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!"# $%& '(# & #) , , ## «*#"& +# &.$ #-. *. . /!+&» 0.1. 2-#3+#, 4.5. +#, 4.1. 6"#+ 789:;?@? A B9 C@. :* !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#& : < #! x1 − 2 x2 + x3 + 5 x4 = 5
- 18 -
1 −2 1
1
1 −2 1
5
4424. $'(-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 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 >478A25 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< 936D410-4=10-4336 >6-4>M 65 < ;4D=8491,, L0 6 -14 8M1269: .234536P6 B>610>8310-8 -:B6.3,/01,. B U 6. 384 2 .6 8 >6. >6 2 6 6.6D36P6 .4943 8 =. 0 L 1 1, , ? V 0 ?B 0 - B V 0 5 3438 =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:-3:C 38 60>47M4 [a, b ], .23453:9 B>610>8310-69 1 6 ;:L3:92 6B4>8A2,92 1.6D432, 2 4>:-3:C Y4>:-365 =83369< 936D410-4=10-4336 B>6-4> M6 ; 1,, L0 6 -14 8M1269: .234 36P6 B>610>8310-8 -:B 6.3,/01,. T610>8310-8 ,-., 401, Y8-38, 3602-6B6.6D3:9 V.49430 69 - 22 -
>6 >8310-4 Y8A2,92 .23453:9 B>610>8310-69? O 1.8794>3610? 2 <M8780? ;8721. 980>2A B6>,=M8 m × n .2345364 7.2. ;>87,96610>8310-6 6036120 4. ?36 6B4>8A2 1.6D432, 980>2A 2 2A: 38 L21.6? O 1.8794>3610? 2 <M8780? ;8721. 5 7.3. O:,1320?, ,-., 401, .2 =83364 936D410-6 Y47M 4 [a, b], .23453:9 B>610>8310-69. 1) T 4B>4>:-36 =2YY4>43A2>47M4. 2) 30 4P>2>47M4. 3) P>832L 433:C 38 =83369 60>47M4. 5 4) :C sup [a ,b ] f ( x ) ≤ 1 . 460>2A80 4.?3:C 38 =83369 60>47M4. 5) T U 8 3:C 32 x = a . 6) U 7) 8-3:C 4=232A4 B>2 x = a . 5 8) :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>< M08 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 Y2A< 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 478310-69, 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- 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;:?: #, + !+$,+(%$,( L , %(.5(=, +,)%(" 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" $23' $/ 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 : ;