Post on 25-Nov-2015
description
.
( )
, 2003
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%/ $ ! / '$!
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'# 1/ $ .
# !, %% # $ 1-
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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