1+x2, 2x1+4x3, x1x2+x3),+;

L $ + % ! %! /

! %! $ %. E+", A L

! R3

A =

1 1 0

2 0 4

1 1 1

! #'!

AT =

1 2 1

1 0 10 4 1

.

64 5. F**;E& E;4&E& ; ;E&

4 AT V%/

1 2 1

0 2 20 4 1

,

1 2 1

0 2 20 0 3

,

"+ L , ! %% % ! R3, #

L +.

" 5.3

5.2. DI*4& *& F**;& E;4&E&, &4*4FS&*4, 55 D&E& 65

! Rn A A1, '. /" A1 A'/:

A1 A : Rn Rn, (A1 A)x = A1(Ax) = (A1A)x = Inx = x,

!$ A # #, +, 5 5.1, +,

# #'! D&(A) = n.

$, D&(A) = n, + + A , +, 1

/ 5 5.1, +. E# ! '

+ A, A1. 4 + A1 A ! Rn A1A, % A1 A !+ Rn " #'!A1A = In. $ ' %/ + AA1 = In.

& +!" + %! !" # -

+ #=. # !+ ! %#,

# !' #. / +%1

, . 6.2.

$ 5.2 # A n n , A Rn,n. & %

(5.1)n

j=1j =i

|aij | < |aii|, i = 1, . . . , n,

A =.

! / (5.1) # !$! #

+ #'! % .

66 5. F**;E& E;4&E& ; ;E&

&.

5.2. DI*4& *& F**;& E;4&E&, &4*4FS&*4, 55 D&E& 67

I#! $

B =

b11 b12 . . . b1n

b21 b22 . . . b2n...

......

bn1 bn2 . . . bnn

! B

xj =

ni=1

bijyi, j = 1, . . . , n,

, + (5.2), " #'!

(5.3)

1...

n

= B

1...

n

.

+ (5.2) (5.3) ! #

(5.4)

1...

n

= BA

1...

n

,

1...

n

= AB

1...

n

.

5+ ! + (5.4) '/ !'+ %/ (1, . . . , n), (1, . . . , n) Rn,!! + B = A1, (5.3)

(5.5)

1...

n

= A1

1...

n

.

! '+ +, +, %!

A1. * !+ "# " #'! ! '"/ !-

#'.

&$! + (5.5) %/

(5.6) A

1...

n

=

1...

n

.

'# ! / + 1, + " %/ !-

#', / !+ ! (1, . . . , n) $ + ! +

68 5. F**;E& E;4&E& ; ;E&

/ !, #! + ; ,

+' + . + " %/ + +

/ %+ %% + ! + / !.

5.3. $%$

5.1. E1 + + L : R3 R4, L(x1, x2, x3) := (x1, x1 +x2 x3, 3x1 + 5x2, 4x2 2x3), . % L # R3 R4, "+ ! /

'$! L(R3).

5.2.

6. !($%

& !+ !, %% # 1$-

1/ '$ # % . # + " ! -

, ! / !' #, #

/ /

, +, ! -

$ ! .

70 6. F**; &H&*

/ (6.1)

(6.1) Ax = b.

6.1. 4*4E F**; &H&* 71

&. 4 $ ! + A : Rn Rm, x Ax. E# !+' ! Rn, / '# % ! # + / % , . I$ 3.1.

&+ $ %$! #"% %/ + /

+ / !. & + / / !-

, %! {u1, . . . , u} ! '$! /$ ! , /,! , " + '$.

D+ + %% ! $ ! +-

!" #:

% 6.1 # A m n S = m m . $

Ax = 0 (SA)x = 0 % % .

&.

72 6. F**; &H&*

%% # %$ "# (i, i) % + '

%!,

(6.4) Qji =

1 0. . .

1 1. . .

1. . .

0 1

,

%% # % %$ "# (i, j) % +

' !. # %$! + ! #-

=, ! (Si()

)1= Si(

1)

(Qji)1

=

1 0. . .

1 1. . .

1. . .

0 1

.

M % (6.3) (6.4), / #!

+ + + A + Si()

+ i ! A , $ + Qji , j = i, +" j ! A i !,

Si()

a1...

ai...

am

=

a1...

ai...

am

,

6.1. 4*4E F**; &H&* 73

Qji

a1...

ai...

aj...

am

=

a1...

ai + aj...

aj...

am

.

; !+ + + ! '$% '/ -

$ i. ii. + + #

. E# %, ! , ! '$% -

'/ $ iii. iv..

(6.3) (6.4) "$ ! / -

% .

! '$% '/ $ ! $

! '$% '/ $.

" 6.2

74 6. F**; &H&*

iii. "/ i ! A " i ! ! j ! # + "+ ,

(. . . ai . . . aj . . . ) (. . . ai + aj . . . aj . . . ) .iv. ! i j ! A,

(. . . ai . . . aj . . . ) (. . . aj . . . ai . . . ) .

&$! +, + '%$ '$ -

$, '$% ' $ iii. iv.

/! -

# '$ i. ii..

E, 1 + '%$ '$ -

$, '$% ' $ i. ii. /

"/ +, ! + %1 , # -

(6.3) (6.4). , + + A + %-

1 Si() + i ! A ,$ + Qij, j = i, +" j ! A i !,

(a1 . . . ai . . . an)Si() = (a1 . . . ai . . . an),

(a1 . . . ai . . . aj . . . an)Qij = (a1 . . . ai + aj . . . aj . . . an),

+! ! , ! , Si(), Qij Rn,n, i, j = 1, . . . , n, j = i - + + + + %1 Qij +' Q

ji .

E# %, ! , ! '$% '-

/ $ iii. iv..

I %/ $ + + + (6.3)

(6.4), + + %1 , "+ !

.

$ 6.2 # A = (aij) mn , Si(), Qji (6.3) (6.4). $ Si()A,ASi(), Q

jiA,AQ

ji %

A.

&. ; ', + ! A + %1 # Si()

Qji #' !# # '$% '+ $ ! A. E#

6.1. 4*4E F**; &H&* 75

%! '$ ! ! ! ASi(), AQji

A !!, !$ % ! %.

# #= mm S, #'!, A = (a1 . . . an), SA =(Sa1 . . . San), , % + S : Rm Rm +,'/

dim < a1 . . . an >= dim < Sa1 . . . San >,

+ ! %/ !+ # +.

I %/ $ + "+ + , !$

"+ ! , "+ ! # !!, +

!#' " / $ "+ + .

$ 6.3 # A = (aij) m n . $ A

.

&.

76 6. F**; &H&*

B =

b1j1 . . .

b2j2 . . .

b3j3 . . .. . .

brjr . . .

0

,

+! $ r # %#, !+ %#, $

ji 1 ' i %#, i = 1, . . . , r. &/ 6.1 6.3, % ! '$! / ! / !-

Bx = 0, , ! , '$ / ! Ax = 0, n r,(6.5) dim = n r. !$, ' + +, !"#-

! + j1 = 1, . . . , jr = r, / !+ !'"

$, %% " $. 4 B " +

B =

b11 . . .

b22 . . .

b33 . . .. . .

brr . . .

0

.

* := n r 1, . . . , R, / %/ + ! ' $ #x Rn x = (x1, . . . , xr, 1, . . . , ) . , 1 r ! !Bx = 0 brrxr +br,r+11+ +brn = 0, , / brr = 0, ! ' $ #xr ! ! '#. + 1 r1 % - xr1 / " 1, , #, + $ 1 %

x1.

6.1. 4*4E F**; &H&* 77

78 6. F**; &H&*

/ Bx = 0, %%

(6.7)

x2 + 2x4 x5 4x6 = 0x3 x4 x5 + 2x6 + x7 = 0

x5 + x6 = 0

x6 + 2x7 = 0 .

* x1 = 1, x4 = 2 x7 = 3 (+ +: $, #

#% ! B % % '

) #'! + !+ + /

(6.8)

x6 = 2x7 = 23x5 = x6 = 23x3 = x4 + x5 2x6 x7 = 2 + 53x2 = 2x4 + x5 + 4x6 = 22 63 .

4 '$ / # + +

G : R3 R7

(1, 2, 3) (1,22 63, 2 + 53, 2, 23,23, 3) , # #. *

G(1, 0, 0) = (1, 0, 0, 0, 0, 0, 0) =: u1

G(0, 1, 0) = (0,2, 1, 1, 0, 0, 0) =: u2G(0, 0, 1) = (0,6, 5, 0, 2,2, 1) =: u3 ,

#'! # =< u1, u2, u3 > .

+ !+ %!$ + 1: &/-

(6.8), !' #! 1, 2 3, / ! ! (6.7)

(1,22 63, 2 + 53, 2, 23,23, 3). $(1,22 63, 2 + 53, 2, 23,23, 3) == 1(1, 0, 0, 0, 0, 0, 0) + 2(0,2, 1, 1, 0, 0, 0) + 3(0,6, 5, 0, 2,2, 1) == 1u1 + 2u2 + 3u3 ,

%% =< u1, u2, u3 > .

6.2. 4*4F555;4 H4AF4 ; * 4*4E F**; &H&* 79

6.2. ;# (#0 % $($%

& ! + " ! , +' , -

! " ! % %+ /$ !.

80 6. F**; &H&*

E+", + + y1 + Y1 = y2 + Y2 !! # +

! ' x Y1 # $ y2 = y1 + x, + y2 y1 = x Y1.&/ / # / $ %$! +-

!" +:

" 6.4

6.2. 4*4F555;4 H4AF4 ; * 4*4E F**; &H&* 81

# $ # + + 6.4 '# %

%' + 6.1.

82 6. F**; &H&*

% 6.3 # A m n . $

Ax = b % b Rm, A m, "+(A) = m.

&. H"#! $ + + / Ax = b #' / "

b Rm. !+ ! + ' + A , +A(Rn) = Rm, !$ "+(A) = dimA(Rn) = dimRm = m.

, "+ ! A m, + dimA(Rn) = m, %% dimA(Rn)

= dimRm, +, ! += + + A(Rn) Rm, !-! + A(Rn) = Rm, %% " b Rm ! ' x Rn # $ Ax = b,

+ + / Ax = b #' / " b Rm.

E/ $ %/ !" #' #

+ / $ /. ! % + / Ax = b #'

/, + "+(A) = "+(A, b). / ! %

# ' # / Ax = 0 #' + # /,

%% ! + A : Rn Rm, x Ax, #' + %+ % !, + + / % !-

dim mA = n, + "+(A) = n. &! %% " #'! "+(A) =

"+(A, b) = n.

% 6.4 # A m n b Rm. $

Ax = b% , "+(A) = "+(A, b) = n.

&. &/ I$ 6.1, + / Ax = b #' /,

+ "+(A) = "+(A, b). E+", '# "+(A) = n

dimA(Rn) = n, + / % % dimKerA = 0,

%% KerA = {0}.H"#! $ + "+(A) = "+(A, b) = n. +, /

/, / Ax = b #' ! ' /,

'$ /$ ! x , + '/ = x + KerA, !$ = {x}, %% / #' $ /.

, + / Ax = b #' $ / x, +

= {x}, + '# = x+KerA % KerA = {0}, !$, / / % , dimA(Rn) = n, "+(A) = n.

6.3. *EI4>4& 54S& 4H GAUSS 83

% # $ !

% " 1$ $. + ! 5 6.4

5.1 +!" #, / # +

/ Ax = b, +! A # n n , #' + / $ +, A #= .

$ 6.5 # A n n b Rn. $

Ax = b% , A =.

6.3. '/* #! $ #"% ! $ ! . -

# #"% ! , !, % # ,

'/ +! ! % ! / !$ -

$ 1"/, '/ / 1, ! +' /

!.

84 6. F**; &H&*

; !+ +, + / min(m,n) 1 , / "! (6.11).

6.3.2. ;#$/*$. ! ! !/, !

/ (6.11) Bx = c $ !1# (B, c) " -

(B, c) =

b1j1 . . . c1

b2j2 . . . c2

b3j3 . . . c3. . .

...

brjr . . . cr

cr+1...

0 cm

,

b1j1 = 0, . . . , brjr = 0. $, "+ ! A r, % (A, b)

(B, c) #'! % "+, !! / + #'!

(A, b) A % "+ %/ cr+1 = = cm = 0.&/ I$ 6.1, '$ / ! / ! Ax = b

!$ $ + +, cr+1 = = cm = 0. & r = m, % "+! cr+1, . . . , cm, , + % 5 6.3,

/ Ax = b #' / " b Rm.H"#! $ + cr+1 = = cm = 0. M -

$ $ ! , # xj j {j1, . . . , jr} /" . !$ !"#! ! +

j1 = 1, . . . , jr = r. %! % / ! / -

/ ! Ax = b, "#! xr+1 = = xn = 0. + r 1 !Bx = c ! + brrxr = cr, + %$ !! xr.

!! xr1, . . . , x1, ! ! %

/ u = (x1, . . . , xr, 0, . . . , 0) ! Bx = c. / (B, c)

/ +

(A, b) + #! "! '$% '/ $,

! ' # #= n n S, # $ (B, c) = S(A, b), %%(B, c) = (SA, Sb). E# #'!

Au = S1SAu = S1Bu = S1c = b,

6.3. *EI4>4& 54S& 4H GAUSS 85

%% u / ! / ! Ax = b.

$, #"%, !! '$ / ! /

/ !Ax = 0, + '$ / !Ax = b = u+.

! 6.2 I/ # + /

(6.13)

x1 2x2 + x3 = 1x1 2x2 x4 = 2

x3 + x4 = 1 .

E+ '$% '/ $ !1#

!/ ! / !,

(A, b) =

1 2 1 0 11 2 0 1 20 0 1 1 1

,

! !

1 2 1 0 10 0 1 1 10 0 1 1 1

,

1 2 1 0 10 0 1 1 10 0 0 0 0

,

1 2 1 0 10 0 1 1 10 0 0 0 0

=: (B, c) .

E% (A, b) A #'! % "+, 2 =: r, / Ax = b #'

/. 4 '$ / #' % %/, dim = n r = 4 2 = 2. E#'! j1 = 1, j2 = 3. * x2 = x4 = 0, #'! x3 = 1 x1 = 1 x3 = 2, u = (2, 0,1, 0) / ! Ax = b.

%! '$ / ! Ax = b, %! $

'$ / ! '! / / ! Ax = 0, !

%/! !+ Bx = 0, %% !

(6.14)x1 2x2 + x3 = 0

x3 + x4 = 0 .

86 6. F**; &H&*

* x2 = 1 x4 = 2 #'! x3 = 2 x1 = 21+2. 4 '$ / # + +

G : R2 R4

(1, 2) (21 + 2, 1,2, 2) , # #, %% #'! #

=< G(1, 0), G(0, 1) >=< (2, 1, 0, 0), (1, 0,1, 1) > .&!$, '$ / ! / ! Ax = b % 1

= u + = (2, 0,1, 0)+ < (2, 1, 0, 0), (1, 0,1, 1) > .* +, / ! / ! (6.13) %/

(2, 0,1, 0) + 1(2, 1, 0, 0) + 2(1, 0,1, 1) = (2 + 21 + 2, 1,1 2, 2) ,+! 1 2 %#'! +! ! / "/.

6.4. $%$

6.1. % ! '$! / 1 $ $ !

x1 + 2x2 x3 + x4 = 02x1 x2 + x3 + x4 = 0x1 x2 + x3 + 2x4 = 0 ,

x1 + 2x2 x3 = 02x1 x2 + x3 = 03x1 + x2 + x3 = 0 .

6.2.

6.4. &;&E& 87

6.3.

88 6. F**; &H&*

6.8. %1 + 6.5.

6.9. %1 + # n n A #=, +

# + / Ax = 0 #' + # /, x = 0.

6.10.

7. ;?($

& !+ " ! # '

, / . ! +,

' " + + # + "-

+. I ! % %+ !$, " %/ $ "

! / %$! = / + / !-

% " 1$ $. % + + !

! %$, "# " ' #-

; .

7.1. ;$, "#) $% ?($

& ! + " %$! # + + !, " %/ +

!+ + ! #, " ! %

%+ !$.

; " A Rn,n " !!

a1...

an

,

+! ai i !.

" 7.1

90 7. 4F_4H&E&

i. ai = ai + ai , +

det i

...

ai...

= det

...

ai...

+ det

...

ai...

.

ii. ai = ai, +

det

...

ai...

= det

...

ai...

.

D2 %/ # ! nn A #' %/ # ! %, + ! ! %#, detA = 0.

D3 ! ! n n %! / %, det In = 1.

4 + % 1, ! , /1 !. & !#'

" %1! /1 %+, %% + ! ' $ +-

%+ D1, D2 D3.

E$ / "/ ! + Kn,n,

+! K # $. 4 ! ! + !

%$, " '"/ ; 9, +!

" '! ! %/ / . E

! %!/! ! =+ + / $

! % " 1$ $, #=

!$.

'! +%1 /1 %+ !,

$! # # %+# !, + !

/ !$.

% 7.1 # A Rn,n,

A =

a1...

an

.

$ %:

7.1. 4F&*4&, HFR ; *44&*4 & 4F_4H&& 91

i. &

, ,

det

...

ai...

aj...

= det

...

aj...

ai...

.

ii. &

,

,

det

...

ai + aj...

aj...

= det

...

ai...

aj...

.

iii. & %

, ,

ai = 0 i {1, . . . , n}, detA = 0.iv. & , n,

det (A) = ndetA.

&.

i.

92 7. 4F_4H&E&

+! $ + ' %/ # D2, %/

D1 ! 1 D2.

ii.

7.1. 4F&*4&, HFR ; *44&*4 & 4F_4H&& 93

!, # ! '! aij . ! '! aij . 4 cij =

(1)i+jdetAij # ! aij.

+ 7.1 (% D : Rn,n R, D1, D2 D3.

&.

@. I/ # !', "+ !#', j {1, . . . , n},

! % +

det : Rn,n R

A = (aik) Rn,n n

i=1

aij(1)i+j detAij ,

det (a) := a, 1 1 ' # + "+ a. I %-1! $, n, + det !" D1, D2 D3.

n = 2,

A =

(a11 a12

a21 a22

),

j {1, 2}, #'! detA = a11a22 a12a21, %$! # + det#' %+ D1, D2 D3.

& + !"#! + det #' %+ D1, D2 D3

n 1, " %1! + #' !# %+ n. *

A =

a1...

am...

an

B =

a1...

am1am

am+1...

an

,

"+ + "+, #'!

94 7. 4F_4H&E&

(7.1)

detB =n

i=1

bij(1)i+j detBij

= bmj(1)m+j detBmj +ni=1i=m

bij(1)i+j detBij .

$, bmj = amj,detBmj = detAmj, , / !+" ,

detBij = detAij, i = m. E# (7.1) %

detB = [amj(1)m+j detAmj +

ni=1i=m

aij(1)i+j detAij]= detA .

7.1. 4F&*4&, HFR ; *44&*4 & 4F_4H&& 95

detC = (amj + bmj)(1)m+j detAmj +ni=1i=m

aij(1)i+j(detAij + detBij

)

=n

i=1

aij(1)i+j detAij +n

i=1

bij(1)i+j detBij

= detA + detB .

&!$, #' $ %1 + det D1 n.

96 7. 4F_4H&E&

%% det D3 n.

0.

7.1. 4F&*4&, HFR ; *44&*4 & 4F_4H&& 97

/ m,m n, # $ %"/ % In,

In =

e1...

en

,

+, / " $ ! + D D, !-

$ , " #'!

(7.5ii)

ej1...

ejn

= (1)m(In) = (1)m(D(In) D(In)) = (1)m(1 1) = 0 .

5 $ ! += (7.5), (7.4) % # (A) = 0, " A Rn,n, !$ D = D.

&! , +, / /, ! + n n A , n, 1: n = 1 A = (a) "#!

detA := a, n > 1, # !' j {1, . . . , n}, "#!

(7.6) detA :=n

i=1

aij(1)i+j detAij ,

+!Aij ! '! aij , %% (n1)(n1) !

/ + A, % =! i ! j !. '# (7.6) # ! ! j.

+ + 1+ 1 ! j, + !

I 6.1.

>! $ # "+ #, + " #= -

!" + '%$ , " '!

+%1 + +.

% 7.2 # A Rn,n = . $ % r % E1, . . . , Er

(7.7) A = E1 Er.&. ! % + A , ! -

'/ $, %% + +

98 7. 4F_4H&E&

Si(), Qji , #= + % %-

$ ', +

E1 EsA = B =

1 . . .

2 . . . . . .

...

0 n

.

$,

S1(1

1) Sn( 1

n)B = B =

1 . . .

1 . . . . . .

...

0 1

.

* ! '/ $,

! ! "/ $ + " +

/ # # $ %/ + ' !

+ + ! '$ ' , /-

/'! 1

E1 EqB = In.

&!%! / ! #

E1 EqS1( 11

) Sn( 1n

)E1 EsA = In.

# $

/ # + $

(E1)1, !' (E2)1, / " 1, # (Es)1, -

! += + + + ! !

'$%.

4# ' %+ !$ % +!" +-

.

$ 7.1 # A Rn,n. $ %:

7.1. 4F&*4&, HFR ; *44&*4 & 4F_4H&& 99

i. & A , aij = 0 i > j,

A =

1 . . .

2 . . . . . .

...

0 n

,

% A, detA =

1 n.ii. & a1, . . . , an

A, A ,

a1, . . . , an

.

iii. # =,

.

iv. < n n . +, A = ,

, detA1, A, detA1 =

1/detA.

v. < , detAT =

detA.

&.

i. +%1 n. n = 2 '#

, %'"/ + '/ n 1 /1! n n ! $ , + + $ + %+ #'!

(n 1) (n 1) ! % , %$! # + '#'/ n.

ii. * ! '/ $ $ iii. iv., .

+ 4.2, + A

/ # + B,

B =

1 . . .

2 . . . . . .

...

0 n

.

!m ' iv., + #'! detA = (1)mdetB, -#, / i., detA = (1)m1 n. H"# $ + detA

100 7. 4F_4H&E&

% ! %+, + ! + + i % ,

%. "+ ! B, !$ ! A, # n, !!-

+ %/ a1, . . . , an 1 . 5 %/

!+ + + ! ! A %, + a1, . . . , an

1#.

iii. &/ + 5.1, # + nn A #-=, + "+ ! A n. &/ ii.,

+, ! ! A % ! %+, + "+

! # n. E% " ! A #

!, !! + ! ! A % %#,

+ A #=.

iv. %/! ' ! %+ ! A

'$% . ;+ '/ !+ # %-

"/ # .

7.1. 4F&*4&, HFR ; *44&*4 & 4F_4H&& 101

det(QjiB

)= det

b1...

bi1bi + bj

bi+1...

bj...

bn

= det

b1...

bi1bi

bi+1...

bn

= detB

$ ' %/ +

det((Qji )

1B)

= detB .

+ # + +

(7.8) det(EB

)= detEdetB ,

+ E '$% . $ A #=,

+, / 5 7.2, + '%$

(7.7), '$

+ # #'!

det (AB) = det (E1 Er B)= detE1det (E2 Er B)= detE1detE2 detEr detB= det (E1 Er)detB = detAdetB .

#, A % #=, +, / iii., detAdetB = 0.

E, dimA(Rn) < n, !$, "+ (AB) = dimAB(Rn) dimA(Rn) < n,

+ det (AB) = 0.

v.

102 7. 4F_4H&E&

A % #=, + "+ ! A #, !$

"+ ! AT , + ! n, !$ AT

% #=, + detA = detAT = 0.

A$ + + ! ! +! + /-

! ! , / %$! + !

+ , . 7.5. ! %+

" '" + +.

7.2. @#$ ?($

7.2. H454&*4& 4F_4H& ; ESF*4E& 103

# (7.9)

/ + Sarrus, %%

1: E ! %/ $ ! ,

+ + +

a11 a12 a13 a11 a12

a21 a22 a23 a21 a22

a31 a32 a33 a31 a32

'! + $ % +

%1 , "+ +, $ % +

%1 , + +, + #, !

" #1 + !

/!. ! + ! Sarrus -

+ + + % ! ' ! / n, n > 3.

+ n, # + !/ ! % !" +

+ . 4 + !+ %+, # 1.

104 7. 4F_4H&E&

+ #'!

(7.11) A(u1, . . . , un) = (Au1, . . . , Aun) = (e1, . . . , en) = In,

!$ (u1, . . . , un) ! A. ! (7.10) -

/ !"/, , #"% ! Gauss. S! #"%

+ !+' + !, / ! #'! %

!$. !+ # + ! '

1, ' # ! ' + + .

7.2. H454&*4& 4F_4H& ; ESF*4E& 105

" 7.3

106 7. 4F_4H&E&

7.3. $%$

7.1. / "/ t s, %1 +

det

1 t s 1

1 t t t

t 1 ts s

t t ts 1

= (t 1)(t s)(1 ts) .

I /, ' ! !/, #= + 1+-

#' "# + ! %/ $! ! % #;

7.2. I/ / "/ t1, . . . , tn. %1 +

det

1 1 . . . 1

t1 t2 . . . tn

t21 t22 . . . t

2n

......

...

tn11 tn12 . . . t

n1n

=

1i

7.3. &;&E& 107

" ! ! cn % + '#

() cn := det

1 1 . . . 1

t1 t2 . . . tn1t21 t

22 . . . t

2n1

......

...

tn21 tn22 . . . t

n2n1

.

$ # # n = 2, , !"# + "/

n 1, + () () !! # + "/ n.]E: H !

1 1 1 1 1

2 3 1 4 5

4 9 1 16 25

8 27 1 64 125

16 81 1 256 625

,

1 1 1 1

x y z

x2 y2 z2 2

x3 y3 z3 3

.

7.3.

108 7. 4F_4H&E&

iii. 1! %/ + , + ! ! +-

.

iv. + "#! #

, + ! # .

v. # n n #' % , + ! ! %#.

vi. a1, . . . , an + nn A, + ! !A %#, + %/ a1, . . . , an 1#.

7.6. >$ / 6.7 " !$ "+%! !

Cramer.

[(: + Ai := (a1, . . . , ai1, ej , ai+1, . . . , an) ,

i > j, + ' ! "# (i, i) %#, !$

! ! %#.]

7.7. % / ! / !

x1 + x2 + x3 + x4 + x5 = 0

x1 + 2x2 + 4x3 + 8x4 + 16x5 = 0

x1 + 3x2 + 9x3 + 27x4 + 81x5 = 0

x1 + 4x2 + 16x3 + 64x4 + 256x5 = 0

x1 + 5x2 + 25x3 + 125x4 + 625x5 = 0

7.8.

8. & $0

+ + #' $ # %!$" -

/ / '$!, / + / !"/ '$!

# !'+ $ K. # !/ ! ! '/!, +

$ $ $ "$ R $ %$

"$ C. I !! + %$ $ K "

/

+ K = R + K = C.

& !+ " ! % ' #, $ %-

# + !# ! ' #' $. ; ' " !-

# +, ! +! $ "$

/ '$!, ' " % ! # +

"+ # # #" + %/. H -

'! + + % /' +

! + # + +, !+ + !+

" '"/ #! '$!. I ! % %+ !$

'$ " %/ + # # # /

#! '$! !"/ / /, $ !+ % + +

+ % + + +.

8.1. ' ( ( $0

* %!+ %! # + '$

% # +. + !

+! / "/. & ! + " !

# ! / #! " %/ # %+ $ !

+ +.

" 8.1

110 8. AF4 *E E&EF;4 4*E4

: X Rx x ,

# (, norm), '/!:

(N1) x X x = 0 x = 0

(N2) K x X x = || x

(N3) x, y X x + y x + y ( +).

# K+ '$ " +, + !+ # % % .

. & '# (N2) '/, K, +! ! ,# ! + / %/ "/.

! 8.1

(i) + 1$ + # + x X '/ x 0 . #'!

0 = x x = x + (x) x + x = x + x = 2 x ,+! + '/ + (N3) ! + + (N2).

(ii) '/ + , %%

x, y X x y x y . #'!

x = (x y) + y x y + y , %% x y x y

' y x y x , + # # /.! 8.1

1.(R, ) x := |x| x R . (C, ) z := |z| z C .

2.(Rn, 1

) x1 :=

ni=1

|xi| , x = (x1, . . . , xn) (1 +).

'/!:

8.1. E4 4H AF4H *E E&EF;4 4*E4 111

(N1) x1 = 0 n

i=1 |xi| = 0 xi = 0, i = 1, . . . , n x = 0 . (N2) R , x Rn #'!

x1 =n

i=1

|xi| =n

i=1

|| |xi| = ||n

i=1

|xi| = || x1 .

(N3) x, y Rn

x + y1 =n

i=1

|xi + yi| n

i=1

(|xi| + |yi|) = ni=1

|xi| +n

i=1

|yi| = x1 + y1 .

3.(Rn,

) x := max

1in|xi| ( +).

+%1 + + Rn / .4.(C[a, b],

) f := max

axb|f(x)| (+ !), +! < a < b < .

'/!:

(N1) f C[a, b]f = 0 max

axb|f(x)| = 0 f = 0 .

(N2) R f C[a, b] #'!f = max

axb|f(x)| = || max

axb|f(x)| = || f .

(N3) f, g C[a, b]f + g = max

axb|f(x) + g(x)| max

axb[|f(x)| + |g(x)|]

maxaxb

|f(x)| + maxaxb

|g(x)| = f + g .

4 / ' ! + + +

+.

" 8.2

112 8. AF4 *E E&EF;4 4*E4

(E2) (x, y) = (x, y) x, y X K

(E3) (x, y) = (y, x) x, y X

(E4) (x, x) > 0 x X \ {0} .

8.1. E4 4H AF4H *E E&EF;4 4*E4 113

% 8.1 (+ CauchySchwarz.) % ,(X, (, )), % CauchySchwarz(8.1) x, y X |(x, y)|

(x, x)

(y, y) .

&. & y = 0, (8.1) '/ $.

114 8. AF4 *E E&EF;4 4*E4

4 !"# + "

ni=1

xiyi ( n

i=1

|xi|2)1/2( n

i=1

|yi|2)1/2

,

%/ "/ xi, yi, i = 1, . . . , n, "$

$

ba

x(s)y(s) ds ( b

a

|x(s)|2 ds)1/2( ba

|y(s)|2 ds)1/2, !' ! # % [a, b], / %# -

$ + CauchySchwarz, !# ! '$!

(Cn, (, )) (x, y) := ni=1 xiyi x = (x1, . . . , xn), y = (y1, . . . , yn) Cn, (C[a, b], (, )) (x, y) := b

ax(s)y(s) ds .

$ 8.1 #(X, (, )) % . $

x :=

(x, x) , x X ,

X, (, ) .

&. 4 (N1) (N2) #. %1!

+, / ' + x, y X #'!x + y2 = (x + y, x + y) = x2 + (x, y) + (y, x) + y2

= x2 + 2Re (x, y) + y2,

" + CauchySchwarz !

x + y2 x2 + 2x y + y2 = (x + y)2,+ # # +.

M + + # '$ + '

%!,

/ + ! + +

+. E, + # '$! +

+,

/ # ! +.

% 8.2 % (X, (, )) %

(8.2) x, y X x + y2 + x y2 = 2 x2 + 2y2 .

8.1. E4 4H AF4H *E E&EF;4 4*E4 115

+, % !

(8.3) x, y X (x, y) = 0 = x + y2 = x2 + y2 , x1, . . . , xn X(8.4) (xi, xj) = 0 , i = j = x1 + + xn2 = x12 + + xn2 .

&. "# # '#

(i) x + y2 = (x + y, x + y) = x2 + (x, y) + (y, x) + y2

(ii) x y2 = (x y, x y) = x2 (x, y) (y, x) + y2

! # (8.2).

(x, y) = 0, (i) (8.3).

%/! $ (8.4). n = 2 '/, /

(8.3).

116 8. AF4 *E E&EF;4 4*E4

%/ '$!.

* " ! E!%! / #! Rn, / $

! / .

" 8.3

8.2. 4FI4;4;44& 117

# % , '$! + +, " %/ +

"% ! ! .

" 8.5

118 8. AF4 *E E&EF;4 4*E4

E% %/ x1, . . . , xi , i n , !$ %/ e1, e2, . . . , ei1,xi , 1 , '/ $ ei = 0 . E# ei+1

$, %% e1, . . . , e

n $ #. E$ 1, %$-

! / +, i {1, . . . , n}, (ei, ej) = 0 , j = 1, . . . , i 1 , +! # # + / {e1, . . . , en} "$. I#! $

ei :=ei

ei, i = 1, . . . , n,

! # "% /, ! '!/

! ".

.

1. +%1 ! !#! " ! % %-

'" $ /1 e1, . . . , en , %+" #"% %/

!.

2. e1, . . . , en 1$ $ + "

x1, . . . , xn .

!# ! !#! I + " '$ Y -

# % + + #' "% %!$-

! !+ # !+ .

$ 8.1 % X, X = {0} , % .

&Rn E!% + +, ! "-

% / Rn, %% "%

! Rn.

8.2.1.

8.2. 4FI4;4;44& 119

" 8.6

120 8. AF4 *E E&EF;4 4*E4

!+ +, y # $ ! x

'$ Y.

I %1! $ + !+ y # #

! x + Y !' + # # ! x +

Y.

8.3. &;&E& 121

* 8.1

122 8. AF4 *E E&EF;4 4*E4

#' # + ' Rn,

i = (x, ei) , i = 1, . . . , n .

!+ +%1 /1 %+ # -

#, "$ '# (8.7).

[(: %1 +

x (1 e1 + + n en)2 = (x, x) + ni=1

{2i 2(x, ei)i

}.]

8.2.

8.3. &;&E& 123

8.5.

9. *', **"$ 0#$

& !+ %/ # # # ,

!# %$ %%! . E " ' "#

% , , + " %/, !%#

" #.

9.1. *' **"$ #$0

& ! + " ! # %$ %%! -

$ # + '$, "$ $ -

. E " ! '+ !$! + , +-

, + % +

.

" 9.1

126 9. >4*E&, >4>H&* ; >44&

. * +

L : X X X1 $ + , ! ' -

+ "+ # $ '/ Lx1 = x1, +, + %$

#, " '/ Lx = x " x X1. !+ + % %/ / $ $, #'! % -

# *"$ S!# E, !$ %$ %%-

! .

& !#' '$ " %,

K = R K = C.

" 9.2 .

& # ! !", " %/ + %%/, ! '/

%/ %# %#, 1 .

% 9.1#X

% , x1, . . . , xm

L : X X 1, . . . , m. $ x1, . . . , xm

.

&. +%1 " %" m. m = 1 '!+ -

"/, $, / x1, %+ % !, 1 .

H"#! $ + '!+ '/ m 1 " %1! + '/ m.

9.1. >4*E& ; >4>H&* F**; E;4&E 127

E#

(9.1) 0 = 1(m 1)x1 + + m1(m m1)xm1.

+ ! '# # #, + 1 x1, . . . ,

xm1 ! + + m %+ 1, . . . , m1, + 1 = =m1 = 0. E#, '# 1x1 + +mxm = 0 mxm = 0

% m = 0. &!$, x1, . . . , xm 1 .

I %/ $ # # %+ %%! .

% 9.2 # X

% K, L : X X

, , K, >A (L, ) := {x X : Lx = x}. $ %:i. ' K, >A (L, ) % X.ii. # K L , >A (L, ) %

, >A (L, ) = {0}.iii. >A (L, ) \ {0} L .iv. $ >A (L, ) L idX , >A (L, ) = Ker (L idX).v. ' 1, 2, >A (L, 1) >A (L, 2) %

, >A (L, 1) >A (L, 2) = {0}.

&. i., ii., iii., iv. . v. # ! + 5

9.1. %1!, !"#! + x >A (L, 1) >A (L, 2). + "#'! Lx = 1x Lx = 2x, !$ 1x = 2x, + (1 2)x = 0, #, + ! + 1 2 = 0, x = 0.

4! $ %# %%/ $ , !-

#' " %/ + %# + !! %# !

'! % , !"# ! + -

+ '$ # % .

" 9.3

128 9. >4*E&, >4>H&* ; >44&

.

9.1. >4*E& ; >4>H&* F**; E;4&E 129

S1 = (sij) #'!

(9.2) yj =n

i=1

sijxi xj =n

i=1

sijyi, j = 1, . . . , n.

E ! A B #'!

(9.3) Lxj =n

i=1

aijxi Lyj =n

i=1

bijyi, j = 1, . . . , n.

%! $ L B2 B1, %

/ % $ , ', %/ +!, %"/

"! '# 1/ A B.

130 9. >4*E&, >4>H&* ; >44&

&.

9.1. >4*E& ; >4>H&* F**; E;4&E 131

#

det (A I3) = (1 + ) (1 + )(2 1 + 1) = (1 + )(2 + 1).

&!$ %# ! 1 = 1, 2 = i, 3 = i. %-! ' %%/, /! !-

(A jI3)x = 0, j = 1, 2, 3.

. / %, A "" +

! R3 !+ !, + + % 1.

" 9.6

132 9. >4*E&, >4>H&* ; >44&

&!$, S1AS

S1AS =

0 . . . . . .

......

...

0 . . .

. . .

0 . . .

. . .

=

(I B

0 C

).

'+ !$! ! A '/, #,

p() = det (A In) = detS1det (A In)detS= det (S1AS In) = ( )det (C In),

+! %/ + ' + + detS1 = 1/detS, !-

$ .

! 9.3 + % ! A,

A =

2 1 0 0

0 2 2 0

0 0 2 3

0 0 0 2

,

2, + #, (2) = 4. / (A2I4)x =0 #' '$ / < (, 0, 0, 0) >, + "+. &!$ -

+ % #, (2) = 1.

9.2. 0#$ # 0

& ! + " '"/ % , " -

! % !" # % "

"/ + ' !$! + / .

9.2. >44& ; 133

" 9.7

134 9. >4*E&, >4>H&* ; >44&

+ " # / , #-

1! , ! Rn.H"#! + A %,

S1AS = D = diag (d1, . . . , dn) = (dij). I/ {Se1, . . . , Sen} ! Rn,+! ! Sei = si i ! S. I %1! # $ +

, ! ' LA ! , % +

%$ D, + , #,

% . #'!

LAsj = Asj = ASej = SS1ASej

= SDej = S(djej) = djSe

j = djsj

(=n

i=1

dijsj), j = 1, . . . , n,

'# , , % "!+ #.

" 9.8

9.2. >44& ; 135

& !#', I$ 9.2, " %1! + !

%. * " %/ +, A Rn,n # !+, + ! ' "$ S, %% # $ ST = S1,

'/ S1AS = diag (1, . . . , n). #! %"/

!+ + #, " %$! #

#

!/ .

" 9.9 >/ A,B Rn,n # , ! ' "-$ S Rn,n # $ B = STAS.

136 9. >4*E&, >4>H&* ; >44&

(xi, xj) = ij, i, j = 1, . . . , n. $, S, S := (x1, . . . , xn), "$-

'/ STAS = ST (Ax1, . . . , Axn) = ST (1x1, . . . , nxn) = STSdiag (1, . . . ,

n) = diag (1, . . . , n).

* + / %/

# $

"# %$! + #. &$! + '/ -

!/ ! #, +%1 ! ! / !-

+, . 9.15.

+ 9.2 / n n

.&. +%1 " %" n. n = 1 #

#. &+ + , !"#! + '!+ '/

(n 1) (n 1) " %1! + '/ n n .

9.2. >44& ; 137

B !+. $, / !+" ,

! ' # "$ T Rn1,n1 # $ T TBT = diag (2, . . . ,n). I#! $

T1 :=

1 0 . . . 0

0... T

0

#'!

(ST1)TA(ST1) = T

T1 (S

TAS)T1 = TT1

1 0 . . . 0

0... B

0

T1 = diag (1, . . . , n).

$, ST1 "$, # !! + '!+

'/ n n !/ .

9.2.1. = $ #(( #.

138 9. >4*E&, >4>H&* ; >44&

A, + % ! " +! "/ " / '#

(p q)(A) = 0, %% p q % " ' !$!, .

% 9.4 # A Kn,n, p % , P % P (A) = 0. $ p P.

&. + + ! p # + "+ ! / +

"+ ! P. E# '/ P = pq + r, !$! q r # $

"+ ! r + ! "/ ! p. I#! %1! +

r %+ !$!. !"#!, , + !+ %

"#, " ' + r(A) = P (A) p(A)q(A) = 0 #! "+! r " + ! "/ ! p, .

& !#' " %/ + + #'! % ' !$!.

% 9.5 E % % .

&.

9.2. >44& ; 139

#'!, ! += + ! '/ !/!,

(9.4) (A In) adj (A In) = p()In.

$ adj (A In) #' ' !$! "/ / n 1, !$

(9.5) adj (A In) = B0 + B1 + + Bn1n1.

+ (9.4) (9.5) !

(9.6) (A In)(B0 + B1 + + Bn1n1) = (1)n(n + n1n1 + + 0)In.

&/ !$ %/ # (9.6), %% / !$

% %! ! , %

AB0 = (1)n0InB0 + AB1 = (1)n1In

...

Bn2 + ABn1 = (1)nn1InBn1 = (1)nIn .

!# '# In, A, . . . , An, ', "#-

# !

0 = (1)n(0In + + n1An1 + An),

%% p(A) = 0.

&!%! 5 9.4 I$ 9.3 %/ # +-

!" #.

$ 9.1 $ % % -

.

140 9. >4*E&, >4>H&* ; >44&

9.3. $%$

9.1. % %# ' %%/ +

L : R3 R3, L(x) := Ax, +!

A :=

1 0 11 2 1

1 1 1

.

9.2.

9.3. &;&E& 141

9.8. % + -

$ %$ !

A :=

7 1 21 7 22 2 10

.

9.9.

142 9. >4*E&, >4>H&* ; >44&

%% " % / %+ %, #-

aii n

j=1j =i

|aij |. 4 / ! # Gerschgorin. *!+ + % %! ! %# + , #'!

% " #"% / .

[(:

. , D. . >!: + & &, % #%, #

E%+ ;, , 2002.

[2] &. A. %% : '

H. E%+ &!, ", 1991.

[3] H. Anton: Elementary Linear Algebra. 4th ed., Wiley, New York, 1984.

[4] G. Fischer: Lineare Algebra, rororo vieweg, Rowohlt, Reinbek, 1975.

[5] W. Greub: Linear Algebra. 3rd ed., SpringerVerlag, New York, 1967.

[6] G. Hadley: Linear Algebra. 3rd printing, AddisonWesley, Reading, 1973.

[7] W. Nef: Lehrbuch der Linearen Algebra, Birkhauser Verlag, Basel, 1966.

[8] G. Strang: Linear Algebra and its Applications. 3rd ed., Harcourt Brace Jovanovich Publishers, 1988.

(E '

H +. #%, #

E%+ ;, , 1999.)

143

(%

%, 7

" $ '$, 29

!"/, 31

+ %, 131, 141

+ !, 93

, 66

, 4

! Fourier, 120

, 53

+

! Bessel, 123

! Gerschgorin, 141

CauchySchwarz, 112114

+, 4

#= , 64, 142

, 103

+ , 52, 99, 140

+, 4

, 4

# #, 4

, 4

!, 5

! !' %$, 65, 142

# #, 118, 119

"+ +, 59

"+ , 62, 8082

#, 47, 75

, 47, 75

"+ #, 47

"+ , 47

"

, 117

"%, 117

/ '$!, 22

, 22

+ %, 131, 141

# , 45

1 %/, 18

1# %/, 18

+, 35

+ /, 69

#, 79

#, 70

+ !%!+, 18

+ '$, 13

% , 29

# % , 29

V%/ , 48

%# ! Kronecker, 20

% , 132

% , 133, 134, 141

%$ !# , 65

%$ , 46, 141

%$ / , 46

% !, 14

%!+ '$, 14

% / '$!, 29

#, 29

%#o /, 3

145

146 EHFEF4

+, 4

+ +, 39

, 93

' !$! , 137139

#, 4

!1# / !,

81

+ , 134

+ +, 111

E!%, 112

+, 112

!"/ " $ '$, 31

E!% + +, 112

E!% '$, 112

#, 4

" # , 116, 122

"$

!"+, 115

CaleyHamilton, 138

%% !

, 125, 126, 140

, 127, 140, 142

%

, 125, 126, 140

, 127, 134, 140142

%/ !, 71

+, 59

+ '$, 59

+ ! !, 114, 115

, 22

+ + +, 112

+ ! Sarrus, 103

+ +, 3

+ , 48

/ / !, 70

#"%

! Gauss, 83, 88, 103, 104

"

GramSchmidt, 117

! Cramer, 105, 108

, 116

# + /, 79

+, 110

!, 111

%, 6

, 7

# + /, 70

+ , 129

!+', 79

"%+, 83, 84

" , 120

"$ % , 135, 136,

142

"$ %/, 112

"$ , 120

"$ + , 135

"$ !, 120

"$ /, 117

"$ , 134

"

, 117

"

, 116

"

+ /, 117

"% , 117

"% /, 117

!, 89, 91, 93, 96, 98, 99, 102, 107, 108

! Vandermonde, 106

% /, 4

% $, 4

, 45

+, 52, 99, 140

#=, 142

, 57

EHFEF4 147

%, 133, 134, 141

%$, 141

+, 134

" #, 116, 122

+, 48

"$ %, 135, 136, 142

"$, 134

'$%, 73

!+, 134136

!$, 69

! Gram, 122

+ , 53

#, 49

+ %

, 131, 141

, 131, 141

%, 2

1 , 52

, 120

# , 105

+" , 52

!"+ "$, 115

! +, 39

# Fourier, 120

", 110

, 45

"+ +, 69

"+ '$, 110

' , 45

'$% ' $, 46

'$% ' $, 73

'$% , 73

!+ , 134136

! , 93

/ ! Kronecker, 20

/", 5

/, 1

/

"$, 117

"

+, 117

"%, 117

/ $ 1$, 122

$, 9

+ , 46

+, 110, 116

, 83

, 49

/ % , 41

!+', 16

'+ !$! , 128, 138,

139

'$

E!%, 112

/ / !, 70

+ +, 109, 112

+, 110

$ , 47

$ , 47