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A ,
. # A m !
"
n , $ m × n , " % aij , $ i = 1, &m j = 1, 'n . "
$ n . ( $
m = n
% $ $ ) $ * , " tr( A) = ∑ a . +
$ % "
* . , , : (A ) = a . -. " α $ % (αA) = αa . ( . $* . / $ $ $ , ( A + B ) = a + b . 0 $ $ $ $* $ m × n n × p , .. " . " . , $ ( A ⋅ B) = ∑ a b . + %$ . $ n -% $
A = A ⋅ A ⋅ 1⋅ A n - . . 2 "* A B . "
AB BA ! " AB ≠ BA . 3 " [AB] = AB − BA
, , A B . , $ * [ AB ] = 0 , % %! . $ $ $ $ % .$
$ , $ 4 . $ $ . 0
$ * " %! "
4 5
, %! $ .
" -
$
) . * . 2 1.
n
i =1
ii
T
ij
ij
ji
ij
ij
ij
n
ij
ik kj
k =1
n
-3-
ij
/ $ , a = λ , %. # λ = 1 , A $ " E $ n E = δ , $ δ . ( " $ a = a , $
" , , a = −a . 3 " * $ . A , " %!
. : (a ) = a , $ "
" . . , $ %!
% . , " $ A = A , $ ! , "* . * ! . 0
$
$ , " $ %! a = −a . ii
i
i
ij
ij
ij
ij
ij
ji
ji
+
+
ij
ji
+
ij
ji
3" $ AB = 1 0 11 11 . 2 % $ . 1× 2 2 × 2 , *
$ C $ 1× 2 . 3" ) : c = a b + a b = 1 ⋅ 1 + 0 ⋅ 1 = 1, c = a b + a b = 1 ⋅ 1 + 0 ⋅ 1 = 1, $ , C = 1 1 .
f ( A) $ 2. 3" " " −1 4 3 f ( x) = x − x . , A= −2 5 3 2 −4 −2
. " " . . ,
f ( x ) = x − x = x( x − 1). 2 " " , "
f ( A) = A( A − E ) , $ E - $ " %! $ . 0$ A, $ %
, " " 1.
11
11 11
12
12 21
11 21
12 22
2
2
−1
4
3 −2
4
3
0 0 0
f ( A) = A( A − E ) = − 2 5 3 −2 4 3 = 0 0 0. 2 −4 −2 2 −4 −3 0 0 0
-4-
. 1.1.
3" % % : 1) 3
3)
1 2 1 2
3 1
+
−
3 2 3 2
−4
1+ i
0
2) 2 2 − 3 5
0 1
4) 2
3
2
0 1
1
0 0 1 1 1 0 1 0 0 1
−
6 1 0 1 1 1 0 1 1 1 1
. . $ m n ? . . n $ m ?
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4
4
1) 2 − 3 0 3
2) 3 2 − 3 0
1 4) 1 0 7)
2 4
1.4.
1
1 1
a b α
5)
1 1
1 2
3)
2
c
d γ
8) 1 2
1
β δ
6)
1 1 3 5 1 1 5 9
1 2 2 1 2 3 4 1
2 2 4 1
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1 1 1
1 1
2) 0 0 0
1 1
5)
1.5.
0 0 0
1 1 0 1 1 lim
n→ ∞
cos ϕ sin ϕ
6)
−
α n
α n 1
3)
1 −1 1 −1
λ1 0 4) 0
− sin ϕ . cos ϕ
, λi –
λn
n
, α - .
1.6. , ! " # ! ! : 1+ i 0 1 1 0 1+ i 1 2 1) 2) 0 1 + i − 1 3) 0 1 − 1 − 2i 2 2 1 −1 2 − i 1 − i − 1 + 2i 2 -5-
4)
1
e iϕ
e −i ϕ − ie −iϕ
2 0
ie iϕ
0 , ϕ – . 3
1.7. ! 1 1) f ( x ) = x 2 − 2 x + 1 , A = 0 1
f (x) ! " A : 1 1 1 2) f ( x ) = x 2 − 2 x + 1 , A = 1 1 1 −2 3
3) f ( x ) = 3x 2 − 2 x + 5 , A = 2 − 4 1 3 −5 2
1.8.
1.9.
T ! " P = E − (e i − e k )(e i − e k ) , e i – i - " ! " E .
, ! ! " A B , E – ! ":
1) ( A + B ) = A 2 + 2 AB + B 2 3) A 2 − B 2 = ( A + B )( A − B ) 2
2) ( A + B )( A − B ) = ( A − B )( A + B ) 3 4) ( A + E ) = A3 + 3 A 2 + 3 A + E
, ! " A B tr ( AB ) = tr (BA) .
1.10.
1.11.
1 0 ! " ! 0 −i , 0 1, σz = σy = 0 −1 1 0 i 0 !! ! ! , ,
# . !!
"
σx =
# ! ": [σ x σ y ] , [σ y σ z ] , [σ z σ x ] . 1.12. " Aˆ = ,
1.11.
1 , σˆ y ! " 1 − σˆ y / 2
! " (α - ), ! " 1.13.
σ x, σ y σ z 1)
1.11.:
exp(iασ x ) ;
2)
exp(iασ y ) ; -6-
3)
exp(iασ z ) .
0 1 0
0
1 1 1 0 1 , L2 = −i 1.14. !! ! " [ L1 L2 ] , L1 = 2
2.
0 1 0
2
0
i
0
0
i .
−i 0
.
!, ! ! det A ! "
A
" #!: ! " det A = a11 , ! " 2- det A = a11a 22 − a12 a 21 , ! " 3- ! ! "
! ! , i : n
det A = ∑ (− 1) n
j =1
i+ j
aij M ij ,
M ij ! #! aij
( ! ), n − 1 , #! ! " A ,
i - j - ". M ij . " ! ! ! , ! , det A = 0 ! ", #! ! ! , ( " ) , . " #!! ! " : " ( ) det A ! , !
( ") det A ! # .
! " ! . ! "
, ! ( ") ! " , #!! #! ( "), . #!! ! " , ! ! , #! ! ! ! " .
n det A n = (det A) , det (αA) = α n det A
det ( AB ) = det A ⋅ det B .
! ! " k - ! A ,
#! k k " # ! " , #! k ≤ min (m, n ). ! , , ! , ! ! " A rang( A) . ! !, ! ! ! ! ! " . " , , ! , ! ! "! ,
! ! .
( )
-7-
1.
5 5 5
5 5 n - : Dn = 5
5 0 5
5 5 0
.
5 5 5 0 . ! ! #! . ! ! " , #! ! :
5 5 −5 0 0. = 5 5 −5 0
5
Dn
5
5
!
5 5 5 −5
#!, ! " , n −1 , . . Dn = 5 ⋅ (− 5) ⋅ (− 5) ⋅ (− 5) = (− 1) ⋅ 5n .
2. , # ! ! "
! !.
. !
T ! " ! A = A , ! , ! " !
,
det A = det AT , ! #! ! " !! , # , ! !,
det ( A ) = det A . !
# !
! "
det A = det ( A ) = det AT = det A , . . , det A ,
!.
-8-
. 2.1.
, !"# $ % &' & ( '! )'&# %' *:
1)
13547 13647 28423 28523
246 427 327 2) 1014 543 443 . − 342 721 621
;
2.2. 3-+ # ,', + α , β , γ – ) - ) && ':
2 sin 2 α
− cos 2 α
2 sin 2 β 2 sin 2 γ
− cos 2 β − cos 2 γ b+c
c+a
b2 + c2
c1 + a1 c2 + a 2
2.3. . ,'!', b1 + c1
2.4.
α2 β2 2 γ δ2
(α + 1)2 (β + 1)2 (γ + 1)2 (δ + 1)2
a+b
a a1 + b1 = 2 a1 a 2 + b2 a2
(α + 2)2 (β + 2)2 (γ + 2)2 (δ + 2)2
b b1
c c1 . c2
b2
(α + 3)2 (β + 3)2 . (γ + 3)2 (δ + 3)2 2 x+2
2.5. / 0 & ') &) & & x : 1
5
−1 −2 >0
1 −3
x
0 0
0
1 1
0 0
0
2 3
2.6. 5-+ # ,': 1 2
4 0 0 1 4 16 0 0 1 6 36 0 0
2.7. det A , !&'#, ) %' * A "%%' , 1&% & % '% ')&' "%% , & 1&% & % '%.
3 2.8. 2 ' ) , % ,& ', "%& 4 & &' , %' * # ,' n & !% & 1 #. -9-
3 3 2.9. 1) 2 ' ) %& ) k -+ # ,' ,)' '& %' * # ,' n , 4'- # ) , )'&& k ,' . 3 3 2) 2 ' ) %& ) k -+ # ,' ,)' '& %' * # ,' n . 2.10. " ) %' * # ,' n ,', - ( n $ % & ) ')& 1, ' '& ')& &". %" % 4 ( ') & %' * A ?
3 3 2.11. 2 ' ) ', "%& 4 & &' , ,)' '& %' * # ,' n & % &# 1 #. 2.12. 0 1 0 0 0 1 1) 0 0 0 1 1 0 0 0 1 0
0 1
4) 0 0 1 2 3
n -+ # ,': 1 2 −1 0 2) − 1 − 2
n n
3 3 0
n
−1 − 2 − 3 n 2 1 0 0 1 2 1 5) 0 1 2 0 1 0 0 1 2
0 0 1 n
0 1 1 1 1 0 3) 1 0 1
)
2.14. '& % && : 1 1 1 x1 x2 x3 W ( x1 , x n ) = x12 x22 x32 x1n−1
x2n−1
x3n−1
1 0 0
3 3 2.13. . ,'!', # ( ) - ) && %' * & ') &) det AAT ≥ 0 .
(
1 0
0 1
A ) &# #
% & ' W ( x , x ) , + x , x – 1 n 1 n 1 xn xn2
x n−1 n
3 3 2.15. 2 ' '&+ ",'!' ('!& %& %' *: 0 2 2 2 −1 3 − 2 4 1 1 1 1 −3 1 1) 2) 3) 4 − 2 5 1 7 −2 0 −4 1 1 −1 2 −1 1 8 2 4 6 14 - 10 -
25 31 17
43
75 94 53 132 4) 75 94 54 134 25 32 20
1
0
−1
0
0
0
0
1
0
1
0
0
0 0
0 1
−1 0
5) 0 − 1 1 0 0 −1
48
0
0
0
−1 −1
1
b = (b , b ) - . 1) A = ba . 2) rangA = 1 . , A . 3. ! " # $% . & $'!() * +!)!$" , - . . A - . A , -/ . AA = A A = E . 0 . a 1-- 2 a = (− 1) M det A . 3 - . , . det (A ) = 1 det A . 4 1 AX = B 4 1 X = A B , A - A .
2.16. " a = (a1 , T
an ) -
T
1
n
T
−1
−1
−1
−1 ij
i+ j
−1 ij
ji
−1
−1
−1
567896 1. :. - 1 2 − 3 . A= 0 1 2 0 0 2 ;9<9=79. - , . , . > , . 4 A = 12 10 00 , −3 2 2 1 . A , 1 det A . 2 −4 7 1 − 2 3. 5 1 / A = 0 1 − 2 . A = 0 2 −2 , T
T
−1
−1
2
0
0
0
1 - 11 -
0
0. 5
, AA , . 567896 2. :. X - 2 5 X = 2 1 . −1
1 3
1 1
;9<9=79. AX = B , X = A B . :1 - A, - . A = 3 − 5 . −1 2
-- X = A B , X = 10 −12 . . , X 1 . AX , B . -. 7 8 9 = 69<9=7. . 3.1. : - A ( xy ) ϕ − sin ϕ ϕ , A = cos . sin ϕ cosϕ . 3.2. : - /1 3- -: −1
−1
−1
1 −2 0
1) A = 0
1
0
2
2) A = 0
−1
0
2
−1
A - , A + A + E = O , E -, O - - . , A / . . 3.4. A B , . . AB = BA , . ,, - - A B = B A . 3.5. A C . 4 - X : 0
0
1
−1 −1
1
2
3.3.
−1
−1
1) AX = O 4) AXC = B
2) AX = B 3) XA = B 5) A( X + C ) = B
- 12 -
−1
−1
−1
. - A = E − λI , – : , I – - -, I = E , – , λ < 1. . 3.7. : X -: 3.6.
2
1) X
1 1
=
1 −1
1 1 1 −1 2 3 1
2)
1 1 0 1
4) X 3 7 2 = 10 3 3 5 4 2 2 2 −1
−1 −1 2
6) X 2
3.8.
2 = 2
X=X
1 1
3) X −1
0 1 5 −6 4
1 1 3
X=
0 0 0 2
5) 3 − 3 2 X = 2 4 −5 2
5 5
1 1
1
2
5 8 −1
. , U . . U = U , - - - ,
/ , . −1
4.
+
. m .1 . n - > m , - Ax = b , -
x = (x1 , , xn )T !"#, !$ b = (b1 , %, bm )T "!& ' ( ) ! *"# +' !, $ ,-- ( " ) (" A . &*$ m × n &!'&/& $ ,-- ( ) . !"#. 0' m = n b ≠ 0 , ) "!& ) 1 $ ) !$ 2 .. 3 ) * ! 4 . 5' ! det A ≠ 0 ) 4 1 . det A = 0 . 6 '5+ det A ≠ 0 7 * ! 1 4 # * & . - )5') 8 ) : x j = ∆ j ∆ , 2* ∆ = det A , ∆ j . *' ' , . '5+" ) 1 j -2 ' ( A ' ( ! *"# +' !. 9 )" ' 1"# 5 ! 1 ) : $: 4 ) * . '* !' 2 $'/+ & !"#, ' ) * ) ;5. <=>?@= 1. A4 E )5 ' 1"# 5 ! 1 2 . &*$, . ' 5& B2 x1 + x2 − x3 = 2 - )5'" 8 ) : DB3x + x − 2 x = 3 . 3 C 1 2 x1 + x3 = 3
- 13 -
@@>@. :* !2 , !"+ ' ) . *' ' )"
2 1 −1 . ∆ = 3 1 − 2 = −2 1 0
1
$ '$5 ∆ ≠ 0 , ) ) * ! 4 . 6"+ '&) 2 1 −1 '" . *' ' : , ∆ 1 = 3 1 − 2 = −4 3 0
2 2
−1
∆2 = 3 3 − 2 = 2, 1 3
1
1
2 1 2
- )5') 8 ) # * ) 4 )": ' ∆ 3 = 3 1 3 = −2 . . 1 0 3
x1 = ∆1 ∆ = 2 ,
x2 = ∆ 2 ∆ = −1 ,
x3 = ∆ 3 ∆ = 1 . *! !
1*" + & !"# ! # *5/ )5 5 ! 1, 5 :*)&, + 5 ! & . ! /& ! :*!. <=>?@= 2. . A4 )5 ' 1"# 5 ! 1 2 . &*$, . ' 5& - )5'" 8 ) : 2 x1 − x2 + 3x3 = 9 . 3 x − 5 x2 + x3 = −4 1 4 x1 − 7 x2 + x3 = 5
2
−1 3
@@>@. 6"+ ' ) . *' ' )" ∆ = 3 − 5 1 = 0. 4 −7 1 * * 1 , )& 5'! 1 ' ( ∆ = 0 , ) &!'&& +' !, 4 & 5!5. 9 ) &!'&& !) 1.
$ '$5 ! *"#
> ? @ =@@> . 4.1. A4 )5 ' 1"# 5 ! 1 2 . &*$ . - )5' ) 8 ) :
x2 + 3 x3 = −1 1) 2 x + 3x + 5 x = 3 1 2 3
3x1 + 5 x2 + 7 x3 = 6
4.2.
2 x − x + 3x3 = 9 1 2 2) 3 x1 − 5 x2 + x3 = −4 4 x1 − 7 x2 + x3 = 5
1 $ ,-- ( " $!* + 2 ) 2 +' f (x ) = ax 2 + bx + c , &, + f (1) = −1 , f (− 1) = 9 f (2) = −3 . - 14 -
4.3.
A4 )5 +! 2 . &*$, . ' 5& - )5'" 8 ) ' ) * $'/+ & !"#: 3 x1 + 2 x2 − x3 + 2 x4 = −5 2 x1 − 3 x2 + 2 x3 + x4 = 11
x − 2 x + 3 x − x = 6 1 2 3 4 2 x + 3 x2 − 4 x3 + 4 x4 = −7 2) 1 3 x1 + x2 − 2 x3 − 2 x4 = 9 x1 − 3 x2 + 7 x3 + 6 x4 = −7
3 x − 2 x − 5 x + x = 3 1 2 3 4
2 x − 3 x2 + x3 + 5 x4 = −3 3) 1 x1 + 2 x2 − 4 x4 = −3 x1 − x2 − 4 x3 + 9 x4 = 22
3 x + 8 x + 3 x − x = 4 1 2 3 4 2 x + 3 x 2 + 4 x3 + x4 = −4 4) 1 x1 − 3 x2 − 2 x3 − 2 x 4 = 3
5 x1 − 8 x2 + 4 x3 + 2 x4 = −8
1)
x1 + 2 x2 + 3x3 − 2 x4 = 1 2 x1 − x2 − 2 x3 − 3 x4 = 2
4.4. !" #$ f ( x ) = ax 3 + bx 2 + cx + d , % & ! f (− 1) = 0 , f (1) = 4 , f (2 ) = 3 f (3) = 16 . 4 4 5. '()*+,- ./0 .1./0-2 3(0+ 0-2 51670+0( 8! 9 : # ; #:;% < : b = 0 ## < 9! : Ax = 0 = : #& %! %. >%! %& ## :# % T #: #, $# "9 9 : !? x = (0, 0, @ 0) ;! A :# 9! :& ## : B%#:. C # 9 :%! ! A $! det A ≠ 0 $ ! 9 D! ! % & E #9A#:9 ?" !: " !? x = 0 . F ;! = , 9 # : & !: " !?& %! % ## # :%! ! &: & #& ! :#: 9 G E $!% &. C ;A # 9 $!& 9 " ! m × n ## E !?&, . . n -
(
)
T x1 , H, x n , $!%#: &G #; $ : ! B#:, &: &GA#& : ! $! #! #: . I# =:# < !?, . . ! = !#" $! #! #: , ! : n − r , % r = rang( A) . J=# : $! #! #: = : #& 9% " ## !? ( KLM). N & < %& M ! B KL : ! A : % &G ;=# # ! ## !:G ?" 9! :&, #%!BA#& : < #! <. N : < : % &G ;=# # ; , $! , #%!BA#& : ;=# < # ;<, # G #:;% $! #& : $! :9G #". F< #: ;% < $! < # : #& n − r ?9. K! !9& = < n − r =:# < # ;: : # n − r , !?G n − r ! = $ 9 :?9G #& ## 9 M ! ! :# $, < %& KL = : ! : {e1 , O, e n− r }. PQRSS T SUSVWS
- 15 -
x OO
V T V W S S WVSV QWV WS S T W
STW n − r T W V VV, . S. x OO =
∑ Ck e k . n −r
k =1
1. W W W QRSS T SUSVWS W S WVS V V T V T VSVW: x1 + x2 + x3 = 0 . x1 + x2 − x3 = 0 ! 1 1 1 . PT SS W V " S T V# TW W S A = . $ W V S WV VV " S
T T T W SSV SVV T , % T # W
1 1 −1
$ T SV . WV T T # T SS# Q , S
Q WV, T SV M = 1 1 = −2 , S T V# TW T SV . 1 −1 &W VSW S V V T T ' , % "W WVS V VS WW T SUSVW S n − r = 1 . ' Q W T T V S T SUSVW, . S. . ( % # " W S SSVVS W S - T , %))W WSV TW T * , x + x3 = − x1 Q WV WV T: + 2 . .TW ST SSVV x V "SVWS x = 1, 1
x2 − x3 = − x1
1
T SU S " SVV* W S T ST # W , V x2 = −1 W x3 = 0 . T / W
QT , W W V # S T e1 = 1, − 1, 0 , QRSS T SUSVWS W S STW V T W V* VV* W WSS W x = Ce1 .
(
)
012341 T 2. . Q e1 = (1, − 1, 0, 0) W e 2 = (0, − 1, 0, 0 ) *
VS T W S V T V WVS V T VSVW . 5 W S % W S ? 6T VSVW 474824. &W VSWSV V T SS W T STV W S T T SUSVW , T VV " S T V " ST ', . S. n = 4 . ( SS, T STV T T V T SUSVW T V "W S T , . S. VV " S . S S V , T V# TW W S r S T S $ T VSVW* 4 − r = 2 , r = 2 . TW , WS*R T V# r = 2 , V WS T VS SVSS , S VS' V Q VS SVSS T . 9 S # W S WS SR W VS RWS T T ' T , Q WV WV T. , W W S W S S S V VSVW T
T T ' T , VV W S W W W W SS VSVW % Q T .
- 16 -
2 3 4 8 147482 . W W 5.1. W VSVW T :
QRSS T SUSVWS W S WVS V
x1 − x2 + x3 = 0 1)
2 x1 + x2 − x3 = 0
5 x1 − 8 x2 + 3 x3 + 3 x4 = 0 3) 4 x1 − 6 x2 + 2 x3 + x4 = 0
2 x + 4 x + 6 x = 0 1 2 3 3x + 6 x 2 + 9 x3 = 0 5) 1 x1 + x2 − 4 x3 = 0 x + 2 x3 = 0 #2 x1 + 3x2 + 2 x3 = 0
V T V
x1 + 3 x2 + 2 x3 = 0 2) 2 x + 4 x 2 + 3 x3 = 0 1 2 x1 − x2 + 3x3 − 2 x4 + 4 x5 = 0 4) 4 x1 − 2 x2 + 5 x3 + x4 + 7 x5 = 0 2 x1 − x2 + x3 + 8 x4 + 2 x5 = 0 x1 + x2 + x3 + x4 = 0 x1 + x2 − x3 − 2 x4 = 0
6) x1 + x2 − 7 x3 − 11x4 = 0 x1 + x2 − 9 x3 − 14 x4 = 0
' $x1 + 2 x2 + 2 x3 = 0 $ & 8) 3 x1 + 2 x 2 + 5 x3 = 0 $ $%7 x1 + 7 x2 + 12 x3 = 0 3 x1 + 16 x2 + 7 x3 = 0.
7) "2 x1 − x2 + 3x3 = 0
3x1 − 5 x2 + 4 x3 = 0 ! x1 + 17 x2 + 4 x3 = 0
8 T 5.2. ()*+, *+-./01 e1 = (1, − 1, 0, 0) 2 e 2 = (0, − 1, 0, 0 ) 34.35+ *3 67 9:;-+-<-= *2*+:>1 -?9-<-?91@ .29:=91@ )
:< +A;-= *2*+:>1, *-*+-3C:= 2D +<E@ )
6. FGHIJKL MJNO M NPNO MLQ R GMJSM LQ T PUVMJMGS
WXYZ[\] [_` _a_` X^[e^bc fa]g^[^Xe Ax = b _hi[j_ gX`], Z. [. Y ^e ^ ^bc d kal\_fj_dm^ _ \]ZaXn[e A k_al`o] m × n X ^[^fd [gb\ YZ_d hn_\ Yg_h_`^bc pd [^_g b , g _hi[\ Ydfp][ ^[ lgdl[ZYl Y_g\[YZ^_e. qdl _ka[`[d [^Xl Y_g\[YZ^_YZX ^[_hc _`X\_ ^]al`f Y \]ZaXn[e YXYZ[\b A a]YY\_Za[Z m Z]o * ^]rbg][\fs a]YtXa[^^fs \]ZaXnf YXYZ[\b A , o_Z_a]l k_dfp][ZYl Xr \]ZaXnb A `_kXYbg]^X[\ Yka]g] YZ_d hn] Yg_h_`^bc pd [^_g b X X\[[Z k_al`_o m × (n + 1) . uaXZ[aXe Y_g\[YZ ^_YZX f YZ]^ ]gdXg][Z Z[_a[\] ua_^[o[a]e u ]k[ddX, jd ]Yli]l, pZ_ YXYZ[\] Ax = b lgdl[ZYl Y_g\[YZ^_ Z_j`] X Z_dmo_ Z_j`], [YdX rang( A) = rang A* . v _Yd [ Z_j_, o]o Y_g\[YZ^ _YZm f YZ]^_gd [^], r]`]p] a[t][ZYl k_ ]^]d _jXX Y a[t[^X[\ _`^_a_`^_e YXYZ[\b. w\[^^_, _hi[[ a[t[^X[ ^[_`^_a_`^_e YXYZ[\b x ] _`XZYl o]o Yf\\] ^[o_Z_a_j_ [x OH ^ c
( )
- 17 -
p]YZ^_j_ a[t[^Xl x X f[ ^]e`[^^_j_ _hi[j_ a[t[^Xl _`^_a_`^_e YXYZ[\b x . qdl ^]c _`[^Xl x `_YZ]Z_p^ _ gbk_d^XZ m _kXY]^^fs g OO ka[`b`fi[\ k]a]ja][ ka_n[`faf ^]c _`[^Xl x kaX`]g gY[\ n − r OO , Yg_h_`^b\ k[a[\[^^b\ g ka]g_e p]YZX ^fd [gb[ r^]p[^Xl. v _dfp[^^]l YXYZ[\] oa]\[a_gYo_j_ ZXk] Y ^[_`^_a_`^_YZ ms b _ka[`[dXZ _`^_ ^[_hc _`X\_[ p]YZ^_[ a[t[^X[ x .
e e 1. ] ZX _hi[[ a[t[^X[ YXYZ[\b fa]g^[^X , j`[ a X b - ka_Xrg_dm^b[ pXYd ]: x1 + x2 + x3 = a
x1 + x2 − x3 = b
e 1 1 . va[`[ gY[j_, ^] `x\ a]^j \]ZaXnb YXYZ[\b A =
a]YtXa[^^_e
1 X a]^j 1 1 −1 gZ_a_j_ k_al`o],
a . X^_ab 1 1 −1 b Y_`[a]iX[Yl g_ gZ_abc X Za[ZmXc YZ_d hn]c _h[Xc \]ZaXn, lgdlsZYl * ^[^fd [gb\X, k_Z_\f rang( A) = rang A = 2 , Z_ [YZm Y_jd ]Y^_ Z[_a[\[ e e ua_^[o[a]-u ]k[ddX YXYZ[\] lgdl[ZYl Y_g\[YZ^_ . hi[[ a[t[^X[ _`^_a_`^_ YXYZ[\b hbd _ a]YY\_Za[^_ g ka[`b`fi[\ k]a]ja][, j`[ hbd k_dfp[^ a[rfdmZ]Z x = C (1, − 1, 0 )T . ]e`x\ p]YZ^_[ a[t[^X[ ^[_`^_a_`^_e OO YXYZ[\b. v _gZ_all a[t[^X[ _`^_a_`^ _e r]`]pX X _YZ]gdll g d [g_e p]YZX k[a[\[^^b[, _Zg[p]siX[ h]rXY^_\f \X^_af, k_dfp][\ YXYZ[\f x2 + x3 = a − x1 ]e`x\ o]o_[ Xh_ [x a[t[ X[ ]kaX\[a kaX ^ , ^ . -d , x1 = 0 ,
\]ZaXnb
A* =
1 1
1
( )
x2 − x3 = b − x1
o_Z_a_[ X\[[Z gX` x = (0,
(a + b ) 2 , (a − b ) 2)T . ! "#
" $%"$ $%"$& '#'( )* '(+ ',))- x ./ # x . OO
012341 # 6-6$) 7"-8 "## 9- -) ( - λ '#'( )- :#" &"*; , -<" "#& 5 2. =@x1 − 2 x2 + x3 + x4 = λ ? =x1 − 2 x2 + x3 − x4 = −1 A<:A ('A '$<) '("$&? > x1 − 2 x2 + x3 + 5 x4 = 5
- 18 -
1 −2 1
1
1 −2 1
5
4424. $'(-<:A ) )-( #, '#'( )* # -'!# "", A = 1 − 2 1 −1 )-( #,
1 −2 1
1
λ
A = 1 − 2 1 −1 −1 . *
1 −2 1
5
& -7) )#"$ - )-( #*
-6'#)-:+"*
5
A , " -<"$$ ",:, -< " %<,). -7#'"*) )#"$ $) A<:A ('A, "-9 #) , $9 % :#( :+, '$'(-<: ""*& #7 : ) "($< 9 <*; %<,; '( $6 # 9$': %"#; %<,; '($: $< )-( #* A . (-6, rang( A) = 2 . -: , $9 % :#) (-6$ 7"-8 "#
9- -) ( - λ , 9 # 6$($ $) (-6 # rang( A* ) = 2 , 8($ A<:A ('A 6 #( # ) '$<) '("$'(# '#'( )*. 7 '( ,6(, * )-( #* A* <#%"$, 8($ "$<*& '($: '9 -<- " 9 #< % ( 6 ,< :#8 "# 9$ A%6- " ",: <$$ )#"$ -, ':# $" ,% ( -< " '$' %" ), '($: ,. -6-A '#(,-#A %$'(#- ('A 9 # 7"-8 "## λ = 1, (. . '#'( )- ,% ( '$<) '("- 9 # λ = 1. 2 3 4 1442 . 6.1.
'9$:+7,A ( $ ), $" 6 ---9 ::#, 9 $< #(+ '$<) '("$'(+ '#'( )*
:#" &"*; , -<" "#& # "-&(# $ # ! "#A '$<) '("*; '#'( ): x1 + x2 + x3 + x4 = 0 1) x 2 + x3 + x 4 + x5 = 0 x1 + 2 x2 + 3x3 = 2 x2 + 2 x3 + 3 x4 = −2 x3 + 2 x 4 + 3 x5 = 2. & #x1 + x2 + x3 + x4 + x5 = 7 # 3) %3 x1 + 2 x2 + x3 + x4 − 3 x5 = −2 #x + 2 x + 2 x + 6 x = 23 3 4 5 #$ 2
" x1 + 2 x2 + 3x3 − x4 = 1 2) ! 3 x1 + 2 x2 + x3 − x4 = 1 2 x1 + 3x2 + x3 + x4 = 1 2 x1 + 2 x2 + 2 x3 − x4 = 1 5 x1 + 5 x2 + 2 x3 = 2.
* '2 x1 + x2 − x3 + x4 = 1 ' 4) )3 x1 − 2 x2 + 2 x3 − 3 x 4 = 2 '5 x + x − x + 2 x = −1 4 '( 1 2 3
5 x1 + 4 x2 + 3 x3 + 3 x4 − x5 = 12.
2 x1 − x2 + x3 − 3 x4 = 4.
- 19 -
6.2.
: 2 x1 − x2 + 3x3 − 2 x4 + 4 x5 = 1 1) x1 + x2 + x3 + x4 = 1 2) 4 x1 − 2 x2 + 5 x3 + x4 + 7 x5 = 1 2 x − x + x + 8 x + 2 x = −1 1
3)
5)
2
3
4
5
3x1 − 2 x2 − x3 − x4 = 1 3x1 − 2 x2 + 5 x3 + 4 x4 = 2 3x1 − 2 x2 − x3 − x4 = 2 4) 9 x1 − 6 x2 + 9 x3 + 7 x4 = 5 3x − 2 x + 5 x + 4 x = 2 3x − 2 x − x − x = 1 1 2 3 4 1 2 3 4 ax + y + z = 4 ). x + by + z = 3 ( a b - x + 2by + z = 4
6.3.
6.4.
! " # , $"% 2, " $ # f (t ) f (t ) &$, f (2) = f (2) = 3 . 1 2 1 2 & & λ $ ( ,' )2 x − x + x + x = 1 + 1 2 3 4 % % % ? x + 2 x − x + 4 x = 2 )* 1 2 3 4 x1 + 7 x2 − 4 x3 + 11x4 = λ
- 20 -
$ # & # &( & # ( # % % % % # . &$ " , ( $ & , " & & $ $ . #$ " $ & % , & $ % % % ( &. & , && , & ( % % % " ( % ( & # " & &. 7. ! "#" "$%& $%'&. (&) $ &)* "$%+ 0.12 3456748.67952 L .8:198/76; 2.5/?759 ,-./ 345-:[email protected] 34-45B1 x, y, z, C, .8:198/21D E@/2/.782- E75F5 6798, -@- 9/?75482, B@; ?57541D 534/B/@/.1 53/48G-- «6@5
1
2
n
n 0 8; = 8G D @ ?5 ; - -./ . 2 -. - ∑ ck x k = 0 @-QA 34- H6@59--, ?5FB8 96/ c k 489.1 k =1 0.5 :89-6-250. R8662574-2 457 5 6@ 8 @M 9 9 .H , 3 - . 2 HI / 6-67 /28 .8:198/76; @-./ 9 B8..52 3456748.679/ L .8=54 -: n @-./0.5 ./:89-6-21D 9/?75459 {e1 , S, e n }. T6@- B@; @M=5F5 x ∈ L .80BO76; 78?50 /B-.679/..10 .8=54 n ?5EUU-G-/.759 ( x1 , Vxn ) , I75 x = ∑ xk e k , 75 6-67/28 {e1 , W, e n }
k =1
XYZ[\Y]^_` aYZb_cd ebX]fXcgc hic_^iYX_^\Y L . jYaci kb_]e (x1 , lxn ) XYZ[\Y]^_` mccinbXY^Ydb, beb mcdhcX]X^Ydb oe]d]X^Y x \ aYZb_] {e1 , p, e n }. qb_ec oe]d]X^c\ \ XYaci] {e , , e }, `\e`st]]_` dYm_bdYeuX[d kb_ecd 1 r n f bX] \ L , XYZ[\Y]^_` iYZd]iXc_^us ebX]fXcgc Xc X]ZY\b_bd[v ]d]X^c\ eic_^iYX_^\Y b c cZXY oeY]^_` h a k dim L . w mccinbXY^Xcd hi]n_^Y\e ]Xbb, mcgnY \ - 21 -
nYXXcd aYZb_] oe]d]X^ x _chc_^Y\e`]^_` XYaci ( x1 , xn ) , es ac] ebX]fXc] hic_^iYX_^\c iYZd]iXc_^b n bn]X^bkXc hc _\c]f _^i m^ i] (bZcdciXc) n d]iXcd \]m^ciXcd hic_^iYX_^\ Rn . ^YXnYi^X[d aYZb_cd \ o^cd hic_^iYX_^\] XYZ[\Y]^_` XYaci oe]d]X^c\ {e1 , , e n } _ mccinbXY^Ydb e1 = (1,0, ,0 ) , e 2 = (0,1,0, 0 ) , …, e n = (0,0, ,0,1) .
L : e1 , , e n f1 , , f n . ! " , # # $
f1 , %, f n = e1 , %e n S , " S & n , & ' {e1 , (, e n } {f1 , ), f n }. * $ , # ! {f1 , +, f n } ,-.,/01, .234536 3478-2129:92 2 ;<=< 0 6;>876-:-80? ;8721, .2@? 41.2 980>2A8 B4>4C6=8 S ,-., 401, 34-:>6D=43365.
{
}
{
}
EFGHIF 1. J., 936D410-8 B6.6D20 4.?3:C -4K410-433:C L214. 6B4>8A22 1.6D432, 2 <936D432, 38 -4K410-43364 L21.6 6B>4=4.43: M8M « x + y » = xy 2
5 « λx »= x λ . N-., 401, .2 <M8783364 936D410-6 1 0 8M2925 6B4>8A2,92 .234 3:9 B>610>8310-69? O 1.8794>3610? 2 <M8780? ;8721. QIRISGI . J., ./;:C -4K410-433:C B6.6D20 4.?3:C L214. x , y 2 λ >47<. ?0 80 6B4>8A25 xy 2 x λ 0 8MD4 ,-., 401, -4K410-433:9 B6.6D20 4. ?3:C 5 L21.69, 0 6 410? B>238=.4D20 <M8783369< 936D410-<. T 4B61>4=10-4336 >6-4>M 65 < ;4D=8491,, L0 6 -14 8M1269: .234536P6 B>610>8310-8 -:B6.3,/01,. B U 6. 3<.4 6P6 .4943 8 2P>84 2 .6 8 >6. >6 2 6 6.6D36P6 .4943 8 =. 0 L 1 1, , ? V 0 ?B 0 - B V 0 5 343<.4 65 .4943 >8 =8336P6 x -:B6.3, 40 L21.6, >8-364 : 34 6 6>: - V 0 1 x. O ; - M 0 a ≠ 1, B6.8-4=.2-6 C B>4=10 8-.4324 b = Ca = a , P=4 C = log a b . W8M29 6;>8769, ./;:4 =-8 5 V.49430 8 =8336P6 B>610>8310-8 .234 36 78-2129: 2, 1.4=6-80 4.?36, 4P6 >8794>3610? >8-38 4=232A4. X 872169 ,-., 401, ./;65 343<.4-65 V.49430, 0. 4. ./;64 -4K410-43364 L21.6 a ≠ 1. EFGHIF 5 2. O:,1320?, ,-., 401, .2 936D410- 6 Y<3MA2 , 34B>4>:-3:C 38 60>47M4 [a, b ], .23453:9 B>610>8310-69 1 6 ;:L3:92 6B4>8A2,92 1.6D432, 2 <936D432, 38 L21.6. QIRISGI 234538, M69 ;238A2, 34B>4>:-3:C Y<3MA25 0 8MD4 ,-., 401, . Z 5 34B>4>:-365 =83369< 936D410-<. Y<3MA24 , 0. 4. B>238=.4D20 5 5 < 4D=849 5 T 4B61>4=10-4336 B>6-4> M6 ; 1,, L0 6 -14 8M1269: .234 36P6 B>610>8310-8 -:B 6.3,/01,. T<.4-:9 V.49430 69 =8336P6 B>610>8310-8 ,-., 401, Y<3MA2,, 0 6D=410-4336 >8-38, 3<./, 8 B>602-6B6.6D3:9 V.49430 69 - 22 -
>6 >8310-4 Y<3MA25 96D36 =., =83365 <=4 <3 A2 f ( x ) ; 0 Y M , − f ( x ) . O B 10 5 1610 8-20? .234 38-3610>8310-6 ,-., 401, ;41M634L3694>3:9.
G H I S FIRISG . 7.1. J., 936D410-8 -4K410-433:C L214. 6B4>8A22 1.6D432, 2 <936D432, 38 -4K410-43364 L21.6 6B>4=4.43: M8M « x + y » = tg (Arctg ( x ) + Arctg ( y )) 2 « λ x »= tg (λ ⋅ Arctg ( x )) . N-., 401, .2 <M8783364 936D410-6 1 0 8M292 6B4>8A2,92 .23453:9 B>610>8310-69? O 1.8794>3610? 2 <M8780? ;8721. 980>2A B6>,=M8 m × n .2345364 7.2. ;>87<40 .2 936D410-6 B>,96610>8310-6 6036120 4. ?36 6B4>8A2 1.6D432, 980>2A 2 <936D432, 5 980>2A: 38 L21.6? O 1.8794>3610? 2 <M8780? ;8721. 5 7.3. O:,1320?, ,-., 401, .2 =83364 936D410-6 Y<3MA2 , 78=833:C 38 60>47M 4 [a, b], .23453:9 B>610>8310-69. 1) T 4B>4>:-36 =2YY4>43A2><49:C 38 =83369 60>47M4. 2) 30 4P>2><49:C 38 =83369 60>47M4. 3) P>832L 433:C 38 =83369 60>47M4. 5 4) <3MA2 , =., M60 6>:C sup [a ,b ] f ( x ) ≤ 1 . 460>2A80 4.?3:C 38 =83369 60>47M4. 5) T U 8 3:C 3<./ B>2 x = a . 6) U 7) 8-3:C 4=232A4 B>2 x = a . 5 8) <3MA2 , =., M60 6>:C lim x→ a + 0 f ( x ) = +∞ . 9) 6360 6336 -67>810 8/K2C 38 [a, b ]. 10) 6360 633:C 38 [a, b ]. 5 7.4. T 8 02 980>2A< B>46;>876-832, 60 ;87218 x1 = (1, 0 ), x 2 = (0, 1) M ;8721< x ' = 1 (1, 1), x ' = 1 (1, − 1) . 2 1 2 2
7.5. O B>610>8310-4 R3 =83: =-8 ;87218 {e} 2 {f } 1 M66>=2380 892 ;87213:C -4M0 6>610 83=8>0369 ;87214 e1 = (1, 1, 1) , e 2 = (2, 1, 1) , 2 e 3 = (1, 1, 3) f1 = (0, 1, 1) , f 2 = (1, 0, 1) , f 3 = (1, 0, 2 ) . 5 1) T 8 02 980>2A< B4>4C6=8 S 60 ;87218 {e} M ;8721< {f }. 5 2) T 8 02 980>2A< 6;>8036P6 B4>4C6=8. 5 3) T 8 02 M66>=2380: V.49430 8 e1 - 6;62C ;87218C.
- 23 -
5 4) T 8 02 M 66>=2380 : X e V.49430 8 x - ;87214 {e}, 41.2 4P6 M66>=2380: ;87214 {f } 410? X f = (5, 3, 1) .
7.6. O B>610>8310-4 R3 =83: =-8 ;87218 {e} 2 {f } 1 M66>=2380 892 ;87213:C -4M0 6>610 83=8>0 369 ;87214 e1 = (0, 1, 1) , e 2 = (2, 1, 1) , e 3 = (1, 0, 1) 2 f1 = (1, 2, 3) , f 2 = (2, 1, 2) , f3 = (0, 1, 1) . 5 1) T 8 02 980>2A< B4>4C6=8 S 60 ;87218 {e} M ;8721< {f }. 5 2) T 8 02 980>2A< 6;>8036P6 B4>4C6=8. 5 3) T 8 02 M66>=2380: V.49430 8 f1 - 6;62C ;87218C. 5 4) T 8 02 M66>=2380: V.49430 8 e 3 - 6;62C ;87218C 4) 66>=2380: X e V.49430 8 x - ;87214 {e}, 41.2 4P6 M 66>=2380 : - ;87214
{f } 410?
X f = (2, 3, − 1) .
7.7. O B>610>8310-4 R4 =83: -4M0 6>8 x1 = (1, 1, 2, 1) , x 2 = (1, − 1, 0, 1) , x 3 = (0, 0, − 1, 1) , x 4 = (1, 2, 2, 0) , 8 0 8MD4 -4M0 6> y = (1, 1, 1, 1) . 6 878 5 M 0?, L0 6 -4M0 6>8 {x1 − x 4 } 6;>87=2380: -4M0 6>8 y . 7.8.
8M 27943201, 10>< M0<>8 980>2A: S B4>4C6=8 60 34M60 6>6P6 ;87218 {e} M =>80369 B6>,=M4?
5 2 7.9. T 8 02 M66>=2380 : 936P6L.438 f ( x ) = a 0 + a1 x + a 2 x + 5 2 n 1) O ;87214 27 Y<3MA2 1, x, x , , x .
{ 5 2) O ;87214 27 Y<3MA2 {1, 2 2>6 83364 2 .6
x − α,
}
+ an x n
(x − α )2 , , (x − α )n }, P=4 α -
Y M1 L 1 . 8 8 >2A< 2 98 B4>4C6=8 94D=< <M87833:92 ;8721892. 3) B 1 0? 0
6=936D4
8.
5 5 10- 6 A .234 36P6 B>610>83 10-8 L 387:-8401, .234 3:9 5 B6=B>610>8310-69 L 6036120 4. ?36 --4=33:C - L 6B4>8A2 1.6D432, 2 <936D432, 38 L21.6, 41.2 =., ./;:C V.49430 6- 27 A 2 ./;:C L214. >47<.?0 80 1.6D432, 2 <936D432, 38 L21.6 -36-? ,-., 401, V.49430 69 A . 0 6 6738L 840, 6 1896 ,-., 401, .23453:9 B>610>8310-69, 294/K29 ;8721 L0 6 B6=B>610>83102 >8794>3610?. >2 V0 69 dim A < dim L , B61M6.?M< B>2 dim A = dim L 5 B6=B>610>8310-6 27696> Y36 -149< B>610>8310-<. 36D410- 6 -14C .234 3:C - 24 -
L .
, , , {x ,! x }. - A A L
, A A . " A + A , x ∈ A x ∈ A , A ∩ A , A A . # A dim(A + A ) + dim(A ∩ A ) = dim A + dim A . A $ x ∈ L x = x + x , x ∈ A , x ∈ A , L A A , L = A ⊕ A . % , , A A $ . &'()*' 1. % , R ,
, , , . +*,*-(*. # : x = (x, . x) y = ( y, / y ). 0 α x + β y α x + β y , , , , R . 1 , , dim P = 1. &'()*' 2. 2 L L , {a ,a } {b ,b }: a = (1, 2, 0, 1) , a = (1, 1, 1, 0 ) ; (1, 0, 1, 0) , b = (1, 3, 0, 1) . 3454674b. #= L L , M M {a ,a } {b ,b }: . 5 M69 ;238A2 -4M0 6>6-
1
{x1 ,
xk }
k
1
1
1
2
2
2
1
1
2
1
2
2
1
2 1
1
2
1
2
1
2
1
2
1
2
2
1
1
1
2
1
2
2
P
n
P
P
P
n
1
1
1
2
1
2
2
1
1
M1 =
1 1 2 1
0 1 1 0
2
1 1
M2 =
0 3 1 0 0 1
- 25 -
1
2
1
2
2
2
1
2
# , L L . # L + L , M , {a ,a } {b ,b }: . # M , , : 1
1
2
3
2
1
1
2
2
1 1 1 1
M3 =
2 1 0 3
3
0 1 1 0 1 0 0 1
0 1 1 0 2 1 0 3,
0 1 1 0
. % , , $ . 1 0 0 1
0 0 0 0 2 1 0 3
0 1 1 0
, ,
, 0 . , L + L . ! L ∩ L
dim ( L + L ) + dim( L ∩ L ) = dim L + dim L , 1 0 0 1
1
1
1
2
dim(L1 ∩ L2 ) = 2 + 2 − 3 = 1.
1
2
2
2
1
2
. 8.1.
8.2.
, !" ! #$ %" &%#' %(" % &)% ) $ $ Rn * &%#&)% ) $ %', "! !" , % $ "+% ) $,'") % . 1) - %(" % ". %)% , &") $ .%%)# $ $ .% %)/ ) $ $ 0!1. 2) - %(" % ". %)% , 0''$ .%%)# $ .% %) / ) $ $ "# 2". 3) - %(" % ". %)% , 0''$ .%%)# $ .% %) / ) $ $ 0!1. , )3/'") %+% &)% ) $ $, .% 2 .% %)/ 4) - %(" % ". %)% * * !"($ $ ".% %)% &)'% . , !" ! #$ %" &%#' %(" % &)% ) $ $ Rn×n . $#) $ / '$ ) 2 &%)#.$ n &%#&)% ) $ %', "! !" , % $* "+% ) $,'") % 0.$,$ 4$, . * * * 1) -$ ) 2 0!" % &") % )%.% . 2) 5 $+% $! / '$ ) 2. 3) ")/ / )"0+%! / '$ ) 2. - 26 -
4) ''" ) / '$ ) 2. 5) ''" ) / '$ ) 2. 6) )%(#" / '$ ) 2. 8.3. &)% ) $ " R3 #$ ". %) $ x1 = (0, 1, 0 ) , x 2 = (1, 2, 1) , * x 3 = (1, 1, 1) x 4 = (3, 4, 3) . $ ) $,'") % 0.$,$ 4$, ! "* %* %4%!% . / ". %)% .
8.4.
8.5.
$*
) $,'") % 0.$,$ 4$, &%#&)% ) $ " A ⊂ Rn , "! .%'&% " ". %)% x ⊂ A 0#% !" %)1 0!% 1 x + + x = 0 . 1 n
$* )$,'") % 0'' &")""" &%#&)% )$ L1 L2 , !1 / ! "* ' %4%!%.$' ". %)% {a , a , a } {b , b , b }: 1 2 3 1 2 3 a1 = (1, 1, 1, 1) , a 2 = (1, − 1, 1, − 1) , a 3 = (1, 3, 1, 3) ; b1 = (1, 2, 0, 2 ) , b 2 = (1, 2, 1, 2 ) . b 3 = (3, 1, 3, 1) . 9.
"* %"
&)% ) $ % L , $ .% %)%' ,$#$ % .$!) %" &)% , "#" ", $, $" " .! #% '. .$!) %" &)% , "#" " (x, y ) %&)"#"!" .$. $.$ !% $ 0 .2 % # 0/ !"'" % x y &)% )$ $ L , .% %)$ 0#% !" %)" ("!"#01 ' $. %'$'. 5! "" " %+% !0 $, .%+#$ (x, y ) ∈ R : 1) 2) 3)
(x, y ) = (y, x )
( ''" ) % );
(α x + β y , z ) = α (x, z ) + β (y , z ) (x, x ) > 0 , ∀x ≠ 0 %&)"#"!3 % ).
(# ) 40 % ); (x, x ) = 0 ⇔ x = 0 (&%!%( "! $
!0 $" .%'&!". %+% , $ " .$!) %+% &)% , "#" , .%+#$ (x, y ) ∈ C , 0 $) %' &)% ) $ " #! &)% ) $ % $, $" 0 $) ' . * &%! " $. %' &%!%( "! % %&)"#"!3 % )"40" ,'" &") 01 , $. %', $4%) .% %)/ +! # !"#01 ' %4) $,%': 1)
(x, y ) = (y, x ) ()' % $ ''" ) % );
(
)
(
)
2) (αx + β y , z ) = α (x, z ) + β (y , z ) , &) %' x, αy = α x, y ; 3) ∀x ≠ 0 (x, x ) > 0 , (x, x ) = 0 ⇔ x = 0 (&%!%( "! $ %&)"#"!3 % ).
- 27 -
" .! #% %' &)% ) $ " '%( % " &% " %)', ! #! !"'" $ x , %&)"#"! "3 .$. x = (x, x ) . "" " %' !0$" (x, y )∈ R '%( % $.(" " &% " 0+! $ '"(#0 !"'" $' x y , #! .% %)%+% %&)"#"!3 cosϕ = (x, y ) ( x y ) . $. " .! #% %', $. 0 $) %'
&)% ) $ " !"'" x y , #! .% %)/ (x, y ) = 0 , $, $1 %) %+% $! ' . 0 " .! #% %' )% ) " #"!" % .$.%" ! 4% % )% ) % & $ & #& $ L * " %) %+% ! * " 0 ". %) ! ! . ( %+% ". %) # $ # $ x∈M $ # $ ' M. y , % % %.0& % "/ ". %)% y , #! .% %)/ (x, y ) = 0 , %4) $,0" % %"
M ⊥ , $, $"'%" %) %+% $! ' #%&%! " "' &%#&)% ) $ % * * &%#&)% ) $ $ M . 1 4%" " .! #% % &)% ) $ % !" &)'% 0''% %"+% &%#&)% ) $ $ "+% %) %+% $! %+% #%&%! " . ! n -'") %' " .! #% %' &)% )$ " "#3 4$, {e1 , , e n } , % .$!) %" &)% , "#" " !"'" % '%(" 4 ) $(" % ")", %! 42 / .%%)# $ X = ( x , x )T Y = ( y , y )T #$ %' 4$, " &) &%'% 1 n 1 n * , . !) %+% (x, y ) = X T Y , +#" '$ ) % $ &)% , "#" $& ''" ) $ '$ ) 2$ = e , e ij i j .
(
)
! " T (x, y ) = X + # Y , X + ≡ ( X ) $ " . % , ij = (ei , e j ) = δ ij , . . , {e , , e } . ' 1 & n R3 ( . ) ! {f , , f }, 1 * n " ! {e1 , +, e n } -, :
(f1 ,f1 ) ; e 2 = g 2 / (g 2 , g 2 ) , g 2 = f 2 − (f 2 , e1 )e1 ;
e1 = f1
……………………………………………….
en = g n
(g n , g n ) , g n = f n − (f n , e n−1 )e n−1 − -− (f n , e1 )e1 .
- 28 -
1. , R
n (x, y ) = x1 y1 − 2 x2 y 2 , n = 2 . . ,
! " , # ! : (x, x ) = x12 − 2 x22 . , # $ x = (0, 1), % & . 2. C n n (x, y ) = ' x y &
M ⊥ k =1
k
k
M , x ∈ M ( ( x1 + ix2 = 0 n = 2 . . ) % y ∈ M ⊥ , x ∈ M ,
x1 y1 + x2 y2 = 0 . * x ∈ M , ( M ! + x2 = ix1 , % x1 y1 + x2 y2 = x1 y1 + ix1 y2 = 0 , y1 + i y2 = 0 . , % y1 − iy2 = 0 , ( M ⊥ . , - % ,
, y1 ,
, .. M ⊥ , M . y1 − iy2 = 0 . .
& , & M ⊥ & ,
, e = (1, − i )T . /01023 145 607869859:4;<8=8 >:?:<35.
@ x1 = (1, 1, 1) , x 2 = (1, 1, 0) x3 = (1, 0, 0) & "- ? ( & ! ! & . ! ! 9.2. A # λ , - x = (7, − 2, λ ) # e1 , e 2 , e 3 : e1 = (2, 3, 5), e = (3, 7, 8) , ! e3 = (1, − 6, 1) . & - e1 ,
9.1.
2
e2, e3 .
- 29 -
, Rn : 1) (x, y ) = 2 x1 y1 + 3 x2 y 2 , n = 2 ; 2) (x, y ) = x1 y1 − x1 y 2 − x 2 y1 + 3 x2 y 2 , n = 2 . 9.4. , Cn : 1) (x, y ) = 3 x1 y1 + 4 x 2 y 2 , ) n = 2 , &) n = 3 ; 9.3.
2) 3)
(x, y ) = x1 y2 ,
n = 2;
(x, y ) = ix1 y2 + ix2 y1 ,
n = 2;
(x, y ) = x1 y1 + (1 + i )x1 y2 + (1 − i )x2 y1 + 3 x2 y2 , 5) (x, y ) = 5 x1 y1 + ix 2 y 2 , n = 2 .
4)
n = 2;
! - $ @
a 2 b2 a1 b1
B = ( A, B ) = a1a2 − b2b1 + c1c2 − d1d 2 , A = ? c2 d 2 c1 d1 ! & $. 9.6. , # ! # , - -, & 9.5.
( f , g ) = f (− 1) ⋅ g (− 1) + f (0) ⋅ g (0) + f (1) ⋅ g (1).
9.7.
, # # + n & ( % $ # n (a0 , a1 , , a n ) (b0 , b1 , , bn ) ( f , g ) = ∑ ai bi . #
2 n
$ & {1, t , t , , t }. 9.8. , # $ !, - [a, b], & i =0
( f , g ) = ∫ f (x )g (x )dx . b
a
- 30 -
9.9.
C2 (x, y ) = x y + (1 + i )x y + (1 − i )x y + 3 x y 1 1 1 2 2 1 2 2,
f = (1, 0) , f = (1, − 1) . 1 2 {f ,f }, (x, y ) 1 2 x y . , (x, y ) = 0
x 2 + y 2 = x + y 2 . (x, y ), , ?
9.10.
: « Rn {e , , e } . ! e 1 n k ", (e k ) = 1. #$ p ≠ k k % % (e k , e p ) = 0 ⋅ 0 + &+ 1 ⋅ 0 + &+ 0 ⋅ 1 + &+ 0 ⋅ 0 = 0 , ..
». R % 9.12. 4 x1 = (1, 0, 1, − 1, 2 ) , x 2 = (1, 0, 1, − 1, − 2 ) ( ' - x 3 = (1, 0, 3, 0, 0 ) . % . $ % 9.13. {1, x, x 2 } ( f , g ) = f (−(1) ⋅ g (− 1) + f (0 ) ⋅ g (0 ) + f (1) ⋅ g (1). ' - . ( 9.14. ' - n + 1- $ {1, t , t 2 , ), t n }, *1 ( f , g ) = f ( x )g ( x )dx . 9.11.
−1
- 31 -
( ' - $ f n ( x ) = Exp(− nx) , n = 0,1,2,... ,
9.15.
( f , g) =
∞
f ( x )g ( x )dx . 0 ' 9.16. , % [0,1]
( f , g ) = f ( x )g ( x )dx $ ∫ 1
0
f n = t n−1 f n +1 = t n " n → ∞ . 9.17.
C n
(x, y ) = ∑ xk yk n
k =1
":
$
M , x ∈ M
x1 + ix 2 + (1 − i )x3 = 0 , n = 3 ; 2) − ix1 + (2 + i )x2 − x3 = 0 , n = 3 . 1)
- 32 -
M⊥
"
!" # $ %, & '$ , ( #!", % %( % . ) &* # & ' * " $ % & !, , &*' && -$ , ! & $ & % +. , & * % $ , , & # ! % , ' ( &(#" *( " & - &, '! & ! - # & $(! # % % !. % . . ? 10. /0123242562 6 78916:58; <806=> 4652 5@A B02189 B1BC 3.
Aˆ , ! *(D ' L , M , % $ , %(D * x ∈ L y ∈ M * '**: Aˆ (α x1 + β x 2 ) = α Aˆ (x1 ) + β Aˆ (x 2 ). E, # $ x1, 2 ∈ L % %( %, $D ' %, !*' ' M , '! Aˆ (0) = 0 . F $ Aˆ (x )∈ M & x ∈ L % $ Aˆ $ # % imAˆ . F & x ∈ L , $ (D % * M , % %! Aˆ $ # % ker Aˆ . G $ L & L , Aˆ % $ L . ) !"H, $ ' , $*! " ! $ % *(D . ! !%( !# Eˆ , $!(D ∀x ∈ L Eˆ x = x . I ! Aˆ Bˆ !% % && , ! *(D * Aˆ Bˆ (x ) = Aˆ (Bˆ (x )). J& *# !% +, * & * $D *# Aˆ Bˆ ≠ Bˆ Aˆ , " [Aˆ , Bˆ ] ≡ Aˆ Bˆ − Bˆ Aˆ % & * ! . ) *# & * * * * $ % Aˆ Bˆ ( % & **(D ( #). - 33 -
I* " n - L $ $ {e , , e }. 1 n I ! * & ! $ & Aˆ , *" #' *# % $ $ $ & {Aˆ e1 , , Aˆ e n }. J ! & ! ' $ " $ * {e1 , , e n }, ˆ $*! #" $+ & ! n . ( $ * & * Ae j ' ! " ) $* n $+ , & , ! " , $ *( & ! *( +* A %!& n , *( + ' Aˆ $ {e1 , , e n }. G!# * * * !#% +. ! , # ! , $ $D" *# $ % n - ' L m - M , *" #' + Aˆ $*! % *' " + %!& m × n . Aˆ −1 , *! %(D H( Aˆ −1 Aˆ = Aˆ Aˆ −1 = Eˆ , % $ !% ! ' Aˆ . $ ! ! # * #% $ ' !% ! ' Aˆ % % % ' % ! # ", . . # ker Aˆ H" * ' . + ! ' , F!+ ! $# ' $ $" H" $ !* ! *% &* , $ "(, $ $ ' % ! & . $ ! * ' + A = 1 0 . I & "&* det A = 0 , !
0 0
$ ' . $! %, # %! Aˆ ! # $ * "& * ' : !% & ! T Aˆ x = 0 , . . x ∈ ker Aˆ . % ! x 0 = α (0, 1) 0 0 ! ' " & x = (x , x )T , *#, # 1 2 % % % ! %! $ ,&& , & im Aˆ = x1 (1, 0 ) . G D ! + S ! Af Aˆ $ {e1 , , e n } & $ * {f1 , , f n },
A
A = S A S . −1
e
f
- 34 -
e
1.
,
x2
ˆ A(x ) =
: , n = 2; x1 − x2 .
R
n
Aˆ (α 1 + β x 2 ) = α Aˆ (x 1 ) + β Aˆ (x 2 ) . : Aˆ (αx + βy ) = ((αx + βy ), (α ( x − x ) + β ( y − y )) )T , 2 2 1 2 1 2 T T α ( x2 , x1 − x2 ) + β ( y2 , y1 − y2 ) = αAˆ (x ) + β Aˆ (y ) , .
. . 2. , = z =0. x . " .
! a ( ).
# e ' . e ' a , " a , .. . $ e ' e ' ,
* ' 1 ( 1 1 1 % 1 ) , & = (1, 1, 1) . + e '= , 3 3 3 3 3 , A = 1 11 11 11 . 1, 1, 1
i
i
1 3
i
i
T
T
i
3
1 1 1 ,-.-/ .01 2-32431405363 1. 10.1. , : <: 9 B@ ? x : 2 877 @ x2 >== ˆ ˆ 1) A(x ) = ; , n = 2; 2) A(x ) = A , n = 2; x1 x2 x1 − x2 HF E <: 9 x2 C C F x : 2 7 ˆ ˆ 87, n = 2 ; 3) A(x ) = ; 4) A(x ) = F x1 − 3 C, n = 3 ; G D x1 x2
x3
- 35 -
Rn
5) Aˆ (x )
2x + x 3 1 = 2 x3 x1 ,
6) Aˆ (x ) = 0 ;
n = 3;
7) Aˆ (x )
x1 − x2
10.2.
= x1
+ 3 x2 , 2
0
x2
M-
n = 3.
L.
ϕˆ : L → M !" ": ϕˆ (x ) = x x ∈ M ϕˆ (x ) = 0 x ∉ M . # $$ ϕˆ %" "?
10.3. &$ '(" )
(x, n ) a ˆ A ( x ) = 2) (a, n ) ,
a ˆ A ( x ) = ( x , a ) 1) 2 , a
n
a
& $ % " .
R3
"
) % ) % * . , + % )& ( ) $ $ $ ", " +
-
%
, ! R3 , !$ ) & & ) - $" ,
$"
Aˆ
10.4. ,
Aˆ )" &.
x1 = x2 = x3 . . "/ 10.5.
) & . " ) $
x + y + z = 0.
10.6.
10.7.
) & . " " /
) $" 1 2 3.
x =x =x
ϕ = 2π 3
& & ) ( ) . * / " + "
10.8.
'( " " / & ) ,
"
0
/
&
** / $
f1 = (0, 1)
Aˆ
f 2 = (1, 0 )
{cos x,
sin x}
Dˆ = d dx .
R2 .
1
"
/
Af
& & &% & %( - " , " ) ˆ ˆ f = 4, 5 . " Af1 = 2, 3 A 2
10.9.
1
$
(
&
)
Rn : - 36 -
(
Aˆ ,
)
&
"
/
−2
25 60 A = 1) , n = 2; 60 144
10.10.
{1, t , t
, , t
2) A = − 2
2
5
3
5 3 , n = 3. −5 −3
}
2 n & % " + & ) & ( " / , $ ! : 1) **/ $ Dˆ = d dt ;
n
∫ f (ξ )dξ . t
ˆ 2) $ I =
0
11.
M
&% L
Aˆ
) ) M ,
L.
λ -
Aˆ
% " " $ , ! ) + , " " M ", $ ( , ) ) ) +. ! "
x
&%
x∈M
Aˆ x = λ x
Aˆ
% ) $ " " !" , + λ - & % %( & " " + ". 1 + & &% ) $ ". L n -" " & ( " / A , 1 n & %( & %( ) + $ + ) % %( %( ) "
$ n $ " ) x , %" "/ . % & & ( + $ λ $ $
$
$ & ) % " , +
+
&% " n $ , $ $ ! $ λ, () ) % + " ". " , ) ) %( () ) " " + + " n ' ) ( ) %( ) % + " " . ! " , " . &%
{e , , e }
( A − λE )
det ( A − λE ) = 0
- 37 -
1. ,
A = 3 2 + 2i . − 2i 1 . ! 2
" # # λ = 5 λ = −1. 3 − λ 2 + 2i , 1 2 =0 2 − 2i 1 − λ
& # , ) ** x ' ,*1 + i )'. %− 2 x + (2 + 2i) x = 0 , # e = + ('= + $ ( (2 − 2i ) x − 4 x = 0 x 1 / # # .4 x + (2 + 2i) x = 0 , # (2 − 2i) x + 2 x = 0 53 2 53 2
( −1 + i) / 2 3x e 2 = 4 1 100= 4 10. x2 1 2. , A = − 1 2 , #6 ! 3 1 ! # 7 . . 8 # − 1 − λ 2 , =0 3 1− λ 7 # λ − 7 = 0 . 9 # 7 , , # :; # : , #; #. 1
2
1
1
1
1
2
2
2
1
2
2
<=>=? >@A B=CBDCAD@E CFC A. , 11.1. A = − 1 2 , #6 ! 3 1
; ! #
. , 11.2.
:
- 38 -
1) A =
3
2−i
2+i
7
2 −1 −1 2) A = 0 − 1
;
0
1 0 0 0
4)
A=
0 0 0 0
0 ; 1
3) A = i
3
0;
0
0
4
1 0 0 0
; 5)
1 0 0 0
A=
0 0 0 1
11.3.
2
3 −i 0
0 0 0 0
.
0 0 0 0 1 0 0 1
L L ^
1
0 1 0
1 L1 = 1 0 1 2 0 1 0
^
2
0 −i 0 1 L2 = i 0 −i . 2 0 i 0
L L . 9# ^
L1
11.4.
1
2
9# σ = 0 ^
y
11.5.
^
σz
i . −i 0
0 1
= 1 0
!" #", $!" ! x1 = x2 = x3 . #' Aˆ , "( # λ1 ≠ λ2 . % &, α x1 + β x 2 !# # α β # Aˆ . )! * − i ∂∂ϕ , '$ 0 ≤ ϕ < 2π .
11.6. %! & x1 x 2 -
11.7.
11.8.
Aˆ Bˆ ! !". % &, + %!! &#! # , ( , .
- 39 -
#$ L # $ (x, y ) $# $' #' Aˆ ' ! * Aˆ , !$# "(' ! (Aˆ x, y ) = (x, Aˆ *y ) $# #" , +# . {e1 , , e n } * ij = (e i , e j ) * −1 * ' ! # ) # A = ( #! * = ( −1 ) !"# $% $"&'(#, )*# '#+,( - .%('(#, !"# $% # $ !+/0#%1#. 2 '($,% $,1, +, % + 1+ (%% -(.1$# A* = AT #3#$, #%% # "1* !+ $,+(%$, # 1 A* = A+ &%1,(+% !+ $,+(%$, #. 4!#+(, +, *"/ , + ) Aˆ * = Aˆ , %(.5 (#,$/ $( $ !+/0#%%5 , 1"1 6+ 1, 5 . 7) (,+18(, ( 6, $"#*&#, 1. 9 + &"5 A*:, / "/#,$/ #3#$, #%% : $1 #,+1'% : "1- !"# $% : 6+ 1, : *"/ (,+18# . 7; $ -$, #%%5# .%('#%1/ #3#$, #%%5, ( $ -$, #%%5# # , +(, /31#$/ +(."1'%5 $ -$, #%%5 .%('#%1/ , +, ) %("<%5 1 -+(.&=, ,% $ -(.1$ !+ $,+(%$, # L . 4!#+(, + Uˆ , $ >+(%/=31: $ ("/+% # !+ 1. #*#%1#, ,. #. &* "#, +/=31: $ ,% ?#%1= (Uˆx,Uˆy ) = (x, y ) *"/ "= -5> # , + !+ $,+(%$, ( L , %(.5 (=, +, ) %("<%5 !#+(, + $"&'(# #3#$, #%% ) !+ $,+(%$, ( 1 &%1,(+%5 !#+(, + $"&'(# !"# $% ) !+ $,+(%$, (. 2 6, $"&'(# (,+18( $ !+/0;%% ) !#+(, +( +, % + 1+ (%% -(.1$# +( %( -+(,% : (,+18#, U * = U −1 . @ -$, #%%5# .%('#%1/ &%1,(+%5> + ! *&"= +( %5 #*1%18#, ( $ -$, #%%5# # , +(, ( 1 *"/ (!#+(, !+/0 $ $ ;%% ) !#+(, +(, -+(.&=, +, % + 1+ (%%5: -(.1$. 12.
ABCDEB 1. F(:,1 $ !+/0;%%5: !#+(, + "1%#:% & !#+(, +& Aˆ , $&3#$, "/=3# & ! + , %( &) " ϕ = 2π 3 +&) !+/ : x1 = x2 = x3 !+ $,+(%$, # R3 . GEHEICE. J+1 & (.(%% .(*('1 ! + ,# -(.1$%5# # , +( *# (+, : $1$,# 5 !+# -+(.&=,$/ $"#*&=31 -+(. : e1 → e 2 , e 2 → e3 , e3 → e1 , 0 0 1 ! 6, & (,+18( !+# -+(. (%1/ 1 ##, 1* . 2 +, % + 1+ (%%
A= 1 0 0 0 1 0
-(.1$# #3#$, #%% ) !+ $,+(%$, ( - 40 -
(,+18(
$ !+/0;%% ) !#+(, +( +( %( ,+(%$! %1+ (%% : 0 1 !#+(, +(, ! 6, & %'(,#"<% 1 ## A = A = 0 0 *
T
0 1
(,+18# 1$> *% ) .
ABCDEB 2. 2 # "1* !+ $,+(%$, # 1 0R 0 $ $,(%*(+,%5 2 +, % + 1+ (%%5 -(.1$ {e1 ,e 2 } .(*(% -(.1$ {f1 ,f 2 } 1 (,+18( A f "1%#:% ) !#+(, +( Aˆ 6, -(.1$#. F(:,1 -(.1$# {f1 ,f 2 } (,+18& * $ !+/0;%% ) !#+(, +( A f , #$"1 f1 = e1 , f 2 = −e1 + e 2 , 1 A f = 1 4 . 1 1 GEHEICE. (,+18( $ !+/0;%% ) !#+(, +( %(> *1,$/ ! 9 + &"# f = (f , f ) #$,< (,+18( +( ( -(.1$# {f ,f }. * −1 A = , )*# 1 2 ij i j 5'1 + "// # ! 3<= &" 1 1 ! 2 $ ; $ 9 $ "<.&/ f1 = e1 , f 2 = −e1 + e 2 +, % + 1+ (%% $,< -(.1$( {e1 ,e 2 }, 1 ## = 1 − 1 , ( -+(,%(/ −1
f
(,+18( 1 ##, 1* A
*
−1 f
=
2
25'1$"// !+ 1. #*#%1# ,+;>
2 1. 1 1
= A , −1
* f
=
0 . 3 −1
(,+18
3
! !" #$ % &"'"$ . * * *. * . . 12.1. () + , , Aˆ , - /01) / n -, , , / L , 2- -)3 * ,)4* . 566)4)) (λ1 , 7, λn ). 8*) ,)4 . ,9:; 3 , , A* , . ) () /. 566)4)+ λi ∈ R ) (<) /. 566)4)+ λ ∈ C . =/ 9.9 ) , , Aˆ . . ,9:;+? . . . .-,+ , ,), /+ 12.2. > / )- / , , / R <2). {e1 ,e 2 } 2- <2). {f1 ,f 2 } ) ,)4 A f )* 3 , , Aˆ / 5 <2).. 8*) / <2). {f1 ,f 2 } ,)4 . ,9:; 3 , , A*f , . ) f = e , f = 2e + e , ) A = 1 2 . 1 3 . . . . . * 12.3. ? / / )- / , , / L 2- ,9:;+ , , ˆ , )01)* <,+*. ? 2, <,+* , , : A 9/ 9.9 . . ,9:;+. i
2
1
- 41 -
1
2
1
2
f
> , .,./ R3 2- , , / , 3 :)9 Aˆ x = [a, x] , 3- a = (a1 , a 2 , a3 ) - 6).), /+* / ,. 8*) . ,9:;+* , , Aˆ * . : ) ,)4 . . ,9:; 3 , <,2 /)9 /1./ 3 12.5. / )- / 3 , .,./ <+ .),) *? . . ˆ 12.6. ? M - )/,) ) , 3 3 , , A , .,./ / )- / 3 , .,./ L . ? 2, - , 3 - ) M : )/,) .) , , Aˆ . . . . . * ˆ 2- 12.7. > / )- / , , / R2 ,9:;+ , , A ./ * ,)4* A . 8*) ,)4 B , , Bˆ 3 , Bˆ 2 = Aˆ , )2 , ,). 1./ ) ,) 2-) - 9 0< 3 (.2- . ,9:; 3 , ,? 12.4.
⊥
1) A =
5
−2
−2
8
;
2) A =
52 32
32 52
- 42 -
.
4.
,. )). * 3 ,)) ,..,)/ ). ,/)9 ,)/+ > / , 3 ,9- . .), : ,/)9 /, .* / , 3 ,9- / , .,./. > 5) ,/)9 / -) . , 3 ). . 3+ / , 3 ,9-, 2+/9 3, * /-,)+ . 3+. < <1) - <+ /+,:)* . * , .,./ 0 < 3 ). * 9/ 9.9 ,)9 /-,)+ ( - 3 ,3) ) <) )*+ )2, ) . * ( -/ ,3 /) 6 ,. ) / 2- )) 3 ,)), / , . 9/ 9.9 /+< , * .).+ ,-), 3- 6 , ) . /+
, . * /)- .+ /-, / . ,- ;+) 566)4)), )< 2+/+* ).) /)- . , 3 , / ,) :)9 ) /: / , . . 9./ 2 - 9 2)*, ,))+ / ,) * 6 , * ,) ,2 )+ ,3, ,)/ -) < -) .) )2)9 ..)6)4)) /-,)+ 6 ,. 13.
*) )* *
!"#$ % &. '" # ! ! $ (! &) "#)
6 , * B(x, y ) , 2- * )* , .,./ L , 2+/.9 ). /9 64)9 -/ 5 / - 3 , .,./,
)*9 :- )2 ), .. B (α x + β z, y ) = α B (x, y ) + β B(z, y ) 3) - 9 / , 3 ,3. + <- ,..,)/ /1./+ )6 B(x, y ) ,) 0<+ 2)9 ,3 / 9/ 9.9 ,+. , 3- 2 ) /1 /+ ). . +. ) B(x, y ) = B(y, x ) , <) )*9 6 , 2+/.9 .),) *. ,4)9 - 3 ,3 A(x ) ≡ B(x, x ),
9 )2 .),) * <) )* * 6 ,+, 2+/.9 /-,) * 6 , *. <, , 2- * /-,) * 6 , A(x ) : <+ .),)9 6 , ., <) )*9 1 . {e1 , -, e n } n -./01232 B (x, y ) = ( A(x + y ) − A(x ) − A(y )). > < 2) 2 < 402560715687 L 9:;:1/ 17= : >87?076:@17= A20.B .23C6 9B6D E74:571B @ /0 /E 562; 9FB >220?:176 X = ( x1 , Gxn )T : Y = ( y1 , H yn )T 582:I 703C./1628 40: 42.2J: .760:@12< E74:5: 9:;:1/<12< A20.B B (x, y ) = X T BY , 3?/ .760:F7 9:;:1/<12< A20.B 240/?/; /17 @/0/E /K E17@/1:= 17 97E:51BI 8/>6207I >7> bij = B(ei , e j ) . L >220?:1767I 8/>62028 x : y E17@/1:= A20.
E74:5B87M65= >7>
B(x,y ) =
bij xi y j : A(x ) = ∑ aij xi x j , 40: N62. ∑ i , j =1 i , j =1 n
n
- 43 -
.760:F7 aij >87?076:@12< A20.B 85/3?7 =8;=/65= 5:../60:@12<. L @7561256:, 5>7;=012/ 402:E8/?/1:/ 8 8/J/568/112. /8>;:?282. 40256071568/ 40/?5678;=/6 5292< 9:;:1/<1CM A20.C 5 5:../60:@12< .760:F/<, =8;=MJ/<5= .760:F/< 07.7. 5;: 40: 42.2J: 1/8B02?/112< .760:FB S 8B42;1/1 4/0/I2? 26 97E:57 {e1 , , e n } > 97E:5C {f1 , , f n }, 62 .760:F7 B f 9:;:1/<12< A20.B B 8 1282. 97E:5/ 58=E717 5 .760:F/< Be 8 56702. 97E:5/ 522612/1:/. B f = S T Be S . 7 :5 8 26202 760:F7 >87?076:@12< A20.B :.//6 ?:73217;D1B< E . . , < > 5>:. 97E:52.. 87?076:@17= A20.7 8 / (>7121:@ /5>: ) 8:?, 17EB87/65= >7121:@ 67>2. 97E:5/ =8;=/65= 5C..2< >87?07628 >220?:176 /K 703C./167, n A(x ) = ∑ λi xi2 , 3?/ >2NAA:F:/16B (λ1 , λ n ) i =1 . ( ! )
# #
! " $ !
% , ,
% % &! ' % , % ! ' ! " # , % ! " ! . (! ! ! " , ! ) " & , %* "
! % " ! &! ! & ! A(x ) . + " ! % ! % ˆ - % &$ % ,
B(x, y ) = Aˆ x, y , "! A % ! ( ! ) B (x, y ) . , % ! $ " % % ! , % , - '
% ! $ " % A ˆ , ( ! )
! " # !, $ ˆ . # % '! ! " % A " ! " - " # % , ! % % ! ! % ! " # % .
(
)
./012/ 2 1. ! ! A = −18 x1 x 2 + 9 x 2
# % R2 % # $ . 3242502 . 6 , % ! ! A(x ) & # % , !
& - 44 -
1 # , % , B (x, y ) = ( A(x + y ) − A(x ) − A(y )). % 2
! ! ! A(x )
B(x, y ) = −9 x1 y2 − 9 x2 y1 + 9 x2 y2 , 0 − 9 . , % % , %!
! B=
−9
9
& A(x ) ≡ B (x, x ) ! ! , % # '! & ! A(x ). ./012/ 2. ! A(x ) = 2 x3 x 4 % R4 ! ! ) " & , ! . % 3242502 . , ' ! " ! - % " ! & . ! , % % ! " ! " % !
! & % # ' " % !
y −y % % x3 = 3 4 % % x , x ,
y = x
x
4
3
2 y3 + y4 = 2
4
1
1
y2 = x2 . , ' ! ' ! %
! A( y1 , y2 , y3 , y4 ) = y32 − y42
( x1 ,
(0,
0, 1, − 1)
T T xn ) = S ( y1 , yn )
% S=
1 0 0 1 0 0
0 0 1
0 0 −1
0 0 2
.
1
2
2 1
2
. 13.1.
# ! % Rn % # - ! : 1) B = x1 y1 , n = 1 ; 2) B = x1 y1 , n = 2 ; 3)
B = 2 x1 y1 − x1 y 2 − x 2 y1 − 5 x 2 y 2 , n = 2 .
- 45 -
13.2. ! !
#
% Rn % # $ :
1) A = −3x1 , 2
3)
n = 1;
2)
A = 2 x12 − 6 x1 x2 − 3 x22 , n = 3 ;
A = x12 + 4 x1 x 2 + 4 x1 x3 + 5 x 22 + 12 x 2 x3 + 7 x32 , n = 3 ;
A = ∑ xi xi +1 . n−1
4)
i =1
13.3. ! % R3 ! ! ) " & , ! % : 2 2 1) A(x ) = 3 x 2 + 3 x3 + 4 x1 x 2 + 4 x1 x3 − 2 x2 x3 ;
2) 3)
A(x ) = x12 + 3 x22 + x32 + 2 x1 x2 − 4 x1 x3 ; A(x ) = x1 x2 + x2 x3 ; 4) A(x ) = x1 x2 + 2 x1 x3 + 4 x2 x3 .
13.4. ! %
R3 !
! " # " % , % :
1) 3)
A(x ) = −2 x2 x3 ;
2)
!
A(x ) = x12 + 3 x22 + 2 x1 x2 ;
A(x ) = 3 x 22 + 3 x32 + 4 x1 x2 + 4 x1 x3 − 2 x2 x3 .
!"# A(x ) $%& ' (") "* &)+ " "(&&), "-, &') & & && & ∀x ≠ 0 # # A(x ) > 0 A(x ) = 0 ⇔ x = 0 . ."" ' ", '# " $ / && ' " ' (" & " 0 &)+ "- "(&&), "' / "- !"#%, ( ) &' "1"1" && ' !"# $%& ' (")2"(&&), "-. !"# (") "' +3 "(&&) & ' 4""# '" 5 / " &'/ 5 /"6!! 0 & " (λ1 , 7λn ) , (")2 & %5 ( 14.
2 ( && &, / 1" )+ "#2 2 A(x) = ∑ λi xi . 8') '& λi > 0 , i =1 & & & # ) ' ) * )+ A(x ) , " !" ", (" " " "(&&), "-. " &'/ n
- 46 -
4$ ' $ & & '&5 λ $ ' " '("'"4 ( && / 1" )+ "#2 i 2, " /" ')" (" )"* &)+ %5 " 0 &)+ %5 / " &'/ 5 /"6!! 0 & " ) & ' "# (&"4$" 4$ '. " 2 &*& & "' $ & $/" &0 / %5 !"#. & & && ' " + $ /""( ), "' + / "- !"#% (" , # 0 aij , & ( " &, / / " &'/"#2 2, ("$") & / & - . )+&' . ) 6 "1" $2 3 ' $ / 21) "%5 # " " ∆ k # 0% aij , (&' ) 3 & '"4"- "(&&) &) k -1" (" /, '"' )& %& $ 6)&#& " )&"# &5 &# 21)2 # 0% aij $#&"# k × k . 8') '& ∆ k > 0 , / !"# ' # 0&- aij ) & ' (")"* &)+ " "(&&), "-. 8') $ / ∆ k &&23 ' , ( ,# ∆1 < 0 , / !"# ) & ' " 0 &)+ " "(&&), "-. #& #, " "4" 5 ')2 5 det A = ∆ n ≠ 0 . /" &0, &') 2/$ %& 2')" & %(") 3 ' , !"# aij & ) & ' $ /""(&&), "-.
1.
& & / / 5 $ 5 ( # λ /
A(x ) = λx12 − 4 x1 x2 + (λ + 3)x 22
4
!"# ) & ' () (")"* &)+ " "(&&), "-
&)+ " "(&&), "-? -
−2 . . ."' # # 02 " / " !"#%: A = λ −2 λ +3 & & ,# , 21)" % # "%: ∆1 = a11 = λ , ∆ 2 = det A = λ (λ + 3) − 4 . & && &4 & & & & & ) (")"* )+ "- "( ), "' " 5" #" " " # " %(") 2')" - ∆ > 0 ∆ > 0 . & (")2 & 23 ' ' && ' 1 2 & & 5" #, " !"# ) & ' (")"* &)+ " # ' )+ " "& & "- ( λ , && "( &), " ( λ > 1. ) , 2')" & " 0 &)+ "- & "(&&&), "&' "$ , " ∆1 < 0 ∆ 2 > 0 , " ( " / 21"#2 3 ' ' #% && ' " "' &)+ " (#& λ , #& ", λ < −4 . /" &0, &*2 /& $ & - − 4 < λ < 1 / !"# & ) & ' ()"#* &)+ " "(&&), "-, ("'/")+/2 / & - . )+&' (" " &)+ ", & " 0 ) %5 $ - λ & %(") & ' .
( ) " 0
- 47 -
&4 & & 2. / /"# " 5 " #"# "' " "# 2') " / %
#% A(x ) − A(x ) #"12 4% + ( && % / " "#2 / " &'/"#2 2? !"
))3' 2&# && & $ ' # #&& )& (" " ( . " . 2#& "# ("' ' & ''#" # / 23 !"#2 A(x ) = x 2 − x 2 , #&323 / " &'/ - . "- !"#% 1 & 2#, 1 ')2" &)+ %5 /"6!! 0 & " ". "# " 0 &)+ %5 (")& "* && & - , " ) 3 -' " − A(x ) = − x12 + x22 /* # / " '/ &1" "4 # $ /"#. ) "1" "4% / " &'/ & % '"() , (& &%2 "45" #" ("&' $#& 2 /"" x ↔ x . &(&+ ''#" # 21"1 2 & && & 2 2 2 3 # # ' # ,# 1 % ( !" " , , : A(x ) = x1 − x2 + x3 . 1/" & +, " '"" & ' 23 !"# − A(x ) = − x 2 + x 2 − x 2 & #"*& 4% + 1 2 3 & & & & & && #2 #2 / %2 / '/ ) 2 A(x ) (2 ,# % ) (4 & ( " " 1" $ / (&&' "/ /"" , ("'/")+/2 #&& & , %- 1, . &. " & , "& ')" & 2)&%5 / " &'/ 5 /"6!! 0 & ". /" &0, ''#" # !"#2 " "1" 1 A(x ) = x12 + x22 , ) /" ""- − A(x ) = − x12 − x22 . ", " ' )2 && ' ') (")"* &)+ %5 (2) " 0 &)+ %5 (0) / " &'/ 5 /"6!! 0 & " 2 !"#% A(x ) (" $&' (&&' "/2 /"" ' 0&)+3 '"(& / " &'/"1" A(x ) − A(x ) /*& & ) & ' "$#"* %#. 4"4& & (")2 & %5 &$2)+ " (&' ' ' )3 4"- $#& "' )3 4%& / %& !"#%, /" "%& ('&"1 #"* " ( &' / / " &'/"#2 2, & (&' ) & $ 2 & -. 4 & ' )& %- 2') " $ "( "' $2 /: / # " $"#, " (" / !"# ")* #& + , %- 1 "& ') " (") "* &)+ %5 " 0 &)+ %5 / " &'/ 5 /"6!! 0 & ". . 14.1.
& & & / / 5 $ & 5 &( & # - λ 4 / & & &!"# - ) ' ) * )+ ), 0 )+ ), ( ) (" 1) 2) 3) 4)
14.2.
"
" "(
"
A(x ) = −9 x12 + 6λx1 x2 − x22 ;
( )"
" "(
"
?
A(x ) = 5 x12 + x22 + λx32 + 4 x1 x2 − 2 x1 x3 − 2 x2 x3 ;
A(x ) = 2 x12 + x 22 + 3 x32 + 2λx1 x2 + 2 x1 x3 ; A(x ) = x12 + x22 + 5 x32 + 2λx1 x2 − 2 x1 x3 + 4 x2 x3 .
/ !"# ("' ' & Rn . ) & ' ) ("("' ' "# Rn # "*&' " M &/ "" x ∈ Rn / 5, "
2' + A(x )
-
- 48 -
/ &' & ( #& !"#2 A(x ) = x12 + x22 − x32 ,5#& "# ("' ' &. & & 14.3. % "* # 0 C (" / n . "/ $ +, " / !"# ' # 0&- B = C T C ) & ' (")"* &)+ " "(&&), "-. # 0 A ) & ' # 0&- (")"* &)+ " 14.4. 2' + / - / "- !"#%. "/$ +, " "4 # 0 A −1 "( &&&), " /* ) & ' # 0&- (")"* &)+ " "(&&), "- / "!"#%. A(x ) ≥ 0 ?
& ''#" +
- 49 -
1.
2. 3.
4. 5.
6.
7.
.
.
&/)&# &, ,
., 2000. . . !, ".#. $%&!'(, ) * , ., 1984. +. . ,-( -. /-01, .2. $-34%0 5, . . 67814%0, 9 : ; < , ., 2003. .. $4%=(74'(%0, 9 : ; < , ., 2002. >.6. ?473 @(1', . . A /( !, ) * B < C : ; C, ., 1985. 9 : ; < , D%E 4-E1(@ -F .G. H-E-!(%, !=(, 1999. I. . ,%4-0 5, J< ; K, ., 1970.
- 50 -
# 101 1. 134 @ , %D4-E- 3- = =3-. !-F! 7410!-! F 1. 134 @ 2. D4 -E- 3 - . 1! .134 @ 3. 8413!1' .134 @1. 134 5! - 7410!-! ' F F 4. G =3 -. !- ! 7410!-! (41.-4 %0=(% % 3 D1 F F 5. G =3 -. %E!%4 %E! !- ! 7410!-! F F 6. G =3 -. !-%E!%4 %E! !- ! 7410!-! # 101 2. + !-F! - D4%=341!=301 F 7. D4 -E- -! - !- !% % D4 %=341!=301. , 1& = 41&.-4!%=3 F 8. $%ED4 %=341!=301 !- ! D4 %=341!=30 9. 0( E%0 D4 %=341!=301 # 101 3. + !-F! - %D-413%4 F 10. D4 -E- -! - .134 5!1' &1D = !- ! %D-413 %4 %0 11. I1E151 % =%8=30-!! &!15 -! ' =%8=30-!! 0-(3 %41
F 12. + !- ! - %D-413 %4 0 -0( E%0 D4 %=341!=301
# 101 4. , !-F! - (01E413 5! - %4. F F (01E413 5!%F %4.. 13. 134 @1 8 !- !% $4 0-E-! - ( (1!%! 5-=(%.7 0 E7 14. ? 1== (1@ ' (01E413 5! %4. + 3-413741
- 51 -
G34. 3 3 7 11 13 15 17 21 21 24 27 33 33 37 40 42 43
46 50
