x0x1x2x3
x0x1cx2 xcxdx13x14x11x1dx1ex1fx20x13x14
x1 x2 5–1
1,x5bx72x36xex15x17x44x42xfx44,x1dx1exdx54x13x29x75x2ax32xex9Rx79x15x74x26x70x71?
(1)x2bxexdZ; (2)x47x5xexdQ; (3)x32xexdR; (4)x40xexdC.
x5,(1)x42(2)x6dx55x13x32xex9Rx79x15x74x26x70x71,x21x22x55x20xfx44x55x12x13,(3)x3a(4)x6dx13Rx79x15x74x26x70
x71.
2,x1eKx22x40xex9C,x1x24x32xex22xbx58x15x48x41n￡nx5dx5ex15xdx54x2x5dx5ex15x17x44x42x55x20xfx44x13x29x75x2a
Kx79x15x74x26x70x71? x22x2fx30?
x5,x29,x2ax3cx55x20xfx44x55x12x13.
3,x42x7dx1dx1exdx54x2x3cx23x1bx15x33x67x13x29x75x2ax32xex9x79x15x74x26x70x71:
(1)x33x10x32x2x75(x4dx75,x79x34x35x36)x5dx5e,x2x3cx5dx5ex15x17x44x42x55x20xfx44;
(2)x48xex56x3cn (n > 1)x15x32x6axex9x3ax29x33x10,x2x3cx9x3ax29x15x17x44x42xfx44;
(3)xcxfx79x33x10x1fx20,x2x3cx1fx20x15x17x44x42x8x1dx1dx4dx15x55x20xfx44:
kfi = fi;
(4)x33x10x72x32xeR+,x17x44x3ax55x20xfx44x1dx4dx22:
a'b = ab;
k–a = ak:
x5,(1)x13; (2)x29,x6fx9x3ax29x55x6bxdx54x16; (3)x29,x21x22x62fi 6= 0x52,0fi 6= 0; (4)x13.
4,x78x67x79x61x16x23x25x31x15x74x26x70x71x15x46xex3ax7a.
x5,(1)x32x2x75,n(n+1)2 x46,x7afEij +Eji j i 6 jg;
x4dx75,n(n?1)2 x46,x7afEij?Eji j i < jg;
x79x34x35x36,n(n+1)2 x46,x7afEij j i 6 jg.
(4) 1x46,x0x8x55x56x3c1x15x72x32xex6dx3ex2fx22x7a.
5,x53x54:x33x10x24x6fx22x1cx73x15x32xex1e
S =
n
fang = (a1;a2;a3;￠￠￠ ;an;￠￠￠) j ai 2R; limn!1an = 0
o
x5bx8x1dx1dx4dx15x17x44x42x55x20xfx44:
fang+fbng = fan +bng;
kfang = fkang
x75x2ax32xex9Rx79x15x48x66x2cx73x46x74x26x70x71.
x3x4,x7dx53x74x26x70x71x69,x22x58x54x38x13x2cx73x46x15,x2x0x1x15x72x2bxen,x47x48x66x43x44x3c0x15xex1e,fin =
f0;￠￠￠ ;0;1(x3dnx3a),0;0;￠￠￠g,x3cx13x2x3cx0x1x3bx15n,x38x47nx66x1fx20fi1;￠￠￠ ;finx74x26x2cx2a.
￠ 1 ￠
6,x18
P =
‰ fi fl
fl fi
flfl
flfl fi;fl 2C
:
(1)x53x54,Px5bx5dx5ex15x17x44x42x55x20xfx44x75x2ax32xex9Rx79x15x48x66x74x26x70x71;
(2)x73Px15x46xex42x7a.
x5,(1)x69,(2) dimRP = 4,x7ax22:
1 0
0 1
,
i 0
0?i
,
0 1
1 0
,
0 i
i 0
.
7,x18Rx22x32xex9x6bx38x67x45x79x15x74x26x70x71,R+x22x3d3x61(4)x16x15x1fx20x70x71,x53x54,Rx42R+x43x75.
x3x4,x49
’, R?! R+
r 7?! 2r
x4a(a) ’x13x46x5d;
(b) ’x13x14x15,x21x222r1 = 2r2 () r1 = r2;
(c) ’x13x2dx15,x21x22x2x0x1x15a 2R+,a = 2log2 a,x25log2 a 2R,x3cx13’(log2 a) = 2log2 a = a;
(d) ’x46x47x33x67:
’(r1 +r2) = 2r1+r2 = 2r12r2 = 2r1 '2r2 = ’(r1)'’(r2);
’(kr1) = 2kr1 = (2r1)k = k–2r1 = k–’(r1):
x23x24’x13x43x75.
8,x18Fx22x33x10x36x8
(x1;x2;x3;￠￠￠ ;xn;￠￠￠); xn = xn?1 +xn?2; n > 3
x15x32xex1ex23x42x2ax15xdx54,x3cx17x44x42x55x20xfx44x15x1dx4dx8x3d5x61.
(1)x53x54,Fx75x2aRx79x15x48x66x62x46x74x26x70x71;
(2)x1bx25Fx15x48x66x4ex56x65xex1ex23x42x2ax15x7a;
(3)x73x48x49x20x45(Fibonacci)xex1e
(0; 1; 1; 2; 3; 5; 8; ￠￠￠)
x15x72x3ax66x29.
x3x4,(1) Fx22Rx79x74x26x70x71x15x53x54x69,x1dxfx73Fx15x46xe.
xbx3xex1efi1 = (0;1;1;2;3;5;￠￠￠)x42fi2 = (1;1;2;3;5;￠￠￠),x10x11fi1;fi2 2 F.
(a)x18k1fi1 +k2fi2 = 0,x4a(k2;k1 +k2;k1 +2k2;2k1 +3k2;￠￠￠) = 0,x23x24k2 = 0,x43x25k1 = 0,x77x58
x54fi1;fi2x74x26x2cx2a.
(b)x2x0x1x15
fl = (a1;a2;a3;￠￠￠ ;an;￠￠￠); an = an?1 +an?2; n > 3
xbx3
= (a2?a1)fi1 +a1fi2?fl 2 F;
x4a = (0;0;x3;x4;￠￠￠),x21x22 2 F,x23x24x3 = 0+0 = 0,x4 = x3 +0 = 0,x4ex2x50x44x3exb = 0,x77x6fx53
x54x72fl = (a2?a1)fi1 +a1fi2,x21x4ffi1;fi2x75x2aFx15x7a,dimF = 2.
(2)x18x47x56x65xex1e
(a;aq;aq2;￠￠￠) 2 F;
x4ax2n > 2x47aqn = aqn?1 +aqn?2,x43x25q2 = q +1,x50x8q = 1§
p5
2,
￠ 2 ￠
xexb
·1 =
0
@1; 1+
p5
2 ;
1+p5
2
!2;￠￠￠
1
A2 F;
·2 =
0
@1; 1?
p5
2 ;
1?p5
2
!2;￠￠￠
1
A2 F:
x51·1;·2x74x26x2cx2a,x25dimF = 2,x23x24·1;·2x75x2aFx15x7a.
(3)x48x49x20x45xex1e
’ = (0; 1; 1; 2; 3; 5; 8; ￠￠￠) 2 F;
x21x4fx31x6bc1;c2 2R,x27
’ = c1·1 +c2·2:
x43x25 (
0 = c1 +c2
1 = c1 1+
p5
2 +c2
1?p5
2x2dx50
c1 =
p5
5 ; c2 =?
p5
5 ;
x4ex4fx3ex50x48x49x20x45xex1ex50x72x3ax66x29x13
Dn =
p5
5
2
4
1+p5
2
!n?1
1?p5
2
!n?13
5:
9,x23x4anx79x4bx4c,x13x2dx3cx28xdx28x1ex24x68x4dx2x35x3ax48x2x35xbx58x39x3ax6dx65x56x15nx79x40x5e,x8
0
@
6 1 8
7 5 3
2 9 4
1
A
x6fx13x48x66x34x79x4ex5e.
(1)x53x54:x32xex9x79x33x10nx79x4ex5ex15xdx54Mnx5bx5dx5ex15x17x44x42x55x20xfx44x75x2aRx79x15x48x66x74x26x70x71;
(2)x73M3x15x46xe.
x5,(2) 3x46,x7ax22:
0
@
1?1 0
1 0 1
0 1?1
1
A;
0
@
0 1?1
1 0 1
1?1 0
1
A;
0
@
1 1 1
1 1 1
1 1 1
1
A:
x1 x2 5–2
1,x18W1,W2x13x74x26x70x71V x15x0x70x71,x53x54x24x1dx34x66x23x7dx13x56x15x15:
(1) W1 W2; (2) W1 \W2 = W1;
(3) W1 +W2 = W2.
x3x4,(1),(2)x24x68(1))(3)x6dx13x10x11x15.
(3))(1),W1 +W2 = W2 ) W1 W1 +W2 = W2.
2,x73x4ex1fx20fiix7x2ax15x0x70x71x3ax4ex1fx20flix7x2ax15x0x70x71x15x3x42x3ax15x7ax42x46xe.
(1)
( fi
1 = (1;3;1;?1)
fi2 = (1;0;1;2);
( fl
1 = (3;?1;?3;?5)
fl2 = (5;?2;?3;?4);
￠ 3 ￠
(2)
( fi
1 = (1;0;1;0)
fi2 = (1;1;0;1);
( fl
1 = (0;1;0:1)
fl2 = (0;1;1;0);
(3)
8>
<
>:
fi1 = (1;0;2;0;)
fi2 = (2;0;1;1)
fi3 = (1;0;?1;1);
( fl
1 = (3;3;1;?2)
fl2 = (1;3;0;?3):
x5,x4ex4ex1fx20fiix7x2ax15x0x70x71x3ax4ex1fx20flix7x2ax15x0x70x71x11x12x1fx22W1;W2.
(1) dim(W1 +W2) = 3,dimW1 \W2 = 1,
W1 +W2x15x7a,fi1;fi2;fl1,
W1 \W2x15x7a,(3;?2;3;8)
= 13 (?2fi1 +11fi2) =?4fl1 +3fl2
·;
(2) dim(W1 +W2) = 4,dimW1 \W2 = 0,
W1 +W2x15x7a,fi1;fi2;fl1;fl2;
(3) dim(W1 +W2) = 3,dimW1 \W2 = 1,
W1 +W2x15x7a,fi1;fi2;fl1,
W1 \W2x15x7a,(2;0;1;1)(= fi2 = fl1?fl2).
3,x18W,W1,W2x6dx13x1fx20x70x71V x15x0x70x71,x3f
W1 W2; W \W1 = W \W2; W +W1 = W +W2:
x53x54,W1 = W2.
x3x4,dimW +dimW1 = dim(W +W1)+dim(W \W1),
dimW +dimW2 = dim(W +W2)+dim(W \W2),
x23x24x79x29x1ex61x65x56.x3ex50dimW1 = dimW2,x51x21W1 W2,x23x24W1 = W2.
4,x18V1,V2x13nx46x74x26x70x71V x15x37x66x0x70x71,x57x3fx2dx2e
dim(V1 +V2) = dim(V1 \V2)+1;
x53x54,V1 V2x44V2 V1.
x3x4,x21x22dim(V1 \ V2) 6 dimV1 6 dim(V1 + V2) = dim(V1 \ V2) + 1,x37x66x56x59x16x40x47x48x66x2a
x2b,x8x1fx1dx38x56x59x2ax2b,x4ax21V1 \V2 V1,x3ex50V1 \V2 = V1,x43x25V1 V2,x8x1fx1ex38x56x59x2ax2b,x4ax21
V1 V1 +V2,x3ex50V1 = V1 +V2,x43x25V2 V1.
5,x18V = K4,fi1 = (1;2;1;2),fi2 = (2;1;2;1),W = L(fi1;fi2),x73x0x70x71W x6bV x16x15x48x66x4fx70
x71.
x5,x18fi3 = (0;0;1;0),fi4 = (0;0;0;1),x4ax21fi1;fi2;fi3;fi4x74x26x2cx2a,x23x24L((0;0;1;0);(0;0;0;1))
x13Wx6bV x16x15x48x66x4fx70x71.
6,x53x54:x73x48x66nx46x74x26x70x71x6dx13nx66x48x46x0x70x71x15x2ex3a.
x3x4,x18V x22nx46x74x26x70x71,fi1;￠￠￠ ;finx13V x15x7a,x49Wi = L(fii),x4aV = W1 + W2 +￠￠￠+ Wn.
x51,n = dimV = Pni=1 dimWi,x23x24
V = W1 'W2 '￠￠￠'Wn:
7,x53x54,nx46x74x26x70x71V x15x73x48x66x5x0x70x71x6dx13x1ex50x66n?1x46x0x70x71x15x3.
x3x4,x18Wx13V x15x5x0x70x71,x4ar = dimW < dimV = n,x7aWx15x48x66x7afi1;￠￠￠ ;fir,x76x3cx1dx30x2a
V x15x7afi1;￠￠￠ ;fin,x7ax8x1dx15n?rx66n?1x46x74x26x0x70x71
Vj = L(fi1;￠￠￠ ;fij?1;fij+1;￠￠￠ ;fin); j = r +1;￠￠￠ ;n:
￠ 4 ￠
x4ax21
fl =
nX
i=1
aifii 2 Vj () aj = 0;
fl =
nX
i=1
aifii 2
nj=r+1
Vj () ar+1 = ￠￠￠ = an = 0 () fl 2 W:
x4W =
nT
j=r+1
Vj.
8,x18V1x42V2x11x12x13x3x48x74x26x40x41x42
x1 +x2 +￠￠￠+xn = 0 x42 x1 = x2 = ￠￠￠ = xn
x15x2dx70x71.
x53x54,Kn = V1 'V2.
x3x4,(a)x2x0x1x15fi = (a1;￠￠￠ ;an) 2 Kn,x49
fl =
a1? 1n
nX
i=1
ai;a2? 1n
nX
i=1
ai;￠￠￠ ;an? 1n
nX
i=1
ai
!;
=
1
n
nX
i=1
ai; 1n
nX
i=1
ai;￠￠￠ ; 1n
nX
i=1
ai
!;
x4afl 2 V1,2 V2,x3ffi = fl +, x23x24Kn = V1 +V2.
(b)x8x1ffi = (a1;￠￠￠ ;an) 2 V1 \V2,x4a
nX
i=1
ai = 0; a1 = a2 = ￠￠￠ = an
x2dx50a1 = a2 = ￠￠￠ = an = 0,x4fi = 0,x23x24V1 \V2 = 0.
x51x79x3ex50Kn = V1 'V2.
9,x18W1 = fA 2 Mn(K) j AT = Ag,W2 = fA 2 Mn(K) j AT =?Ag.
x53x54,Mn(K) = W1 'W2.
x3x4,(a)x2x0x1x15nx79x5dx5eA 2 Mn(K),x47
A = 12 (A+AT)+ 12 (A?AT);
x25 12 (A+AT) 2 W1,12 (A?AT) 2 W2,x23x24Mn(K) = W1 +W2.
(b)x18A 2 W1 \W2,x4a
A = AT = A;
x4e2A = 0x3ex50A = 0,x23x24W1 \W2 = 0.
x6x2x50x8Mn(K) = W1 'W2.
10,x18A 2 Mn(K)x3fA2 = A,x49
V1 = fX 2 Kn j AX = 0g; V2 = fX 2 Kn j AX = Xg:
x53x54,Kn = V1 'V2.
x3x4,(a)x18fi 2 Kn,x4afi = (fi?Afi)+Afi,x25
A(fi?Afi) = Afi?A2fi = Afi?Afi = 0; x23x24 fi?Afi 2 V1;
A(Afi) = A2fi = Afi; x23x24 Afi 2 V2;
x43x25Kn = V1 +V2.
￠ 5 ￠
(b)x18fi 2 V1 \V2,x4ax21fi 2 V1,x47Afi = 0,x4efi 2 V2,x47Afi = fi,x3cx13fi = 0,x4V1 \V2 = 0.
x21x4fKn = V1 'V2.
11,x18Kn = V1 'V2,x3cx16V1,V2x22Knx15x37x66x6excx78x15x0x70x71.
x53x54:x48x1dx31x6bx2cx48x15x52x56x5dx5e(x4A2 = Ax15x5dx5e)A 2 Mn(K),x27
V1 = fX 2 Kn j AX = 0g; V2 = fX 2 Kn j AX = Xg:
x3x4,x7aV1x15x48x66x7afi1;￠￠￠ ;firx24x68V2x15x48x66x7afir+1;￠￠￠ ;fin,x4afi1;￠￠￠ ;finx13Knx15x7a,x1dx4d
Knx79x15x74x26x3dx4ax41 x22:
x41(fii) =
( 0; 1 6 i 6 r
fii; r +1 6 i 6 n
x4ex74x26x3dx4ax41 x6bKnx15x67x11x7ax1dx15x5dx5ex1fx22A,x4ex41 x15x1dx4dx3ex50x412 = x41,x65x2cx6dx47A2 = A.
x2x0x1x15X 2 Kn,x47X =
nP
i=1
aifii,x4a
AX = x41
nX
i=1
aifii
!
=
nX
i=1
aix41(fii) =
nX
i=r+1
aifii:
x21x4f
AX = 0 ()
nX
i=r+1
aifii = 0 () ai = 08r +1 6 i 6 n () X 2 V1;
AX = X ()
nX
i=r+1
aifii =
nX
i=1
aifii () ai = 081 6 i 6 r () X 2 V2:
x23x24Ax13x2dx2ex31x32x15x52x56x5dx5e.
x12x53x2cx48x26,x8x1fB 2 Mn(K),x27x50
BX = 0 8X 2 V1; BX = X 8X 2 V2;
x4ax21Kn = V1 'V2,x3ex50
(A?B)X = 0; 8X 2 Kn:
x23x24A?B = 0,x43x25A = B.
12,x18A 2 Mn(K),Ex22nx79x14x2fx40x5e,x49
V1 = fX 2 Kn j (A?E)X = 0g;V2 = fX 2 Kn j (A+E)X = 0g:
x53x54,Kn = V1 'V2 () A2 = E.
x3x4,()) Kn = V1 'V2 =) n = dimV1 +dimV2 =) n = (n?rank(A?E))+(n?rank(A+
E)) =) n = rank(A?E)+rank(A+E) =) A2 = E (x60x614–8.12).
(()x2x0x1x15fi 2 Kn,
fi = 12 (A+E)fi? 12 (A?E)fi:
x21x22
(A?E)
1
2 (A+E)fi
= 12 (A2?E)fi = 0;
x23x24 12 (A+E)fi 2 V1,x51x21
(A+E)
12 (A?E)fi
=? 12 (A2?E)fi = 0;
x23x24? 12 (A?E)fi 2 V2.
x21x4fKn = V1 +V2.
￠ 6 ￠
x62fi 2 V1 \V2x52x51x47
fi = 12 (A+E)fi? 12 (A?E)fi = 0+0 = 0;
x21x4fV1 \V2 = 0,x43x25Kn = V1 'V2.
x1 x2 5–3
1,x6bx74x26x70x71R2x16,x2x0x1x37x66x1fx20fi = (a1;a2),fl = (b1;b2),x1dx4d
(fi;fl) = 5a1b1 +2a1b2 +2a2b1 +a2b2:
x7dx53x6bx4fx1dx4dx1dR2x75x2ax48x66x53x27x1ex50x70x71.
x3x4,x69.
2,x6bx74x26x70x71Mn(R)x16,x1dx4d
f(A;B) = Tr(ATB) 8A;B 2 Mn(R):
x19x1,fx13x29Mn(R)x15x48x66x7bx10?
x5,x13,x18A = (aij),B = (bij),x4a
(a) f(A;B) = Tr(ATB) =
nP
k=1
nP
i=1
akibki =
nP
k=1
nP
i=1
bkiaki = f(B;A).
(b) f(A+B;C) = Tr((A+B)TC) = Tr(ATC+BTC) = Tr(ATC)+Tr(BTC) = f(A;C)+f(B;C).
(c) f(kA;B) = Tr((kA)TB) = Tr(kATB) = kTr(ATB) = kf(A;B).
(d) f(A;A) = Tr(ATA) =
nP
k=1
nP
i=1
a2ki > 0,x3f
f(A;A) = 0 () aki = 0; k;i = 1;￠￠￠ ;n () A = 0:
x23x24fx13Mn(R)x15x48x66x7bx10.
3,x18
A =
0
BB
B@
1 0 ￠￠￠ 0
0 2 ￠￠￠ 0.
..,..,..,..
0 0 ￠￠￠ n
1
CC
CA:
x54x1d
(X;Y) = XTAY 8X;Y 2Rn:
(1)x53x54,Rnx2ax3cx4fx1dx4dx75x2ax48x66x53x27x1ex50x70x71;
(2)x73x1fx20"1 = (1;0;￠￠￠ ;0),"2 = (0;1;0;￠￠￠ ;0);￠￠￠,"n = (0;0;￠￠￠ ;0;1)x15x77x20x5dx5e,
(3)x7fx10x7ex25x77x66x70x71x15x55x56–x57x58x55x59x64x7ax55x56x29.
x5,(1)x69.
(2)x77x20x5dx5ex22A.
(3)x18fi = (a1;￠￠￠ ;an),fl = (b1;￠￠￠ ;bn),x4a
flfl
flfl
fl
nX
k=1
kakbk
flfl
flfl
fl6
vu
ut nX
k=1
ka2k ￡
vu
ut nX
k=1
kb2k:
4,x18Cx13x48x66nx79x32x3excx5dx5e,x6bRnx16,x2x0x1x37x66x1ex1fx20X,Y,x54x1d
(X;Y) = XTCTCY
x53x54,Rnx2ax3cx4fx1dx4dx75x2ax48x66x53x27x1ex50x70x71.
￠ 7 ￠
x3x4,x69.
5,x6bx55x12x53x27x1ex50x70x71x7bx78x67x1bx1dx1fx20x15x7bx10,x57x73x38x15x39x71x15x15x35:
(1) fi = (1;1;1;1),fl = (?1;2;4;3);
(2) fi =
1
2 ;?1;
1
3 ;
1
6
·
,fl = (3;?1;2;2);
(3) fi = (3;?1;1;?1),fl = (?2;2;?2;2);
(4) fi = (?1;1;?1;2;1),fl = (3;1;?1;0;1).
x5,(1) (fi;fl) = 8,hfi;fli = arccos 2
p30
15,
(2) (fi;fl) = 72,hfi;fli = arccos 710,
(3) (fi;fl) =?12,hfi;fli = 5…6,
(4) (fi;fl) = 0,hfi;fli = …2,
6,x18x2 +y2 +z2 = 1,(x;y;z 2R),x19x73
x2
1?x2 +
y2
1?y2 +
z2
1?z2
x15x6x5cx4.
x5,x4bx29=?3+ 11?x2 + 11?y2 + 11?z2, x25x4ex55x56–x57x58x55x59x64x7ax55x56x29,
vu
ut 1
p1?x2
2
+
1p
1?y2
!2
+
1
p1?z2
2
￠
q
(
p
1?x2)2 +(
p
1?y2)2 +(
p
1?y2)2
>
1p
1?x2 ;
1p
1?y2 ;
1p
1?z2
!0
@
p1?x2)
p1?y2
p1?y2
1
A = 3;
x4 s
1
1?x2 +
1
1?y2 +
1
1?z2 ￠
p2 > 3;
x23x24
1
1?x2 +
1
1?y2 +
1
1?z2 >
9
2,
x2
1?x2 +
y2
1?y2 +
z2
1?z2 >?3+
9
2 =
3
2,
x51x62x = y = z =
p3
3 x52x79x29x7ax56x59.x21x4bx29x15x6x5cx4x22
3
2,
7,x18a;b;c;x;y;z 2R,x1ea2 +b2 +c2 = 25,x2 +y2 +z2 = 36,ax+by +cz = 30,x73 a+b+cx+y +z x15
x4.
x5,x4ex55x56–x57x58x55x59x64x7ax55x56x29,
30 = ax+by +cz = (a b c)
0
@
x
y
z
1
A
6
p
a2 +b2 +c2 ￠
p
x2 +y2 +z2 = 30:
x21x56x59x2ax2bx52,(a;b;c)x42(x;y;z)x2ax65x6a,x18(a;b;c) = t(x;y;z),x51x52x50
30 = t(x2 +y2 +z2) = 36t;
x2dx25t = 56, x43x25 a+b+cx+y +z = 56,
￠ 8 ￠
8,x6bx55x12x53x27x1ex50x70x71R3x16,x73x7afi1 = (1;0;1),fi2 = (1;1;0),fi3 = (0;1;1)x15x77x20x5dx5e.
x5,A =
0
@
2 1 1
1 2 1
1 1 2
1
A.
9,x6b4x46x53x27x1ex50x70x71V x16,x18x7afi1 = (1;1;?1;?1),fi2 = (1;1;1;0),fi3 = (?1;1;1;1),fi4 =
(1;0;0;?1)x15x77x20x5dx5ex22
A =
0
BB
B@
2 1 0 0
1 2 1 0
0 1 2 1
0 0 1 2
1
CC
CA
(1)x73x7a"1 = (1;0;0;0),"2 = (0;1;0;0),"3 = (0;0;1;0),"4 = (0;0;0;1)x15x77x20x5dx5e;
(2)x73x1fx20fl1 = (1;?1;1;?1),fl2 = (0;1;1;0)x15x7bx10;
(3)x73x48x14x2fx1fx20x42fi1,fi2,fi3x72x3.
x5,(1) (fi1;fi2;fi3;fi4) = ("1;"2;"3;"4)B,
B =
0
BB
B@
1 1?1 1
1 1 1 0
1 1 1 0
1 0 1?1
1
CC
CA; B?1 = 12
0
BB
B@
0 1?1 0
2 0 0 2
2 1 1?2
2 0 2?4
1
CC
CA:
x21x4fx7a"1;"2;"3;"4x50x77x20x5dx5ex2cx22
G = B?TAB?1 =
0
BB
BB
@
6? 12? 92 9
12 1 12?1
92 12 4?7
9?1?7 14
1
CC
CC
A
:
(2) (fl1;fl2) = 6.
(3)x18fi =
4P
i=1
xifiix42fi1;fi2;fi3x72x3,x4a
fij;
4X
i=1
xifii
!
= 0; j = 1;2;3:
x43x25
0
@
2 1 0 0
1 2 1 0
0 1 2 1
1
A
0
BB
B@
x1
x2
x3
x4
1
CC
CA = 0:
x2dx50(x1;x2;x3;x4) = (1;?2;3;?4),x23x24
fi = fi1?2fi2 +3fi3?4fi4 = (?8;2;0;6):
(fi;fi) = (fi4;fi) = 20,x21x4fx23x73x15x14x2fx1fx20x22§
p5
5 (?4;1;0;3).
10,x18fi1;fi2;￠￠￠ ;fimx13x53x27x1ex50x70x71V x15mx66x1fx20,x75x5dx5e
G(fi1;fi2;￠￠￠ ;fim) =
0
BB
B@
(fi1;fi1) (fi1;fi2) ￠￠￠ (fi1;fim)
(fi2;fi1) (fi2;fi2) ￠￠￠ (fi2;fim)
:::::::::::::::::::::::::::::::::::
(fim;fi1) (fim;fi2) ￠￠￠ (fim;fim)
1
CC
CA
x22x1fx20x42fi1;fi2;￠￠￠ ;fimx15x5ax60x5b(Gram)x5dx5e.
x53x54,fi1;fi2;￠￠￠ ;fimx74x26x2cx2ax62x3fx63x62jG(fi1;fi2;￠￠￠ ;fim)j6= 0.
￠ 9 ￠
x3x4,x18x47x74x26x2ax6ax29
x1fi1 +x2fi2 +￠￠￠+xmfim = 0:
x4ex77x66x56x29x11x12x42fi1;￠￠￠ ;fimx2fx7bx10,x3ex24x50x8x3dx20x1;￠￠￠ ;xmx15x48x66x3x48x74x26x40x41x42:
8>
>>><
>>>
>:
(fi1;fi1)x1 +(fi1;fi2)x2 +￠￠￠+(fi1;fim)xm = 0
(fi2;fi1)x1 +(fi2;fi2)x2 +￠￠￠+(fi2;fim)xm = 0
::::::::::::::::::::::::::::::::::::::::
(fim;fi1)x1 +(fim;fi2)x2 +￠￠￠+(fim;fim)xm = 0
x3cx6axex5dx5ex6fx13x5ax60x5bx5dx5eG(fi1;￠￠￠ ;fim),x12x33x1ax3x48x74x26x40x41x42x47x6ex6fx2dx15x30x11x40x26x31x32x3ex50:
jG(fi1;fi2;￠￠￠ ;fim)j6= 0 () x3x48x74x26x40x41x42x7bx47x6fx2dx1 = ￠￠￠ = xm = 0 () fi1;￠￠￠ ;fimx74x26x2cx2a:
11,x18"1;"2;"3x13x34x46x53x27x1ex50x70x71V x15x48x66x54x3dx72x3x7a.
x53x54,fi1 = 13 (2"1 + 2"2?"3),fi2 = 13 (2"1?"2 + 2"3),fi3 = 13 ("1?2"2?2"3)x67x13V x15x48x66x54
x3dx72x3x7a.
x3x4,x2ex11x7dx53x3exb,fi1;fi2;fi3x6dx13x14x2fx1fx20,x3fx37x37x72x3,x21x38x15x13V x15x14x2fx72x3x1fx20x42.x51x21
dimV = 3,x38x15x75x2aV x15x54x3dx72x3x7a.
12,x76x55x12x53x27x1ex50x70x71R4x15x7afi1=(1;1;0;0),fi2=(1;0;1;0),fi3 = (?1;0;0;1),fi4 = (1;1;1;?1)
x4cx22x54x3dx72x3x7a.
x5:
p2
2 (1;1;0;0),
p6
6 (1;?1;2;0),
p3
6 (?1;1;1;3),
1
2 (?1;1;1;?1).
13,x73x3x48x74x26x40x41x42 ‰
x1? x2 +x3 +3x4? x5 =0
x1 +2x2?x3 +2x5 =0
x15x2dx70x71(x2fx22x55x12x53x27x1ex50x70x71R5x15x0x70x71)x15x48x66x54x3dx72x3x7a.
x5,x1cx3x48x74x26x40x41x42x15x48x66x7ax8x2dx6ax22
fi1 =
0
BB
BB
B@
1
2
3
0
0
1
CC
CC
CA; fi2 =
0
BB
BB
B@
2
1
0
1
0
1
CC
CC
CA; fi3 =
0
BB
BB
B@
0
1
0
0
1
1
CC
CC
CA:
x72x3x4cx50:
0
BB
BB
B@
1
2
3
0
0
1
CC
CC
CA;
0
BB
BB
BB
@
127
3
7
67
1
0
1
CC
CC
CC
A;
0
BB
BB
BB
B@
517
2334
6
173
341
1
CC
CC
CC
CA
:
x14x2fx4cx7x50x54x3dx72x3x7a:
p14
14 (?1;2;3;0;0);
p238
238 (?12;3;?6;7;0);
p1938
1938 (?10;?23;12;3;34):
14,x53x54:x6bx53x27x1ex50x70x71V x16,x7a"1;"2;￠￠￠ ;"nx13x54x3dx72x3x7ax15x30x11x40x26x31x32x13,x2V x15x0x1x1f
x20fi = a1"1 +a2"2 +￠￠￠+an"n,x38x47
(fi;"i) = ai (i = 1;2;￠￠￠ ;n):
￠ 10 ￠
x3x4,())x8"1;￠￠￠ ;"nx13x54x3dx72x3x7a,x4ax2x0x1x15fi = Pai"i,x47
(fi;"i) =
0
@
nX
j=1
aj"j;"i
1
A =
nX
j=1
aj("j;"i) = ai:
(()x8x2x0x1x15fi = Pai"i,x47
(fi;"i) = ai;
x4a"j =
nP
k=1
ak"k,x3cx16ak = –kj,x21x4f
("j;"i) = ai = –ij:
x43x25"1;￠￠￠ ;"nx13x54x3dx72x3x7a.
15,x72x4ex2x9x16xcxfx7bx72x40x36x24x68x27x8x70x71x7bx2bx40x10x15x5cx3,x2x50x25x38x15x15x7ex17x57x55x15x69x5d,x43
x25x5excx25nx46x70x71x15x2bx40x10x15x7ex17x66xex66x29,x12x78x674x46x70x71x16x15x2bx40x10x47x9x46x663x46x15x44xf,x9x46x66
2x46x15x44xfx421x46x15x14? x77x664x46x2bx40x10x47x9x46x62x55x43x3bx77x15x2x35x74? x19x73x38x15x15x3bx77x24x68x42x14x15x15
x35,x5ex63x29x4ex77x48x22x1fx5ex1fx8nx46x70x71x15xfx36?
x74
x74
x74
x70x10
x74
x74
x74
x70x10
x75x75x75x72x20
x75x75x75x72x20
x7x1fx3ax9x21x46
x7ax2fxfx1x11x6
(0;0) (1;0)
(0;1) (1;1)
x74
x74
x74
x70x10
x74
x74
x74
x70x10
x75x75x75x72x20
x75x75x75x72x20
x7x1fx3ax9x21x46
x7ax2fxfx1x11x6x71x71
x71x20
x4x75x75x75x72x20
x4x20x71
x71x71
x71x71
x71x20
x4x74
x74
x74
x70x10
x3x16x5x2a
x40x18 x40x18
x40x18 x40x18
x40x18
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20
(0;0;0) (1;0;0)
(0;0;1)
(1;1;0)(0;1;0)
(0;1;1) (1;1;1)
(1;0;1)
x015x1x2
x5,nx46x70x71x15x2bx40x10x16,mx46x0x2bx40x10x472n?mCmn x66.
x62m = 0x52x22x7ex17x66xe= 2n;
x62m = 1x52x22x14xe= 2n?1n;
x62m = 2x52x22xfxe= 2n?3n(n?1);,::
x3cx55x43x3bx77x15x2x35x74x47n?1x62,x3bx77x11x12x22p2;p3;￠￠￠ ;pn.
x3bx77x22pkx15x2x35x74x42x14x15x15x35x22 …2 x44arccos 1pk,
x1 x2 5–4
1,x19x73x72x4ex17A(1;1;1)x42B(1;0;2)x3fx22x2ex3cxcxfx+2y?z?6 = 0x15xcxfx40x41.
x5,x18x23x73xcxfx15x44x1fx20x22” = (A;B;C),x4a”!AB,”x67x42xcxfx + 2y?z?6 = 0x15x44x1fx20x22
x2e,x21x4fx47x40x41x42 (
0￠A+(?1)B +C = 0
A+2B?C = 0:
x2dx50A, B, C = 1,?1,?1,x3ex50x17x44x29x40x41(x?1)?(y?1)?(z?1) = 0,x4x?y?z +1 = 0.
2,xcxfΠx4e3x66x17M1(3;?1;5),M2(4;?1;1)x3aM3(2;0;2),x73xcxfΠx15x48x66x44x1fx20,x57x73x25Πx15
x40x41.
x5,xcxfΠx15x48x66x44x1fx20x3ex7ax22” =!M1M2 ￡!M1M3 = (4;7;1),x3ex50x17x44x29x40x414(x?2) + 7y +
(z?2) = 0,x44x+7y +z?10 = 0.
￠ 11 ￠
3,xcxfΠx4ex17M0(2;3;1),x3fx3ax37xcxfΠ1, x+3y?z +3 = 0,Π2, 2x+y?2z +1 = 0x6dx22x2e,
x73Πx15x40x41.
x5,x33x1ax6a4.5xbxcxfx15x40x41x22fl
flfl
flfl
fl
x?2 y?3 z?1
1 3?1
2 1?2
flfl
flfl
flfl =?5x?5z +15 = 0;
x4cx6bx50x+z?3 = 0.
4,xcxfΠx6bx;y;zx45x79x15xfx32x11x12x13?1; 32 ;3,x73x67x4bx17x2dx1fxcxfx15x14x2fx44x1fx20x15x40x1fx24x25.
x5,x33x1axcxfx15xfx32x29x40x41x50x8x1cxcxfx15x40x41:
x
1 +
y
3
2
+ z3 = 1:
x4cx6bx7x50?3x+2y +z?3 = 0,x38x15x44x1fx20x3ex7ax22§(?3;2;1),x21x22x17P0(?1;0;0)x6bx4fxcxfx79,x25!OP0
x42x29x61x23x26x73x15x44x1fx20x39x71x15x15x35x2cx1cx5cx3c …2,x4x7bx10x3bx3c0,x21x2cx7ax44x1fx20x22(?3;2;1),x38x15x40x1fx24
x25x22
3
p14
14 ;
p14
7 ;
p14
14
.
5,x73x4ezx45x3fx42xcxf2x+y?p5z?7 = 0x2a60–x35x15xcxfΠx15x40x41.
x5,x4ezx45x15xcxfx15x44x1fx20x2cx22” = (A;B;0),x38x2cx42xaxbxcxfx2a60–x35,x23x24 2A+BpA2 +B2p10 =
§ 12, x5ex503A = Bx44A =?3B,x21xcxfx40x41x22x+3y = 0x443x?y = 0.
6,xaxbxcxfΠ, 4x?4y?2z +3 = 0,x17Px42xcxfΠx15x32x33x222,x73x17Px15x42x43.
x5,x18x2dx2ex31x32x15x17x22P(x;y;z),x4ax47
j4x?4y?2z +3j
6 = 2;
x5ex504x?4y?2z +3 = §12,x4x17Px15x42x43x13x37x66xcxdxcxf:
4x?4y?2z?9 = 0x3a 4x?4y?2z +15 = 0:
7,xaxbx37x66xcxfx4ex1dx29x64x1d,x73x38x15x15x3x35,x57x64x1dx17(0;0;1)x23x6bx15x37xfx35x15x3bx5c:
(x+2y +4z?3)(?3x+y?z?1) = 0:
x5,x37xcxfx15x44x1fx20x11x12x22(1;2;4)x42(?3;1;?1),x4ax3x35 x2dx2ecos = §?5p21p11 = § 5
p231
231,
x23x24 = arccos 5
p231
231 x44…?arccos
5p231
231,
x22x64x1dx17(0;0;1)x23x6bx15x37xfx35,x78x67x4fx17x2ax3cx37x66xcxfx15x33x5bx11x12x22 1p21 x42?2p11,x4ex3cx38x15x5f
x59,x21x4fx23x73x37xfx35x15x3bx5cx22…?arccos 5
p231
231,
8,x6bx2ex35x57x55x6ax1d,x73x1dx1ex17x8xcxfx15x32x33.
(1)x17(2;1;4),xcxf2x?y +4z?12 = 0;
(2)x17(?1;0;5),xcxfx?3y +5z?2 = 0.
x5,(1) d = j4?1+16?12jp21 =
p21
3,
(2) d = j?1+25?2jp35 = 22
p35
35,
9,x6bx2ex35x57x55x6ax16,x18xcxfΠix15x40x41x22
Aix+Biy +Ciz +Di = 0; i = 1;2:
￠ 12 ￠
x3fx77x37xcxfx65x3,x73x38x15x3x2ax15x37xfx35x15x35xcx11xfx15x40x41.
x5,x17P(x;y;z)x6bΠ1x42Π2x15x6ex66x37xfx35x15x35xcx11xfx79x62x3fx63x62x1cx17x8x77x37x66xcxfx15x32x33x65x56.
x21x4fx17Px2cx2dx2ex40x41
jA1x+B1y +C1z +D1jp
A21 +B21 +C21 =
jA2x+B2y +C2z +D2jp
A22 +B22 +C22,
x23x24x35xcx11xfx15x40x41x22
A1x+B1y +C1z +D1p
A21 +B21 +C21 = §
A2x+B2y +C2z +D2p
A22 +B22 +C22,
10,x73x8x37x66x1bx1dx15xcxfx15x32x33x22x1dx65kx15x17x15x42x43x40x41.
x5,x18x37x66x1bx1dxcxfx15x40x41x22
Πi, Aix+Biy +Ciz +Di = 0; i = 1;2:
x18P(x;y;z)x17x8Π1x42Π2x15x32x33x39x65x22k,x4ax47
jA1x+B1y +C1z +D1jp
A21 +B21 +C21 = k
jA2x+B2y +C2z +D2jp
A22 +B22 +C22,
x21x4fPx17x15x42x43x40x41x22
A1x+B1y +C1z +D1p
A21 +B21 +C21 = §k
A2x+B2y +C2z +D2p
A22 +B22 +C22,
11,x6bx2ex35x57x55x6ax1d,xaxbxcxfΠx15x40x41x22Ax+By +Cz +D = 0,x73Πx2ax3cxOyxcxfx15x2x75xf
Π0x15x40x41x3ax2ax3cx57x55x4bx17x15x2x75xfΠ00x15x40x41.
x5,x18x17P0(x0;y0;z0) 2 Π0,x4aP0x2ax3cxOyxcxfx15x2x75x17x13(x0;y0;?z0),x1cx17x2cx6bxcxfΠx79,x21x47
Ax0 +By0?Cz0 +D = 0,x23x24Π0x15x40x41x22Ax+By?Cz +D = 0.
x43x5,x18x17P00(x00;y00;z00) 2 Π00,x4aP00x2ax3cx4bx17x15x2x75x17x13(?x00;?y00;?z00),x1cx17x2cx6bxcxfΠx79,
x21x47?Ax00?By00?Cz00 +D = 0,x23x24Π00x15x40x41x22Ax+By +Cz?D = 0.
x1 x2 5–5
1,x7cx7dx1dx1ex2ex74x42xcxfx15x2fx59x2ax6a,x8x1fx65x3,x4ax73x38x15x15x3x17x42x15x35.
(1)x2ex74 x?12 = y +3?1 = z +25 x42xcxf4x+3y?z +3 = 0;
(2)x2ex74
8<
:
x = 1?2t
y = 2?4t
z =?1+5t
x42xcxfx+2y +2z?7 = 0;
(3)x2ex74
‰ x?y +z = 5
x+y?z =?1 x42xcxf2x+y +z?5 = 0.
x5,(1)x2ex74x15x40x1fx1fx20x13? = (2;?1;5),xcxfx15x44x1fx20x13” = (4;3;?1),x21x22(?;”) = 0,x2ex74x79x15
x17(1;?3;?2)x51x2dx2excxfx40x41,x23x24x2ex74x6bxcxfx7b.
(2)x2ex74x15x40x1fx1fx20x13? = (?2;?4;5),xcxfx15x44x1fx20x13” = (1;2;2),x21x22(?;”) = 0,x2ex74x79x15x17
(1;2;?1)x55x2dx2excxfx40x41,x23x24x2ex74x42xcxfxcxd.
(3)x2ex74x15x55x12x40x41x13 x?20 = y +32 = z2,x42xcxfx40x41x60x2bx7x73x50x3x17(2;?1;2),x2ex74x15x40x1f
x1fx20x13? = (0;2;2),xcxfx15x44x1fx20x13” = (2;1;1),x18x2ex74x42xcxfx15x3x35x22,x4asin = 44p3 =
p3
3,x23
x24x15x35 = arcsin
p3
3,
2,x73x4ex17A(3;?1;1)x3fx42xcxfΠ, x+y +z = 1x22x2ex15x2ex74x40x41.
￠ 13 ￠
x5,x21x23x73x2ex74x42xcxfx22x2e,x2ex74x15x40x1fx1fx20x13? = (1;1;1),x21x2ex74x40x41x22 x?31 = y +11 =
z?1
1,
3,x7cx7dx1dx1ex2ex74x71x15x2ax6a,x57x73x38x15x15x15x35,x2x3cx65x3x15x2ex74x57x73x3x17:
(1) L1, x+14 = y?32 = z?13,L2, x?51 = y +13 = z +12 ;
(2) L1, x+14 = y?12 = z?23,L2, x+26 = y +310 = z7,
x5,(1)x3dx3ex3dx34x2cx37x619.2,xdx1ex29
=
flfl
flfl
flfl
6 4 1
4 2 3
2 3 2
flfl
flfl
flfl =?30 6= 0;
x23x24x37x2ex74x5fxf,x18x15x35x22,x4acos = 16p29p14 = 8
p406
203,x21 = arccos
8p406
203,
(2)xdx1ex29
=
flfl
flfl
flfl
1 4 6
4 2 10
2 3 7
flfl
flfl
flfl = 0;
x23x24x37x2ex74x28xf,x51x21x38x15x15x40x1fx1fx20x55xcxd,x21x37x2ex74x65x3,x18x15x35x22,x4acos = 13
p5365
1073,x21
= arccos 13
p5365
1073, x2dx60x2bx40x41,x73x50x3x17x22
1;2; 72
·
.
4,xaxbx2ex74Lx15x40x41x13
‰ x+y +z?4 = 0
2x?y?z +1 = 0 x73x17A(3;2;?1)x8Lx15x32x33.
x5,x2ex74x15x40x1fx1fx20x13? =
flfl
flfl 1 1
1?1
flfl
flfl;?
flfl
flfl1 1
2?1
flfl
flfl;
flfl
flfl1 1
2?1
flfl
flfl
= (0;3;?3),x3fx17(1;2;1)x6bx2ex74
x79,x21x17Ax8x2ex74x15x32x33x22
d = j(2;0;?2)￡(0;1;?1)jp2 = p6:
5,x19x53x2ex74
L1,
8
<
:
x = 3+3t
y = t
z = 1
x42 L2,
8
<
:
x =?1+u
y = 2
z = u
(x3cx16t;ux13x5exe)
x13x5fxfx2ex74,x57x73x38x15x15x66x22x74x3ax37x2ex74x71x15x32x33.
x5,x2ex74L1x72x4ex17M1(3;0;1),x40x1fx1fx20x13?1 = (3;1;0),x2ex74L2x72x4ex17M2(?1;2;0),x40x1fx1fx20x13
2 = (1;0;1),x21x4f
1 ￡?2 = (1;?3;?1):
x3cx13L1x42L2x15x32x33x13
d = j(
!M
1M2;?1 ￡?2)j
j?1 ￡?2j =
9p
11 =
9p11
11,
x4eM1;?1;?1 ￡?2x64x1dx15xcxfΠ1x15x40x41x13
flfl
flfl
flfl
x?3 3 1
y 1?3
z?1 0?1
flfl
flfl
flfl =?x+3y?10z +13 = 0:
x4eM2;?2;?1 ￡?2x64x1dx15xcxfΠ2x15x40x41x13
flfl
flfl
flfl
x+1 1 1
y?2 0?3
z 1?1
flfl
flfl
flfl = 3x+2y?3z?1 = 0:
￠ 14 ￠
x6bx4cx7x3ex50x66x22x74x40x41x22 (
x?3y +10z?13 = 0
3x+2y?3z?1 = 0:
6,xaxbx2ex74L,
‰ x?y?4z +12 = 0
2x+y?2z +3 = 0 x68x1dx17P0(2;0;?1),x73P0x2ax3cLx15x2x75x17.
x5,x14x15x72x4exbxcx17x2ax3cxcxfΠ1, x?y?4z + 12 = 0x68Π2, 2x + y?2z + 3 = 0x15x33x5bx19x64
x1dP0x2ax3cLx15x2x75x17P00,P0(2;0;?1)x2ax3cΠ1x15x33x5b–1 = 18p18 = p18,x2ax3cΠ2x15x33x5b–2 = 93 = 3.
x23x24x2x75x17 P00(x0;y0;z0) x2ax3c Π1 x15x33x5b –01 = x0?y0?4z0 +12p18 =?–1 =?p18,x2ax3c Π2 x15x33x5b
–02 = 2x0 +y0?2z0 +33 =?–2 =?3,x21x4f
( x?y?4z +12 =?18
2x+y?2z +3 =?9; x4
( x?y?4z =?30
2x+y?2z =?12,(*)
x51x21x22!P0P00x2cx42x2ex74Lx22x2e,x4
flfl
flfl
flfl
x0?2 1 2
y0?1 1
z0 +1?4?2
flfl
flfl
flfl = 6x0?6y0 +3z0?9 = 0; (**)
x60x2bx4f3x66x40x41x2dx50,x0 = 0,y0 = 2,z0 = 7,x4x2x75x17x15x57x55x22(0;2;7).
7,x2ex74x4ex17(2;?3;5)x3fx42x34x31x57x55x45x15x72x1fx3x2ax56x35,x73x17P(1;?2;3)x8x4fx2ex74x15x32x33.
x5,x10x11x2ex74x40x41x22
x+2
1 =
y +3
1 =
z?5
1,
x23x24x17P(1;?2;3)x8x77x31x2ex74x15x32x33x22
d = j(1;?1;2)￡(1;1;1)jp3 =
p42
3,
8,x73x72x4ex37x2ex74 x?1?1 = y8 = z?5?3 x3a
8<
:
x = 3+4t
y = 21+5t
z =?11?10t
x15x3x17,x3fx42x77x37x2ex74x6dx22x2ex15x2ex74x40x41.
x5,x21x1cx2ex74x42xaxbx37x2ex74x6dx22x2e,x25xaxbx2ex74x15x40x1fx11x12x22?1 = (?1;8;?3),?2 = (4;5;?10),x21
x23x73x2ex74x15x40x1fx1fx20x22? =?1 ￡?2 = (?65;?22;?37),x77x37x31x2ex74x15x3x17x3ex73x50x22(?1;16;?1),x21x23
x73x2ex74x15x40x41x22
x+1
65 =
y?16
22 =
z +1
37,
9,x73x6bxcxf2x+3y +4z?9 = 0x79x6ax4ex17(1;1;1)x3fx42xOyxcxfx3x2ax6x3bx35x15x2ex74.
x5,x18x23x73x2ex74x15x40x1fx1fx20x22? = (A;B;C),x21x2ex74x6bxcxfΠ, 2x + 3y + 4z? 9 = 0x79,x23
x242A + 3B + 4C = 0,xOy xcxfx15x40x41x22z = 0,x14x15x6bx27x8x16xax6axbx60,x62x3fx63x62xcxfx79x15x2ex74
x42x37x66xcxfx15x3x74x22x2ex52,x77x31x2ex74x42x16x48xcxfx15x3x35x50x8x1cx3b,x25x77x37x66xcxfx15x3x74x15x40x1fx1fx20x13
(2;3;4)￡(0;0;1) = (3;?2;0),x21x4f?x42(3;?2;0)x22x2e,x43A?2B = 0,x2dx50A = 23 B,C =? 1312 B,x21
x23x73x2ex74x15x40x41x22
x?1
8 =
y?1
12 =
z?1
13,
10,x73x1dx1ex37x2ex74x71x15x32x33,x8x37x2ex74x47x28x22x74,x73x25x38x15x15x66x22x74x15x40x41.
￠ 15 ￠
(1)
‰ 2x+2y?z?10 = 0
x?y?z?22 = 0; x42
x+7
3 =
y?5
1 =
z?9
4 ;
(2)
8<
:
x = 3t?5
y = 2t?5
z =?2t+1;
x42
‰ x?y +z +5 = 0
x+2y?z?14 = 0:
x5,(1)x3dx48x31x2ex74x4cx2ax55x12x40x41x22
x?6
3 =
y +6
1 =
z +10
4 ;
x21x4fx37x2ex74xcxd,x32x33x22
d = j(13;?11;?19)￡(3;?1;4)jp26 =
p16250
p26 = 25:
(2)x2ex74L1x72x4ex17M1(?5;?5;1),x40x1fx1fx20x13?1 = (3;2;?2),x2ex74L2x72x4ex17M2(4;1;?8),x40x1fx1f
x20x13?2 = (?1;2;3),x21x4f
1 ￡?2 = (10;?7;8):
x3cx13L1x42L2x15x32x33x13
d = j(
!M
1M2;?1 ￡?2)j
j?1 ￡?2j =
24p
213 =
24p213
213,
x4eM1;?1;?1 ￡?2x64x1dx15xcxfΠ1x15x40x41x13
flfl
flfl
flfl
x+5 3 10
y +5 2?7
z?1?2 8
flfl
flfl
flfl = 2x?44y?41z?169 = 0:
x4eM2;?2;?1 ￡?2x64x1dx15xcxfΠ2x15x40x41x13
flfl
flfl
flfl
x?4?1 10
y?1 2?7
z +8 3 8
flfl
flfl
flfl = 37x+38y?13z?290 = 0:
x6bx4cx7x3ex50x66x22x74x40x41x22 (
2x?44y?41z?169 = 0
37x+38y?13z?290 = 0:
11,xaxbx48x17P(a;b;c) (abc 6= 0),x4ePx17x1fx28x66x57x55xfx2fx22x74,x22x2ex11x12x22L;M;N,x73x53,OPx42
x28x66xfOMN;ONL;OLMx15x3x35x65x56.
x3x4,x3dx3ex31x18,x22x2ex11x12x22L(a;b;0),M(a;0;c),N(0;b;c),x4axfOMN x15x44x1fx20x3ex7ax22”1 =
!OM ￡!ON = (?bc;?ac;ab),x6bx6cx6d,xfOLMx15x44x1fx20x3ex7ax22”
2 = (bc;?ac;?ab),xfONLx15x44x1fx20x3e
x7ax22”3 = (?bc;ac;?ab),x25?!OP = (a;b;c),x4ex3c
!OP ￠”
1 =
!OP ￠”
2 =
!OP ￠”
3 =?abc;
x3exbOPx42x773x66xfx15x3x35x65x56.
12,xaxbx37x31x5fxfx2ex74L1x3aL2,x73x53x4cx11L1x79x0x48x17x3aL2x79x0x48x17x15x74x78x15x16x17x42x43x13x66x22
x74x78x15x22x2excx11xf.
x3x4,x3bx62x7bx7cx57x55x6ax3ex27L1x22xx45,x40x41x22
( y = 0
z = 0; L2x15x40x41x4ax22
( kx?y = 0
z = a; (ak 6= 0),x18
P1(x1;0;0)x22L1x79x0x1x48x66x17,P2(x2;kx2;a)x13L2x79x0x1x17,x4aP1P2x15x16x17x57x55x22
x
1 +x22 ; kx22 ; a2
·
.
￠ 16 ￠
x4x4fx42x43x2dx2ex5exex40x41 8
>><
>>:
x = 12 u+ 12 v
y = k2 u
z = a2 ;
u;vx22x5exe.
x10x11x13xcxfz = a2, x77x67x13x37x31x5fxfx2ex74x15x66x22x74x78x15x22x2excx11xf.
13,xaxbx2ex74Lx72x4ex17(1;1;0)x3fx42x2ex74
L1, x?14 = y?2 = z1 ; L2, x4 = y +3?3 = z?1?2
x22x2e,x73x2ex74Lx6bx28x66x57x55xfx79x15x5dx17x15x40x41.
x5,x21x22Lx42L1;L2x6dx22x2e,x23x24x3ex7aLx15x40x1fx1fx20x22? = (4;?2;1)￡(4;?3;?2) = (7;12;?4),x4a
Lx15x40x41x22
x?1
7 =
y?1
12 =
z
4,
x38x6bxOy xcxfz = 0x79x15x16x17x40x41x22 x?17 = y?112 = z0,x6byOz xcxfx = 0x79x15x16x17x40x41x22
x
0 =
y?1
12 =
z
4,x6bxOzxcxfy = 0x79x15x16x17x40x41x22
x?1
7 =
y
0 =
z
4,
14,x73x4ex17(2;?3;?1)x3fx42x2ex74
x?1
2 =
y +1
1 =
z
1
x22x2ex65x3x15x2ex74.
x5,x18x23x73x2ex74x15x40x1fx1fx20x22? = (A;B;C),x21x38x40x41x42xaxbx2ex74x22x2e,x21x47?2A?B + C = 0.
x12x7ex25x23x73x2ex74x15x5exex40x41
8
><
>:
x = 2+At
y =?3+Bt
z =?1+Ct;
x51x52xaxbx2ex74x15x40x41,x50
1+At
2 =
2+Bt
1 =
1+Ct
1,
x4ex79x29x3ex50(3A?B+5C)t = 0,x4ex3ct = 0x10x11x55x13x2d,x25x37x2ex74x65x3x58x54tx48x1dx47x2d,x21x4f3A?B+5C =
0,x6x7x2dx50A, B, C = 4,?13,?5,x43x25x2ex74x40x41x22
x?2
4 =
y +3
13 =
z +1
5,
x1 x2 5–6
1,x6bx55x12x53x27x1ex50x70x71R4x16,x73x1fx20flx6bx4ex1fx20fi1,fi2,fi3x7x2ax15x0x70x71Wx79x15x72x3x16x17,x18
(1) fi1 = (2;2;?3;1),fi2 = (?2;1;?2;3),fi3 = (1;2;?3;2),fl = (1;1;?2;1);
(2) fi1 = (?1;2;?1;1),fi2 = (2;?1;1;0),fi3 = (0;1;?1;2),fl = (1;2;?1;0).
x5,(1) x18fl = x1fi1 + x2fi2 + x3fi3 + fl2,x3cx16fl2 = fl? x1fi1? x2fi2? x3fi3 2 W?,x4ex56x29
(fl2;fii) = 0,i = 1;2;3,x3ex24xcx25x24x1dx3x48x74x26x40x41x42:
8>
<
>:
18x1 +7x2 +17x3 = 11
7x1 +18x2 +12x3 = 6
17x1 +12x2 +18x3 = 11
x2dx50(x1;x2;x3) =
1
2 ;
1
12 ;
1
12
·
,x21x4fflx6bWx79x15x72x3x16x17x22:
1
2 fi1 +
1
12 fi2 +
1
12 fi3 =
11
12 ;
5
4 ;?
23
12 ;
11
12
:
￠ 17 ￠
(2)
1
2 ;2;0;
1
2
·
.
2,x18A 2 Mn(R),B 2Rn,x53x54:x32x6axex74x26x40x41x42AX = Bx47x2dx15x30x11x40x26x31x32x13Bx42x40x41x42
ATX = 0x15x2dx70x71x72x3.
x3x4,()) x1eAX = Bx47x2d,x4ax47C = (c1 c2 ￠￠￠ cn)Tx27x50B = AC,x3cx13x2ATX = 0x15x0x1x2d
D = (d1 d2 ￠￠￠ dn)T,x47
DTB = DTAC = (ATD)TC = 0;
x23x24Bx42ATX = 0x15x2dx70x71x72x3.
(()x18ATX = 0x15x2dx70x71x22W1,Ax15x1ex1fx20x42x2ex2ax15x0x70x71x22W2,x4aW1? W2,x51x21dimW1 =
n?rankA = n?dimW2,x23x24V = W1 'W2,x43x25W2 = W?1, xaxbB? W1,x3ex50B 2 W2,x4Bx3ex4e
Ax15x1ex1fx20x42x74x26x1bx1c,x3cx13x31x6bC 2Rnx27x50B = AC.
3,x18V1,V2x13x53x27x1ex50x70x71V x15x37x66x0x70x71,x3fV1x15x46xex5cx3cV2x15x46xe,x53x54,V2x16x40x47x48x6e
x6fx1fx20x72x3x3cV1x16x23x47x1fx20.
x3x4,x4ex37x616.2,V = V1 'V?1,dimV?1 = n?dimV1.
dim(V2 \V?1 ) = dimV2 +dimV?1?dim(V2 +V?1 )
> n?dimV1 +dimV2?n
= dimV2?dimV1 > 1:
x23x24V2 \V?1 6= 0,x31x6bx6ex6fx1fx20fi 2 V2 \V?1,x4fi 2 V2,fi? V1.
4,x18Ux22nx46x53x27x1ex50x70x71V x15x0x70x71,x53x54,?U?￠? = U.
x3x4,x21x22Ux15x1fx20x6dx42U?x72x3,x21x4fU (U?)?,x51x21
dim(U?)? = n?dimU? = n?(n?dimU) = dimU;
x21x4fU = (U?)?.
5,x18V1,V2x22nx46x53x27x1ex50x70x71V x15x37x66x0x70x71,x53x54:
(1) (V1 +V2)? = V?1 \V?2 ;
(2) (V1 \V2)? = V?1 +V?2,
x3x4,(1)x1efi 2 (V1 +V2)?,x4afi? V1x3ffi? V2,x43x25fi 2 V?1 \V?2, x23x24(V1 +V2)? V?1 \V?2,
x8x1ffi 2 V?1 \V?2,x4afi? V1x3ffi? V2,fi? V1 + V2,x23x24fi 2 (V1 + V2)?,x77x58x54V?1 \V?2
(V1 +V2)?.
x51x79x4x47(V1 +V2)? = V?1 \V?2,
(2) (V1 \V2)? = ￡(V?1 )? \(V?2 ) =
h?
V?1 +V?2 ￠?
i?
= V?1 +V?2,
6,x18Wx22x53x27x1ex50x70x71V x15x0x70x71,fix13V x15x48x66x1fx20,x1dx4dfix8Wx15x32x33
d(fi;W) = jfi?fi0j;
x3cx16,fi0x22fix6bWx79x15x72x3x16x17.
x53x54:x8x1ffi1;fi2;￠￠￠ ;fimx22Wx15x7a,x4a
d(fi;W) =
s
jG(fi1;fi2;￠￠￠ ;fim;fi)j
jG(fi1;fi2;￠￠￠ ;fim)j,
x77x1ex15G(￠￠￠)x13x1fx20x42x15x5ax60x5bx5dx5e(x5fx60x615–3.10).
￠ 18 ￠
x3x4,x18fi0 =
mP
i=1
xifii,x43
(fi?fi0;fij) =
fi?
mX
i=1
xifii;fij
!
= 0; j = 1;￠￠￠ ;m;
x3ex50 0
B@
(fi;fi1)
...
(fi;fim)
1
CA = G(fi
1;￠￠￠ ;fim)
0
B@
x1
...
xm
1
CA;
x4ex3cfi1;￠￠￠ ;fimx74x26x2cx2a,x23x24G = G(fi1;￠￠￠ ;fim)x3exc(x5ex5fx61x605–3.10),x21x4f
0
B@
x1
...
xm
1
CA = G?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA:
fi0 = (fi1 ￠￠￠ fim)
0
B@
x1
...
xm
1
CA = (fi
1 ￠￠￠ fim)G?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA:
x23x24
d(fi;W)2 = (fi?fi0;fi?fi0)
=
0
B@fi?(fi
1 ￠￠￠ fim)G?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA;fi?(fi
1 ￠￠￠ fim)G?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA
1
CA
= (fi;fi)?2((fi;fi1) ￠￠￠ (fi;fim))G?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA
+((fi;fi1) ￠￠￠ (fi;fim))G?TGG?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA
= (fi;fi)?((fi;fi1) ￠￠￠ (fi;fim))G?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA
= 1jGj
2
64(fi;fi)jGj?((fi;fi
1) ￠￠￠ (fi;fim))G?
0
B@
(fi;fi1)
...
(fi;fim)
1
CA
3
75
= 1jGj
flfl
flfl
flfl
flfl
fl
(fi1;fi1) ￠￠￠ (fi1;fim) (fi1;fi)
(fi2;fi1) ￠￠￠ (fi2;fim) (fi2;fi)
...,..,..,..
(fi;fi1) ￠￠￠ (fi;fim) (fi;fi)
flfl
flfl
flfl
flfl
fl
= jG(fi1;￠￠￠ ;fim;fi)jjGj,
) d(fi;W) =
s
jG(fi1;fi2;￠￠￠ ;fim;fi)j
jG(fi1;fi2;￠￠￠ ;fim)j,
7,x18V1,V2x22x53x27x1ex50x70x71V x15x37x66x0x70x71,x;y 2 V,x74x26x74x36L1 = x+V1,L2 = y +V2x39x71
￠ 19 ￠
x15x32x33x1dx4dx22
d(L1;L2) = minjfi?flj; 8fi 2 L1; fl 2 L2:
x53x54,d(L1;L2) = d(x?y;V1 +V2).
x3x4,x4eV = (V1+V2)'(V1+V2)?,x3ex50x?y = fl1?fi1+–,x3cx16fi1 2 V1,fl1 2 V2,– 2 (V1+V2)?.
x3cx13
d(x?y;V1 +V2) = j–j = j(x+fi1)?(y +fl1)j> d(L1;L2):
x4dx39,x2x0x1x15fi = x+fi1 2 L1,fl = y +fl1 2 L2,x49
fi?fl = (x?y)+(fi1?fl1) = +–;
x3cx16 2 V1 +V2,– 2 (V1 +V2)?,x4a
x?y = (?fi1 +fl1)+–:
x3cx13
jfi?flj2 = j +–j2 = j j2 +j–j2 >j–j2 = d(x?y;V1 +V2)2:
(x3cx16j +–j2 = j j2 +j–j2x13x21x22? –.) x23x24
d(L1;L2) = minjfi?flj> d(x?y;V1 +V2):
x6x2x3ex50d(L1;L2) = d(x?y;V1 +V2).
8,x73x37x66xcxfL1 = x+L(fi1;fi2)x42L2 = y +L(fl1;fl2)x39x71x15x32x33,x3cx16
fi1 = (1;?2;0;?3); fi2 = (2;?2;1;2); x = (4;5;3;2);
fl1 = (1;0;1;1); fl2 = (1;?2;0;?1); y = (1;?2;1;?3):
x5,W = L(fi1;fi2)+L(fl1;fl2) = L(fi1;fi2;fl1),x23x24
d(L1;L2) = d(x?y;W) =
s
jG(fi1;fi2;fl1;(x?y))j
jG(fi1;fi2;fl1)j =
r 324
36 = 3:
9,x73x1dx1ex40x41x15x6x5cx62xfx2d,8
>>>>
<
>>>>
:
3:4x?1:6y = 1
3:3x?1:7y = 1
3:2x?1:5y = 1
2:6x?1:1y = 1:
x5,A =
0
BB
B@
3:4?1:6
3:3?1:7
3:2?1:5
2:6?1:1
1
CC
CA,B =
0
BB
B@
1
1
1
1
1
CC
CA.
x4ax6x5cx62xfx2d(x;y)x22x74x26x40x41ATAX = ATBx15x2d,x2dx77x66x40x41,x50(
x … 0:69
y … 0:78
x1 x2 5–7
1,x18A = (aij) 2 Mn(R)x22x72x3x5dx5e,x3fjAj = 1.
x53x54,aij = Aij,x3cx16Aijx22aijx15x51xex24x0x29.
x3x4,x21x22jAj = 1,x23x24AA? = E,x43x25
A? = A?1 = AT;
￠ 20 ￠
x37x38x65x62x7x3ex50
aij = Aij; i;j = 1;￠￠￠ ;n:
2,x18x41 x13x53x27x1ex50x70x71V x15x48x66x3dx4a.
x53x54:x8x1fx41 x46x47x7bx10x55x3d,x4x2x23x47x15fi;fl 2 V,(x41fi;x41fl) = (fi;fl),x20x30x38x48x1dx13x74x26x15,x21x25
x13x72x3x3dx4a.
x3x4,x2x0x1x15fi;fl 2 V,x47
(x41(fi+fl)?x41fi?x41fl;x41(fi+fl)?x41fi?x41fl)
= (x41(fi+fl);x41(fi+fl))?2(x41(fi+fl);x41fi)?2(x41(fi+fl);x41fl)
+(x41fi;x41fi)+2(x41fi;x41fl)+(x41fl;x41fl)
= (fi+fl;fi+fl)?2(fi+fl;fi)?2(fi+fl;fl)+(fi;fi)+2(fi;fl)+(fl;fl)
= ((fi+fl)?fi?fl;(fi+fl)?fi?fl) = 0:
x23x24x41(fi+fl) = x41fi+ x41fl.
x6bx6cx6d,
(x41(kfi)?kx41fi;x41(kfi)?kx41fi)
= (x41(kfi);x41(kfi))?2k(x41(kfi);x41fi)+k2(x41fi;x41fi)
= (kfi;kfi)?2k(kfi;fi)+k2(fi;fi) = 0:
x23x24x41(kfi) = kx41fi.
x21x4fx41 x13x74x26x3dx4a.
3,x18"1;"2;￠￠￠ ;"nx42fi1;fi2;￠￠￠ ;finx13x53x27x1ex50x70x71x15x37x66x54x3dx72x3x7a,x53x54,x31x6bx72x3x3dx4ax41,
x27
x41("i) = fii (i = 1;2;￠￠￠ ;n):
x3x4,x4ex3c"1;"2;￠￠￠ ;"nx13x74x26x70x71x15x7a,x21x4fx2dx2ex61x18x31x32x15x74x26x3dx4ax41 x48x1dx31x6b,x2x3cx0x1
x15x37x66x1fx20fi =
nP
i=1
ai"i,fl =
nP
i=1
bi"i,x47x41(fi) =
nP
i=1
aix41("i) =
nP
i=1
aifii,x41(fl) =
nP
i=1
bifii,x21x4f
(x41(fi);x41(fl)) =
nX
i=1
aifii;
nX
i=1
bifii
!
=
nX
i=1
aibi = (fi;fl):
x23x24x41 x13x72x3x3dx4a.
4,x73x1dx1ex72x3x40x5ex15x53x60x35:
(1)
0
BB
@
1
2
p3
2 0
p3
2
1
2 00 0 1
1
CC
A; (2)
0
@
0 0 1
0?1 0
1 0 0
1
A; (3)
0
BB
B@
p2
4
p6
4
p2
2
p2
4?
p6
4
p2
2p
3
2?
1
2 0
1
CC
CA.
x5,(1) = 0,`+? = 5…3 ;
(2) = …2,` = …2,? = …2 ;
(3)x4er33 = cos = 0,2 [0;…],x3ex50 = …2, x12x4er31 = sin? =
p3
2 x24x68r32 = cos? =?
1
2 x3ex50
= 2…3, x6x7x4er13 = sin` =
p2
2 x24x68r23 =?cos` =
p2
2 x3ex50` =
3…
4,
5,x18x17Px15x57x55x22(1;1;0),x73x22x45?!OPx5bx1ex32x40x1fx1bx1c …
6 x15x72x3x3dx4a.
￠ 21 ￠
x5,x5ex68x6a7.5,x1bx1cx45x15x14x2fx1fx20x13? =
p
2
2 ;
p2
2 ;0
,x49· =!k =
p
2
2 ;
p2
2 ;?1
,x63·
x40x1fx15x64x5dx1fx22x53,x4ex3c8
>>>>
>><
>>>
>>>:
x53 (?!i ) =?!i?2 (
!i ;·)
(·;·) · =
1
2
!i? 1
2
!j + p2
2
!k
x53 (?!j ) =?!j?2 (
!j ;·)
(·;·) · =?
1
2
!i + 1
2
!j + p2
2
!k
x53 (?!k ) =?!k?2 (
!k ;·)
(·;·) · =
p2
2
!i + p2
2
!j
x21x4fx53 x6bx7a?!i ;?!j ;?!k x1dx15x5dx5ex22
S =
0
BB
B@
1
2?
1
2
p2
2
12 12
p2
2p
2
2
p2
2 0
1
CC
CA:
x16x48x40xfx1bx1cx52?!k ;?…
6
x15x5dx5ex13
R =
0
BB
@
p3
2
1
2 0
12
p3
2 00 0 1
1
CC
A:
x6x7x50x8x23x73x72x3x3dx4ax15x5dx5ex22
SRS =
0
BB
B@
p3
4 +
1
2?
p3
4 +
1
2
p2
4
p3
4 +
1
2
p3
4 +
1
2?
p2
4
p2
4
p2
4
p3
2
1
CC
CA:
6,x73x72x3x3dx4a
x41(X) =
0
BB
B@
p2
2
p2
2 0
1
2?
1
2
p2
2
1
2?
1
2?
p2
2
1
CC
CAX
x15x1bx1cx45x42x1bx1cx35.
x5,x73x1bx1cx45x65x62x3cx73AX = Xx15x2dx1fx20X 2R3,x2dx50x1bx1cx45x15x40x1fx1fx20x13? = (p2+1;1;p2?1).
x22x73x1bx1cx35,x7ax48x66x42?x72x3x15x1fx20fi = (1;?2;?1),x4ax1bx1cx35 = hfi;x41(fi)i.
cos = (fi;x41(fi))jfij2 =? 34,
x51x21x3ex54x10
(?;fi;x41(fi)) =? 212 < 0;
x23x24x1bx1cx35 = … +arccos 34,
￠ 22 ￠
x0x1cx2 xcxdx13x14x11x1dx1ex1fx20x13x14
x1 x2 5–1
1,x5bx72x36xex15x17x44x42xfx44,x1dx1exdx54x13x29x75x2ax32xex9Rx79x15x74x26x70x71?
(1)x2bxexdZ; (2)x47x5xexdQ; (3)x32xexdR; (4)x40xexdC.
x5,(1)x42(2)x6dx55x13x32xex9Rx79x15x74x26x70x71,x21x22x55x20xfx44x55x12x13,(3)x3a(4)x6dx13Rx79x15x74x26x70
x71.
2,x1eKx22x40xex9C,x1x24x32xex22xbx58x15x48x41n￡nx5dx5ex15xdx54x2x5dx5ex15x17x44x42x55x20xfx44x13x29x75x2a
Kx79x15x74x26x70x71? x22x2fx30?
x5,x29,x2ax3cx55x20xfx44x55x12x13.
3,x42x7dx1dx1exdx54x2x3cx23x1bx15x33x67x13x29x75x2ax32xex9x79x15x74x26x70x71:
(1)x33x10x32x2x75(x4dx75,x79x34x35x36)x5dx5e,x2x3cx5dx5ex15x17x44x42x55x20xfx44;
(2)x48xex56x3cn (n > 1)x15x32x6axex9x3ax29x33x10,x2x3cx9x3ax29x15x17x44x42xfx44;
(3)xcxfx79x33x10x1fx20,x2x3cx1fx20x15x17x44x42x8x1dx1dx4dx15x55x20xfx44:
kfi = fi;
(4)x33x10x72x32xeR+,x17x44x3ax55x20xfx44x1dx4dx22:
a'b = ab;
k–a = ak:
x5,(1)x13; (2)x29,x6fx9x3ax29x55x6bxdx54x16; (3)x29,x21x22x62fi 6= 0x52,0fi 6= 0; (4)x13.
4,x78x67x79x61x16x23x25x31x15x74x26x70x71x15x46xex3ax7a.
x5,(1)x32x2x75,n(n+1)2 x46,x7afEij +Eji j i 6 jg;
x4dx75,n(n?1)2 x46,x7afEij?Eji j i < jg;
x79x34x35x36,n(n+1)2 x46,x7afEij j i 6 jg.
(4) 1x46,x0x8x55x56x3c1x15x72x32xex6dx3ex2fx22x7a.
5,x53x54:x33x10x24x6fx22x1cx73x15x32xex1e
S =
n
fang = (a1;a2;a3;￠￠￠ ;an;￠￠￠) j ai 2R; limn!1an = 0
o
x5bx8x1dx1dx4dx15x17x44x42x55x20xfx44:
fang+fbng = fan +bng;
kfang = fkang
x75x2ax32xex9Rx79x15x48x66x2cx73x46x74x26x70x71.
x3x4,x7dx53x74x26x70x71x69,x22x58x54x38x13x2cx73x46x15,x2x0x1x15x72x2bxen,x47x48x66x43x44x3c0x15xex1e,fin =
f0;￠￠￠ ;0;1(x3dnx3a),0;0;￠￠￠g,x3cx13x2x3cx0x1x3bx15n,x38x47nx66x1fx20fi1;￠￠￠ ;finx74x26x2cx2a.
￠ 1 ￠
6,x18
P =
‰ fi fl
fl fi
flfl
flfl fi;fl 2C
:
(1)x53x54,Px5bx5dx5ex15x17x44x42x55x20xfx44x75x2ax32xex9Rx79x15x48x66x74x26x70x71;
(2)x73Px15x46xex42x7a.
x5,(1)x69,(2) dimRP = 4,x7ax22:
1 0
0 1
,
i 0
0?i
,
0 1
1 0
,
0 i
i 0
.
7,x18Rx22x32xex9x6bx38x67x45x79x15x74x26x70x71,R+x22x3d3x61(4)x16x15x1fx20x70x71,x53x54,Rx42R+x43x75.
x3x4,x49
’, R?! R+
r 7?! 2r
x4a(a) ’x13x46x5d;
(b) ’x13x14x15,x21x222r1 = 2r2 () r1 = r2;
(c) ’x13x2dx15,x21x22x2x0x1x15a 2R+,a = 2log2 a,x25log2 a 2R,x3cx13’(log2 a) = 2log2 a = a;
(d) ’x46x47x33x67:
’(r1 +r2) = 2r1+r2 = 2r12r2 = 2r1 '2r2 = ’(r1)'’(r2);
’(kr1) = 2kr1 = (2r1)k = k–2r1 = k–’(r1):
x23x24’x13x43x75.
8,x18Fx22x33x10x36x8
(x1;x2;x3;￠￠￠ ;xn;￠￠￠); xn = xn?1 +xn?2; n > 3
x15x32xex1ex23x42x2ax15xdx54,x3cx17x44x42x55x20xfx44x15x1dx4dx8x3d5x61.
(1)x53x54,Fx75x2aRx79x15x48x66x62x46x74x26x70x71;
(2)x1bx25Fx15x48x66x4ex56x65xex1ex23x42x2ax15x7a;
(3)x73x48x49x20x45(Fibonacci)xex1e
(0; 1; 1; 2; 3; 5; 8; ￠￠￠)
x15x72x3ax66x29.
x3x4,(1) Fx22Rx79x74x26x70x71x15x53x54x69,x1dxfx73Fx15x46xe.
xbx3xex1efi1 = (0;1;1;2;3;5;￠￠￠)x42fi2 = (1;1;2;3;5;￠￠￠),x10x11fi1;fi2 2 F.
(a)x18k1fi1 +k2fi2 = 0,x4a(k2;k1 +k2;k1 +2k2;2k1 +3k2;￠￠￠) = 0,x23x24k2 = 0,x43x25k1 = 0,x77x58
x54fi1;fi2x74x26x2cx2a.
(b)x2x0x1x15
fl = (a1;a2;a3;￠￠￠ ;an;￠￠￠); an = an?1 +an?2; n > 3
xbx3
= (a2?a1)fi1 +a1fi2?fl 2 F;
x4a = (0;0;x3;x4;￠￠￠),x21x22 2 F,x23x24x3 = 0+0 = 0,x4 = x3 +0 = 0,x4ex2x50x44x3exb = 0,x77x6fx53
x54x72fl = (a2?a1)fi1 +a1fi2,x21x4ffi1;fi2x75x2aFx15x7a,dimF = 2.
(2)x18x47x56x65xex1e
(a;aq;aq2;￠￠￠) 2 F;
x4ax2n > 2x47aqn = aqn?1 +aqn?2,x43x25q2 = q +1,x50x8q = 1§
p5
2,
￠ 2 ￠
xexb
·1 =
0
@1; 1+
p5
2 ;
1+p5
2
!2;￠￠￠
1
A2 F;
·2 =
0
@1; 1?
p5
2 ;
1?p5
2
!2;￠￠￠
1
A2 F:
x51·1;·2x74x26x2cx2a,x25dimF = 2,x23x24·1;·2x75x2aFx15x7a.
(3)x48x49x20x45xex1e
’ = (0; 1; 1; 2; 3; 5; 8; ￠￠￠) 2 F;
x21x4fx31x6bc1;c2 2R,x27
’ = c1·1 +c2·2:
x43x25 (
0 = c1 +c2
1 = c1 1+
p5
2 +c2
1?p5
2x2dx50
c1 =
p5
5 ; c2 =?
p5
5 ;
x4ex4fx3ex50x48x49x20x45xex1ex50x72x3ax66x29x13
Dn =
p5
5
2
4
1+p5
2
!n?1
1?p5
2
!n?13
5:
9,x23x4anx79x4bx4c,x13x2dx3cx28xdx28x1ex24x68x4dx2x35x3ax48x2x35xbx58x39x3ax6dx65x56x15nx79x40x5e,x8
0
@
6 1 8
7 5 3
2 9 4
1
A
x6fx13x48x66x34x79x4ex5e.
(1)x53x54:x32xex9x79x33x10nx79x4ex5ex15xdx54Mnx5bx5dx5ex15x17x44x42x55x20xfx44x75x2aRx79x15x48x66x74x26x70x71;
(2)x73M3x15x46xe.
x5,(2) 3x46,x7ax22:
0
@
1?1 0
1 0 1
0 1?1
1
A;
0
@
0 1?1
1 0 1
1?1 0
1
A;
0
@
1 1 1
1 1 1
1 1 1
1
A:
x1 x2 5–2
1,x18W1,W2x13x74x26x70x71V x15x0x70x71,x53x54x24x1dx34x66x23x7dx13x56x15x15:
(1) W1 W2; (2) W1 \W2 = W1;
(3) W1 +W2 = W2.
x3x4,(1),(2)x24x68(1))(3)x6dx13x10x11x15.
(3))(1),W1 +W2 = W2 ) W1 W1 +W2 = W2.
2,x73x4ex1fx20fiix7x2ax15x0x70x71x3ax4ex1fx20flix7x2ax15x0x70x71x15x3x42x3ax15x7ax42x46xe.
(1)
( fi
1 = (1;3;1;?1)
fi2 = (1;0;1;2);
( fl
1 = (3;?1;?3;?5)
fl2 = (5;?2;?3;?4);
￠ 3 ￠
(2)
( fi
1 = (1;0;1;0)
fi2 = (1;1;0;1);
( fl
1 = (0;1;0:1)
fl2 = (0;1;1;0);
(3)
8>
<
>:
fi1 = (1;0;2;0;)
fi2 = (2;0;1;1)
fi3 = (1;0;?1;1);
( fl
1 = (3;3;1;?2)
fl2 = (1;3;0;?3):
x5,x4ex4ex1fx20fiix7x2ax15x0x70x71x3ax4ex1fx20flix7x2ax15x0x70x71x11x12x1fx22W1;W2.
(1) dim(W1 +W2) = 3,dimW1 \W2 = 1,
W1 +W2x15x7a,fi1;fi2;fl1,
W1 \W2x15x7a,(3;?2;3;8)
= 13 (?2fi1 +11fi2) =?4fl1 +3fl2
·;
(2) dim(W1 +W2) = 4,dimW1 \W2 = 0,
W1 +W2x15x7a,fi1;fi2;fl1;fl2;
(3) dim(W1 +W2) = 3,dimW1 \W2 = 1,
W1 +W2x15x7a,fi1;fi2;fl1,
W1 \W2x15x7a,(2;0;1;1)(= fi2 = fl1?fl2).
3,x18W,W1,W2x6dx13x1fx20x70x71V x15x0x70x71,x3f
W1 W2; W \W1 = W \W2; W +W1 = W +W2:
x53x54,W1 = W2.
x3x4,dimW +dimW1 = dim(W +W1)+dim(W \W1),
dimW +dimW2 = dim(W +W2)+dim(W \W2),
x23x24x79x29x1ex61x65x56.x3ex50dimW1 = dimW2,x51x21W1 W2,x23x24W1 = W2.
4,x18V1,V2x13nx46x74x26x70x71V x15x37x66x0x70x71,x57x3fx2dx2e
dim(V1 +V2) = dim(V1 \V2)+1;
x53x54,V1 V2x44V2 V1.
x3x4,x21x22dim(V1 \ V2) 6 dimV1 6 dim(V1 + V2) = dim(V1 \ V2) + 1,x37x66x56x59x16x40x47x48x66x2a
x2b,x8x1fx1dx38x56x59x2ax2b,x4ax21V1 \V2 V1,x3ex50V1 \V2 = V1,x43x25V1 V2,x8x1fx1ex38x56x59x2ax2b,x4ax21
V1 V1 +V2,x3ex50V1 = V1 +V2,x43x25V2 V1.
5,x18V = K4,fi1 = (1;2;1;2),fi2 = (2;1;2;1),W = L(fi1;fi2),x73x0x70x71W x6bV x16x15x48x66x4fx70
x71.
x5,x18fi3 = (0;0;1;0),fi4 = (0;0;0;1),x4ax21fi1;fi2;fi3;fi4x74x26x2cx2a,x23x24L((0;0;1;0);(0;0;0;1))
x13Wx6bV x16x15x48x66x4fx70x71.
6,x53x54:x73x48x66nx46x74x26x70x71x6dx13nx66x48x46x0x70x71x15x2ex3a.
x3x4,x18V x22nx46x74x26x70x71,fi1;￠￠￠ ;finx13V x15x7a,x49Wi = L(fii),x4aV = W1 + W2 +￠￠￠+ Wn.
x51,n = dimV = Pni=1 dimWi,x23x24
V = W1 'W2 '￠￠￠'Wn:
7,x53x54,nx46x74x26x70x71V x15x73x48x66x5x0x70x71x6dx13x1ex50x66n?1x46x0x70x71x15x3.
x3x4,x18Wx13V x15x5x0x70x71,x4ar = dimW < dimV = n,x7aWx15x48x66x7afi1;￠￠￠ ;fir,x76x3cx1dx30x2a
V x15x7afi1;￠￠￠ ;fin,x7ax8x1dx15n?rx66n?1x46x74x26x0x70x71
Vj = L(fi1;￠￠￠ ;fij?1;fij+1;￠￠￠ ;fin); j = r +1;￠￠￠ ;n:
￠ 4 ￠
x4ax21
fl =
nX
i=1
aifii 2 Vj () aj = 0;
fl =
nX
i=1
aifii 2
nj=r+1
Vj () ar+1 = ￠￠￠ = an = 0 () fl 2 W:
x4W =
nT
j=r+1
Vj.
8,x18V1x42V2x11x12x13x3x48x74x26x40x41x42
x1 +x2 +￠￠￠+xn = 0 x42 x1 = x2 = ￠￠￠ = xn
x15x2dx70x71.
x53x54,Kn = V1 'V2.
x3x4,(a)x2x0x1x15fi = (a1;￠￠￠ ;an) 2 Kn,x49
fl =
a1? 1n
nX
i=1
ai;a2? 1n
nX
i=1
ai;￠￠￠ ;an? 1n
nX
i=1
ai
!;
=
1
n
nX
i=1
ai; 1n
nX
i=1
ai;￠￠￠ ; 1n
nX
i=1
ai
!;
x4afl 2 V1,2 V2,x3ffi = fl +, x23x24Kn = V1 +V2.
(b)x8x1ffi = (a1;￠￠￠ ;an) 2 V1 \V2,x4a
nX
i=1
ai = 0; a1 = a2 = ￠￠￠ = an
x2dx50a1 = a2 = ￠￠￠ = an = 0,x4fi = 0,x23x24V1 \V2 = 0.
x51x79x3ex50Kn = V1 'V2.
9,x18W1 = fA 2 Mn(K) j AT = Ag,W2 = fA 2 Mn(K) j AT =?Ag.
x53x54,Mn(K) = W1 'W2.
x3x4,(a)x2x0x1x15nx79x5dx5eA 2 Mn(K),x47
A = 12 (A+AT)+ 12 (A?AT);
x25 12 (A+AT) 2 W1,12 (A?AT) 2 W2,x23x24Mn(K) = W1 +W2.
(b)x18A 2 W1 \W2,x4a
A = AT = A;
x4e2A = 0x3ex50A = 0,x23x24W1 \W2 = 0.
x6x2x50x8Mn(K) = W1 'W2.
10,x18A 2 Mn(K)x3fA2 = A,x49
V1 = fX 2 Kn j AX = 0g; V2 = fX 2 Kn j AX = Xg:
x53x54,Kn = V1 'V2.
x3x4,(a)x18fi 2 Kn,x4afi = (fi?Afi)+Afi,x25
A(fi?Afi) = Afi?A2fi = Afi?Afi = 0; x23x24 fi?Afi 2 V1;
A(Afi) = A2fi = Afi; x23x24 Afi 2 V2;
x43x25Kn = V1 +V2.
￠ 5 ￠
(b)x18fi 2 V1 \V2,x4ax21fi 2 V1,x47Afi = 0,x4efi 2 V2,x47Afi = fi,x3cx13fi = 0,x4V1 \V2 = 0.
x21x4fKn = V1 'V2.
11,x18Kn = V1 'V2,x3cx16V1,V2x22Knx15x37x66x6excx78x15x0x70x71.
x53x54:x48x1dx31x6bx2cx48x15x52x56x5dx5e(x4A2 = Ax15x5dx5e)A 2 Mn(K),x27
V1 = fX 2 Kn j AX = 0g; V2 = fX 2 Kn j AX = Xg:
x3x4,x7aV1x15x48x66x7afi1;￠￠￠ ;firx24x68V2x15x48x66x7afir+1;￠￠￠ ;fin,x4afi1;￠￠￠ ;finx13Knx15x7a,x1dx4d
Knx79x15x74x26x3dx4ax41 x22:
x41(fii) =
( 0; 1 6 i 6 r
fii; r +1 6 i 6 n
x4ex74x26x3dx4ax41 x6bKnx15x67x11x7ax1dx15x5dx5ex1fx22A,x4ex41 x15x1dx4dx3ex50x412 = x41,x65x2cx6dx47A2 = A.
x2x0x1x15X 2 Kn,x47X =
nP
i=1
aifii,x4a
AX = x41
nX
i=1
aifii
!
=
nX
i=1
aix41(fii) =
nX
i=r+1
aifii:
x21x4f
AX = 0 ()
nX
i=r+1
aifii = 0 () ai = 08r +1 6 i 6 n () X 2 V1;
AX = X ()
nX
i=r+1
aifii =
nX
i=1
aifii () ai = 081 6 i 6 r () X 2 V2:
x23x24Ax13x2dx2ex31x32x15x52x56x5dx5e.
x12x53x2cx48x26,x8x1fB 2 Mn(K),x27x50
BX = 0 8X 2 V1; BX = X 8X 2 V2;
x4ax21Kn = V1 'V2,x3ex50
(A?B)X = 0; 8X 2 Kn:
x23x24A?B = 0,x43x25A = B.
12,x18A 2 Mn(K),Ex22nx79x14x2fx40x5e,x49
V1 = fX 2 Kn j (A?E)X = 0g;V2 = fX 2 Kn j (A+E)X = 0g:
x53x54,Kn = V1 'V2 () A2 = E.
x3x4,()) Kn = V1 'V2 =) n = dimV1 +dimV2 =) n = (n?rank(A?E))+(n?rank(A+
E)) =) n = rank(A?E)+rank(A+E) =) A2 = E (x60x614–8.12).
(()x2x0x1x15fi 2 Kn,
fi = 12 (A+E)fi? 12 (A?E)fi:
x21x22
(A?E)
1
2 (A+E)fi
= 12 (A2?E)fi = 0;
x23x24 12 (A+E)fi 2 V1,x51x21
(A+E)
12 (A?E)fi
=? 12 (A2?E)fi = 0;
x23x24? 12 (A?E)fi 2 V2.
x21x4fKn = V1 +V2.
￠ 6 ￠
x62fi 2 V1 \V2x52x51x47
fi = 12 (A+E)fi? 12 (A?E)fi = 0+0 = 0;
x21x4fV1 \V2 = 0,x43x25Kn = V1 'V2.
x1 x2 5–3
1,x6bx74x26x70x71R2x16,x2x0x1x37x66x1fx20fi = (a1;a2),fl = (b1;b2),x1dx4d
(fi;fl) = 5a1b1 +2a1b2 +2a2b1 +a2b2:
x7dx53x6bx4fx1dx4dx1dR2x75x2ax48x66x53x27x1ex50x70x71.
x3x4,x69.
2,x6bx74x26x70x71Mn(R)x16,x1dx4d
f(A;B) = Tr(ATB) 8A;B 2 Mn(R):
x19x1,fx13x29Mn(R)x15x48x66x7bx10?
x5,x13,x18A = (aij),B = (bij),x4a
(a) f(A;B) = Tr(ATB) =
nP
k=1
nP
i=1
akibki =
nP
k=1
nP
i=1
bkiaki = f(B;A).
(b) f(A+B;C) = Tr((A+B)TC) = Tr(ATC+BTC) = Tr(ATC)+Tr(BTC) = f(A;C)+f(B;C).
(c) f(kA;B) = Tr((kA)TB) = Tr(kATB) = kTr(ATB) = kf(A;B).
(d) f(A;A) = Tr(ATA) =
nP
k=1
nP
i=1
a2ki > 0,x3f
f(A;A) = 0 () aki = 0; k;i = 1;￠￠￠ ;n () A = 0:
x23x24fx13Mn(R)x15x48x66x7bx10.
3,x18
A =
0
BB
B@
1 0 ￠￠￠ 0
0 2 ￠￠￠ 0.
..,..,..,..
0 0 ￠￠￠ n
1
CC
CA:
x54x1d
(X;Y) = XTAY 8X;Y 2Rn:
(1)x53x54,Rnx2ax3cx4fx1dx4dx75x2ax48x66x53x27x1ex50x70x71;
(2)x73x1fx20"1 = (1;0;￠￠￠ ;0),"2 = (0;1;0;￠￠￠ ;0);￠￠￠,"n = (0;0;￠￠￠ ;0;1)x15x77x20x5dx5e,
(3)x7fx10x7ex25x77x66x70x71x15x55x56–x57x58x55x59x64x7ax55x56x29.
x5,(1)x69.
(2)x77x20x5dx5ex22A.
(3)x18fi = (a1;￠￠￠ ;an),fl = (b1;￠￠￠ ;bn),x4a
flfl
flfl
fl
nX
k=1
kakbk
flfl
flfl
fl6
vu
ut nX
k=1
ka2k ￡
vu
ut nX
k=1
kb2k:
4,x18Cx13x48x66nx79x32x3excx5dx5e,x6bRnx16,x2x0x1x37x66x1ex1fx20X,Y,x54x1d
(X;Y) = XTCTCY
x53x54,Rnx2ax3cx4fx1dx4dx75x2ax48x66x53x27x1ex50x70x71.
￠ 7 ￠
x3x4,x69.
5,x6bx55x12x53x27x1ex50x70x71x7bx78x67x1bx1dx1fx20x15x7bx10,x57x73x38x15x39x71x15x15x35:
(1) fi = (1;1;1;1),fl = (?1;2;4;3);
(2) fi =
1
2 ;?1;
1
3 ;
1
6
·
,fl = (3;?1;2;2);
(3) fi = (3;?1;1;?1),fl = (?2;2;?2;2);
(4) fi = (?1;1;?1;2;1),fl = (3;1;?1;0;1).
x5,(1) (fi;fl) = 8,hfi;fli = arccos 2
p30
15,
(2) (fi;fl) = 72,hfi;fli = arccos 710,
(3) (fi;fl) =?12,hfi;fli = 5…6,
(4) (fi;fl) = 0,hfi;fli = …2,
6,x18x2 +y2 +z2 = 1,(x;y;z 2R),x19x73
x2
1?x2 +
y2
1?y2 +
z2
1?z2
x15x6x5cx4.
x5,x4bx29=?3+ 11?x2 + 11?y2 + 11?z2, x25x4ex55x56–x57x58x55x59x64x7ax55x56x29,
vu
ut 1
p1?x2
2
+
1p
1?y2
!2
+
1
p1?z2
2
￠
q
(
p
1?x2)2 +(
p
1?y2)2 +(
p
1?y2)2
>
1p
1?x2 ;
1p
1?y2 ;
1p
1?z2
!0
@
p1?x2)
p1?y2
p1?y2
1
A = 3;
x4 s
1
1?x2 +
1
1?y2 +
1
1?z2 ￠
p2 > 3;
x23x24
1
1?x2 +
1
1?y2 +
1
1?z2 >
9
2,
x2
1?x2 +
y2
1?y2 +
z2
1?z2 >?3+
9
2 =
3
2,
x51x62x = y = z =
p3
3 x52x79x29x7ax56x59.x21x4bx29x15x6x5cx4x22
3
2,
7,x18a;b;c;x;y;z 2R,x1ea2 +b2 +c2 = 25,x2 +y2 +z2 = 36,ax+by +cz = 30,x73 a+b+cx+y +z x15
x4.
x5,x4ex55x56–x57x58x55x59x64x7ax55x56x29,
30 = ax+by +cz = (a b c)
0
@
x
y
z
1
A
6
p
a2 +b2 +c2 ￠
p
x2 +y2 +z2 = 30:
x21x56x59x2ax2bx52,(a;b;c)x42(x;y;z)x2ax65x6a,x18(a;b;c) = t(x;y;z),x51x52x50
30 = t(x2 +y2 +z2) = 36t;
x2dx25t = 56, x43x25 a+b+cx+y +z = 56,
￠ 8 ￠
8,x6bx55x12x53x27x1ex50x70x71R3x16,x73x7afi1 = (1;0;1),fi2 = (1;1;0),fi3 = (0;1;1)x15x77x20x5dx5e.
x5,A =
0
@
2 1 1
1 2 1
1 1 2
1
A.
9,x6b4x46x53x27x1ex50x70x71V x16,x18x7afi1 = (1;1;?1;?1),fi2 = (1;1;1;0),fi3 = (?1;1;1;1),fi4 =
(1;0;0;?1)x15x77x20x5dx5ex22
A =
0
BB
B@
2 1 0 0
1 2 1 0
0 1 2 1
0 0 1 2
1
CC
CA
(1)x73x7a"1 = (1;0;0;0),"2 = (0;1;0;0),"3 = (0;0;1;0),"4 = (0;0;0;1)x15x77x20x5dx5e;
(2)x73x1fx20fl1 = (1;?1;1;?1),fl2 = (0;1;1;0)x15x7bx10;
(3)x73x48x14x2fx1fx20x42fi1,fi2,fi3x72x3.
x5,(1) (fi1;fi2;fi3;fi4) = ("1;"2;"3;"4)B,
B =
0
BB
B@
1 1?1 1
1 1 1 0
1 1 1 0
1 0 1?1
1
CC
CA; B?1 = 12
0
BB
B@
0 1?1 0
2 0 0 2
2 1 1?2
2 0 2?4
1
CC
CA:
x21x4fx7a"1;"2;"3;"4x50x77x20x5dx5ex2cx22
G = B?TAB?1 =
0
BB
BB
@
6? 12? 92 9
12 1 12?1
92 12 4?7
9?1?7 14
1
CC
CC
A
:
(2) (fl1;fl2) = 6.
(3)x18fi =
4P
i=1
xifiix42fi1;fi2;fi3x72x3,x4a
fij;
4X
i=1
xifii
!
= 0; j = 1;2;3:
x43x25
0
@
2 1 0 0
1 2 1 0
0 1 2 1
1
A
0
BB
B@
x1
x2
x3
x4
1
CC
CA = 0:
x2dx50(x1;x2;x3;x4) = (1;?2;3;?4),x23x24
fi = fi1?2fi2 +3fi3?4fi4 = (?8;2;0;6):
(fi;fi) = (fi4;fi) = 20,x21x4fx23x73x15x14x2fx1fx20x22§
p5
5 (?4;1;0;3).
10,x18fi1;fi2;￠￠￠ ;fimx13x53x27x1ex50x70x71V x15mx66x1fx20,x75x5dx5e
G(fi1;fi2;￠￠￠ ;fim) =
0
BB
B@
(fi1;fi1) (fi1;fi2) ￠￠￠ (fi1;fim)
(fi2;fi1) (fi2;fi2) ￠￠￠ (fi2;fim)
:::::::::::::::::::::::::::::::::::
(fim;fi1) (fim;fi2) ￠￠￠ (fim;fim)
1
CC
CA
x22x1fx20x42fi1;fi2;￠￠￠ ;fimx15x5ax60x5b(Gram)x5dx5e.
x53x54,fi1;fi2;￠￠￠ ;fimx74x26x2cx2ax62x3fx63x62jG(fi1;fi2;￠￠￠ ;fim)j6= 0.
￠ 9 ￠
x3x4,x18x47x74x26x2ax6ax29
x1fi1 +x2fi2 +￠￠￠+xmfim = 0:
x4ex77x66x56x29x11x12x42fi1;￠￠￠ ;fimx2fx7bx10,x3ex24x50x8x3dx20x1;￠￠￠ ;xmx15x48x66x3x48x74x26x40x41x42:
8>
>>><
>>>
>:
(fi1;fi1)x1 +(fi1;fi2)x2 +￠￠￠+(fi1;fim)xm = 0
(fi2;fi1)x1 +(fi2;fi2)x2 +￠￠￠+(fi2;fim)xm = 0
::::::::::::::::::::::::::::::::::::::::
(fim;fi1)x1 +(fim;fi2)x2 +￠￠￠+(fim;fim)xm = 0
x3cx6axex5dx5ex6fx13x5ax60x5bx5dx5eG(fi1;￠￠￠ ;fim),x12x33x1ax3x48x74x26x40x41x42x47x6ex6fx2dx15x30x11x40x26x31x32x3ex50:
jG(fi1;fi2;￠￠￠ ;fim)j6= 0 () x3x48x74x26x40x41x42x7bx47x6fx2dx1 = ￠￠￠ = xm = 0 () fi1;￠￠￠ ;fimx74x26x2cx2a:
11,x18"1;"2;"3x13x34x46x53x27x1ex50x70x71V x15x48x66x54x3dx72x3x7a.
x53x54,fi1 = 13 (2"1 + 2"2?"3),fi2 = 13 (2"1?"2 + 2"3),fi3 = 13 ("1?2"2?2"3)x67x13V x15x48x66x54
x3dx72x3x7a.
x3x4,x2ex11x7dx53x3exb,fi1;fi2;fi3x6dx13x14x2fx1fx20,x3fx37x37x72x3,x21x38x15x13V x15x14x2fx72x3x1fx20x42.x51x21
dimV = 3,x38x15x75x2aV x15x54x3dx72x3x7a.
12,x76x55x12x53x27x1ex50x70x71R4x15x7afi1=(1;1;0;0),fi2=(1;0;1;0),fi3 = (?1;0;0;1),fi4 = (1;1;1;?1)
x4cx22x54x3dx72x3x7a.
x5:
p2
2 (1;1;0;0),
p6
6 (1;?1;2;0),
p3
6 (?1;1;1;3),
1
2 (?1;1;1;?1).
13,x73x3x48x74x26x40x41x42 ‰
x1? x2 +x3 +3x4? x5 =0
x1 +2x2?x3 +2x5 =0
x15x2dx70x71(x2fx22x55x12x53x27x1ex50x70x71R5x15x0x70x71)x15x48x66x54x3dx72x3x7a.
x5,x1cx3x48x74x26x40x41x42x15x48x66x7ax8x2dx6ax22
fi1 =
0
BB
BB
B@
1
2
3
0
0
1
CC
CC
CA; fi2 =
0
BB
BB
B@
2
1
0
1
0
1
CC
CC
CA; fi3 =
0
BB
BB
B@
0
1
0
0
1
1
CC
CC
CA:
x72x3x4cx50:
0
BB
BB
B@
1
2
3
0
0
1
CC
CC
CA;
0
BB
BB
BB
@
127
3
7
67
1
0
1
CC
CC
CC
A;
0
BB
BB
BB
B@
517
2334
6
173
341
1
CC
CC
CC
CA
:
x14x2fx4cx7x50x54x3dx72x3x7a:
p14
14 (?1;2;3;0;0);
p238
238 (?12;3;?6;7;0);
p1938
1938 (?10;?23;12;3;34):
14,x53x54:x6bx53x27x1ex50x70x71V x16,x7a"1;"2;￠￠￠ ;"nx13x54x3dx72x3x7ax15x30x11x40x26x31x32x13,x2V x15x0x1x1f
x20fi = a1"1 +a2"2 +￠￠￠+an"n,x38x47
(fi;"i) = ai (i = 1;2;￠￠￠ ;n):
￠ 10 ￠
x3x4,())x8"1;￠￠￠ ;"nx13x54x3dx72x3x7a,x4ax2x0x1x15fi = Pai"i,x47
(fi;"i) =
0
@
nX
j=1
aj"j;"i
1
A =
nX
j=1
aj("j;"i) = ai:
(()x8x2x0x1x15fi = Pai"i,x47
(fi;"i) = ai;
x4a"j =
nP
k=1
ak"k,x3cx16ak = –kj,x21x4f
("j;"i) = ai = –ij:
x43x25"1;￠￠￠ ;"nx13x54x3dx72x3x7a.
15,x72x4ex2x9x16xcxfx7bx72x40x36x24x68x27x8x70x71x7bx2bx40x10x15x5cx3,x2x50x25x38x15x15x7ex17x57x55x15x69x5d,x43
x25x5excx25nx46x70x71x15x2bx40x10x15x7ex17x66xex66x29,x12x78x674x46x70x71x16x15x2bx40x10x47x9x46x663x46x15x44xf,x9x46x66
2x46x15x44xfx421x46x15x14? x77x664x46x2bx40x10x47x9x46x62x55x43x3bx77x15x2x35x74? x19x73x38x15x15x3bx77x24x68x42x14x15x15
x35,x5ex63x29x4ex77x48x22x1fx5ex1fx8nx46x70x71x15xfx36?
x74
x74
x74
x70x10
x74
x74
x74
x70x10
x75x75x75x72x20
x75x75x75x72x20
x7x1fx3ax9x21x46
x7ax2fxfx1x11x6
(0;0) (1;0)
(0;1) (1;1)
x74
x74
x74
x70x10
x74
x74
x74
x70x10
x75x75x75x72x20
x75x75x75x72x20
x7x1fx3ax9x21x46
x7ax2fxfx1x11x6x71x71
x71x20
x4x75x75x75x72x20
x4x20x71
x71x71
x71x71
x71x20
x4x74
x74
x74
x70x10
x3x16x5x2a
x40x18 x40x18
x40x18 x40x18
x40x18
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20
(0;0;0) (1;0;0)
(0;0;1)
(1;1;0)(0;1;0)
(0;1;1) (1;1;1)
(1;0;1)
x015x1x2
x5,nx46x70x71x15x2bx40x10x16,mx46x0x2bx40x10x472n?mCmn x66.
x62m = 0x52x22x7ex17x66xe= 2n;
x62m = 1x52x22x14xe= 2n?1n;
x62m = 2x52x22xfxe= 2n?3n(n?1);,::
x3cx55x43x3bx77x15x2x35x74x47n?1x62,x3bx77x11x12x22p2;p3;￠￠￠ ;pn.
x3bx77x22pkx15x2x35x74x42x14x15x15x35x22 …2 x44arccos 1pk,
x1 x2 5–4
1,x19x73x72x4ex17A(1;1;1)x42B(1;0;2)x3fx22x2ex3cxcxfx+2y?z?6 = 0x15xcxfx40x41.
x5,x18x23x73xcxfx15x44x1fx20x22” = (A;B;C),x4a”!AB,”x67x42xcxfx + 2y?z?6 = 0x15x44x1fx20x22
x2e,x21x4fx47x40x41x42 (
0￠A+(?1)B +C = 0
A+2B?C = 0:
x2dx50A, B, C = 1,?1,?1,x3ex50x17x44x29x40x41(x?1)?(y?1)?(z?1) = 0,x4x?y?z +1 = 0.
2,xcxfΠx4e3x66x17M1(3;?1;5),M2(4;?1;1)x3aM3(2;0;2),x73xcxfΠx15x48x66x44x1fx20,x57x73x25Πx15
x40x41.
x5,xcxfΠx15x48x66x44x1fx20x3ex7ax22” =!M1M2 ￡!M1M3 = (4;7;1),x3ex50x17x44x29x40x414(x?2) + 7y +
(z?2) = 0,x44x+7y +z?10 = 0.
￠ 11 ￠
3,xcxfΠx4ex17M0(2;3;1),x3fx3ax37xcxfΠ1, x+3y?z +3 = 0,Π2, 2x+y?2z +1 = 0x6dx22x2e,
x73Πx15x40x41.
x5,x33x1ax6a4.5xbxcxfx15x40x41x22fl
flfl
flfl
fl
x?2 y?3 z?1
1 3?1
2 1?2
flfl
flfl
flfl =?5x?5z +15 = 0;
x4cx6bx50x+z?3 = 0.
4,xcxfΠx6bx;y;zx45x79x15xfx32x11x12x13?1; 32 ;3,x73x67x4bx17x2dx1fxcxfx15x14x2fx44x1fx20x15x40x1fx24x25.
x5,x33x1axcxfx15xfx32x29x40x41x50x8x1cxcxfx15x40x41:
x
1 +
y
3
2
+ z3 = 1:
x4cx6bx7x50?3x+2y +z?3 = 0,x38x15x44x1fx20x3ex7ax22§(?3;2;1),x21x22x17P0(?1;0;0)x6bx4fxcxfx79,x25!OP0
x42x29x61x23x26x73x15x44x1fx20x39x71x15x15x35x2cx1cx5cx3c …2,x4x7bx10x3bx3c0,x21x2cx7ax44x1fx20x22(?3;2;1),x38x15x40x1fx24
x25x22
3
p14
14 ;
p14
7 ;
p14
14
.
5,x73x4ezx45x3fx42xcxf2x+y?p5z?7 = 0x2a60–x35x15xcxfΠx15x40x41.
x5,x4ezx45x15xcxfx15x44x1fx20x2cx22” = (A;B;0),x38x2cx42xaxbxcxfx2a60–x35,x23x24 2A+BpA2 +B2p10 =
§ 12, x5ex503A = Bx44A =?3B,x21xcxfx40x41x22x+3y = 0x443x?y = 0.
6,xaxbxcxfΠ, 4x?4y?2z +3 = 0,x17Px42xcxfΠx15x32x33x222,x73x17Px15x42x43.
x5,x18x2dx2ex31x32x15x17x22P(x;y;z),x4ax47
j4x?4y?2z +3j
6 = 2;
x5ex504x?4y?2z +3 = §12,x4x17Px15x42x43x13x37x66xcxdxcxf:
4x?4y?2z?9 = 0x3a 4x?4y?2z +15 = 0:
7,xaxbx37x66xcxfx4ex1dx29x64x1d,x73x38x15x15x3x35,x57x64x1dx17(0;0;1)x23x6bx15x37xfx35x15x3bx5c:
(x+2y +4z?3)(?3x+y?z?1) = 0:
x5,x37xcxfx15x44x1fx20x11x12x22(1;2;4)x42(?3;1;?1),x4ax3x35 x2dx2ecos = §?5p21p11 = § 5
p231
231,
x23x24 = arccos 5
p231
231 x44…?arccos
5p231
231,
x22x64x1dx17(0;0;1)x23x6bx15x37xfx35,x78x67x4fx17x2ax3cx37x66xcxfx15x33x5bx11x12x22 1p21 x42?2p11,x4ex3cx38x15x5f
x59,x21x4fx23x73x37xfx35x15x3bx5cx22…?arccos 5
p231
231,
8,x6bx2ex35x57x55x6ax1d,x73x1dx1ex17x8xcxfx15x32x33.
(1)x17(2;1;4),xcxf2x?y +4z?12 = 0;
(2)x17(?1;0;5),xcxfx?3y +5z?2 = 0.
x5,(1) d = j4?1+16?12jp21 =
p21
3,
(2) d = j?1+25?2jp35 = 22
p35
35,
9,x6bx2ex35x57x55x6ax16,x18xcxfΠix15x40x41x22
Aix+Biy +Ciz +Di = 0; i = 1;2:
￠ 12 ￠
x3fx77x37xcxfx65x3,x73x38x15x3x2ax15x37xfx35x15x35xcx11xfx15x40x41.
x5,x17P(x;y;z)x6bΠ1x42Π2x15x6ex66x37xfx35x15x35xcx11xfx79x62x3fx63x62x1cx17x8x77x37x66xcxfx15x32x33x65x56.
x21x4fx17Px2cx2dx2ex40x41
jA1x+B1y +C1z +D1jp
A21 +B21 +C21 =
jA2x+B2y +C2z +D2jp
A22 +B22 +C22,
x23x24x35xcx11xfx15x40x41x22
A1x+B1y +C1z +D1p
A21 +B21 +C21 = §
A2x+B2y +C2z +D2p
A22 +B22 +C22,
10,x73x8x37x66x1bx1dx15xcxfx15x32x33x22x1dx65kx15x17x15x42x43x40x41.
x5,x18x37x66x1bx1dxcxfx15x40x41x22
Πi, Aix+Biy +Ciz +Di = 0; i = 1;2:
x18P(x;y;z)x17x8Π1x42Π2x15x32x33x39x65x22k,x4ax47
jA1x+B1y +C1z +D1jp
A21 +B21 +C21 = k
jA2x+B2y +C2z +D2jp
A22 +B22 +C22,
x21x4fPx17x15x42x43x40x41x22
A1x+B1y +C1z +D1p
A21 +B21 +C21 = §k
A2x+B2y +C2z +D2p
A22 +B22 +C22,
11,x6bx2ex35x57x55x6ax1d,xaxbxcxfΠx15x40x41x22Ax+By +Cz +D = 0,x73Πx2ax3cxOyxcxfx15x2x75xf
Π0x15x40x41x3ax2ax3cx57x55x4bx17x15x2x75xfΠ00x15x40x41.
x5,x18x17P0(x0;y0;z0) 2 Π0,x4aP0x2ax3cxOyxcxfx15x2x75x17x13(x0;y0;?z0),x1cx17x2cx6bxcxfΠx79,x21x47
Ax0 +By0?Cz0 +D = 0,x23x24Π0x15x40x41x22Ax+By?Cz +D = 0.
x43x5,x18x17P00(x00;y00;z00) 2 Π00,x4aP00x2ax3cx4bx17x15x2x75x17x13(?x00;?y00;?z00),x1cx17x2cx6bxcxfΠx79,
x21x47?Ax00?By00?Cz00 +D = 0,x23x24Π00x15x40x41x22Ax+By +Cz?D = 0.
x1 x2 5–5
1,x7cx7dx1dx1ex2ex74x42xcxfx15x2fx59x2ax6a,x8x1fx65x3,x4ax73x38x15x15x3x17x42x15x35.
(1)x2ex74 x?12 = y +3?1 = z +25 x42xcxf4x+3y?z +3 = 0;
(2)x2ex74
8<
:
x = 1?2t
y = 2?4t
z =?1+5t
x42xcxfx+2y +2z?7 = 0;
(3)x2ex74
‰ x?y +z = 5
x+y?z =?1 x42xcxf2x+y +z?5 = 0.
x5,(1)x2ex74x15x40x1fx1fx20x13? = (2;?1;5),xcxfx15x44x1fx20x13” = (4;3;?1),x21x22(?;”) = 0,x2ex74x79x15
x17(1;?3;?2)x51x2dx2excxfx40x41,x23x24x2ex74x6bxcxfx7b.
(2)x2ex74x15x40x1fx1fx20x13? = (?2;?4;5),xcxfx15x44x1fx20x13” = (1;2;2),x21x22(?;”) = 0,x2ex74x79x15x17
(1;2;?1)x55x2dx2excxfx40x41,x23x24x2ex74x42xcxfxcxd.
(3)x2ex74x15x55x12x40x41x13 x?20 = y +32 = z2,x42xcxfx40x41x60x2bx7x73x50x3x17(2;?1;2),x2ex74x15x40x1f
x1fx20x13? = (0;2;2),xcxfx15x44x1fx20x13” = (2;1;1),x18x2ex74x42xcxfx15x3x35x22,x4asin = 44p3 =
p3
3,x23
x24x15x35 = arcsin
p3
3,
2,x73x4ex17A(3;?1;1)x3fx42xcxfΠ, x+y +z = 1x22x2ex15x2ex74x40x41.
￠ 13 ￠
x5,x21x23x73x2ex74x42xcxfx22x2e,x2ex74x15x40x1fx1fx20x13? = (1;1;1),x21x2ex74x40x41x22 x?31 = y +11 =
z?1
1,
3,x7cx7dx1dx1ex2ex74x71x15x2ax6a,x57x73x38x15x15x15x35,x2x3cx65x3x15x2ex74x57x73x3x17:
(1) L1, x+14 = y?32 = z?13,L2, x?51 = y +13 = z +12 ;
(2) L1, x+14 = y?12 = z?23,L2, x+26 = y +310 = z7,
x5,(1)x3dx3ex3dx34x2cx37x619.2,xdx1ex29
=
flfl
flfl
flfl
6 4 1
4 2 3
2 3 2
flfl
flfl
flfl =?30 6= 0;
x23x24x37x2ex74x5fxf,x18x15x35x22,x4acos = 16p29p14 = 8
p406
203,x21 = arccos
8p406
203,
(2)xdx1ex29
=
flfl
flfl
flfl
1 4 6
4 2 10
2 3 7
flfl
flfl
flfl = 0;
x23x24x37x2ex74x28xf,x51x21x38x15x15x40x1fx1fx20x55xcxd,x21x37x2ex74x65x3,x18x15x35x22,x4acos = 13
p5365
1073,x21
= arccos 13
p5365
1073, x2dx60x2bx40x41,x73x50x3x17x22
1;2; 72
·
.
4,xaxbx2ex74Lx15x40x41x13
‰ x+y +z?4 = 0
2x?y?z +1 = 0 x73x17A(3;2;?1)x8Lx15x32x33.
x5,x2ex74x15x40x1fx1fx20x13? =
flfl
flfl 1 1
1?1
flfl
flfl;?
flfl
flfl1 1
2?1
flfl
flfl;
flfl
flfl1 1
2?1
flfl
flfl
= (0;3;?3),x3fx17(1;2;1)x6bx2ex74
x79,x21x17Ax8x2ex74x15x32x33x22
d = j(2;0;?2)￡(0;1;?1)jp2 = p6:
5,x19x53x2ex74
L1,
8
<
:
x = 3+3t
y = t
z = 1
x42 L2,
8
<
:
x =?1+u
y = 2
z = u
(x3cx16t;ux13x5exe)
x13x5fxfx2ex74,x57x73x38x15x15x66x22x74x3ax37x2ex74x71x15x32x33.
x5,x2ex74L1x72x4ex17M1(3;0;1),x40x1fx1fx20x13?1 = (3;1;0),x2ex74L2x72x4ex17M2(?1;2;0),x40x1fx1fx20x13
2 = (1;0;1),x21x4f
1 ￡?2 = (1;?3;?1):
x3cx13L1x42L2x15x32x33x13
d = j(
!M
1M2;?1 ￡?2)j
j?1 ￡?2j =
9p
11 =
9p11
11,
x4eM1;?1;?1 ￡?2x64x1dx15xcxfΠ1x15x40x41x13
flfl
flfl
flfl
x?3 3 1
y 1?3
z?1 0?1
flfl
flfl
flfl =?x+3y?10z +13 = 0:
x4eM2;?2;?1 ￡?2x64x1dx15xcxfΠ2x15x40x41x13
flfl
flfl
flfl
x+1 1 1
y?2 0?3
z 1?1
flfl
flfl
flfl = 3x+2y?3z?1 = 0:
￠ 14 ￠
x6bx4cx7x3ex50x66x22x74x40x41x22 (
x?3y +10z?13 = 0
3x+2y?3z?1 = 0:
6,xaxbx2ex74L,
‰ x?y?4z +12 = 0
2x+y?2z +3 = 0 x68x1dx17P0(2;0;?1),x73P0x2ax3cLx15x2x75x17.
x5,x14x15x72x4exbxcx17x2ax3cxcxfΠ1, x?y?4z + 12 = 0x68Π2, 2x + y?2z + 3 = 0x15x33x5bx19x64
x1dP0x2ax3cLx15x2x75x17P00,P0(2;0;?1)x2ax3cΠ1x15x33x5b–1 = 18p18 = p18,x2ax3cΠ2x15x33x5b–2 = 93 = 3.
x23x24x2x75x17 P00(x0;y0;z0) x2ax3c Π1 x15x33x5b –01 = x0?y0?4z0 +12p18 =?–1 =?p18,x2ax3c Π2 x15x33x5b
–02 = 2x0 +y0?2z0 +33 =?–2 =?3,x21x4f
( x?y?4z +12 =?18
2x+y?2z +3 =?9; x4
( x?y?4z =?30
2x+y?2z =?12,(*)
x51x21x22!P0P00x2cx42x2ex74Lx22x2e,x4
flfl
flfl
flfl
x0?2 1 2
y0?1 1
z0 +1?4?2
flfl
flfl
flfl = 6x0?6y0 +3z0?9 = 0; (**)
x60x2bx4f3x66x40x41x2dx50,x0 = 0,y0 = 2,z0 = 7,x4x2x75x17x15x57x55x22(0;2;7).
7,x2ex74x4ex17(2;?3;5)x3fx42x34x31x57x55x45x15x72x1fx3x2ax56x35,x73x17P(1;?2;3)x8x4fx2ex74x15x32x33.
x5,x10x11x2ex74x40x41x22
x+2
1 =
y +3
1 =
z?5
1,
x23x24x17P(1;?2;3)x8x77x31x2ex74x15x32x33x22
d = j(1;?1;2)￡(1;1;1)jp3 =
p42
3,
8,x73x72x4ex37x2ex74 x?1?1 = y8 = z?5?3 x3a
8<
:
x = 3+4t
y = 21+5t
z =?11?10t
x15x3x17,x3fx42x77x37x2ex74x6dx22x2ex15x2ex74x40x41.
x5,x21x1cx2ex74x42xaxbx37x2ex74x6dx22x2e,x25xaxbx2ex74x15x40x1fx11x12x22?1 = (?1;8;?3),?2 = (4;5;?10),x21
x23x73x2ex74x15x40x1fx1fx20x22? =?1 ￡?2 = (?65;?22;?37),x77x37x31x2ex74x15x3x17x3ex73x50x22(?1;16;?1),x21x23
x73x2ex74x15x40x41x22
x+1
65 =
y?16
22 =
z +1
37,
9,x73x6bxcxf2x+3y +4z?9 = 0x79x6ax4ex17(1;1;1)x3fx42xOyxcxfx3x2ax6x3bx35x15x2ex74.
x5,x18x23x73x2ex74x15x40x1fx1fx20x22? = (A;B;C),x21x2ex74x6bxcxfΠ, 2x + 3y + 4z? 9 = 0x79,x23
x242A + 3B + 4C = 0,xOy xcxfx15x40x41x22z = 0,x14x15x6bx27x8x16xax6axbx60,x62x3fx63x62xcxfx79x15x2ex74
x42x37x66xcxfx15x3x74x22x2ex52,x77x31x2ex74x42x16x48xcxfx15x3x35x50x8x1cx3b,x25x77x37x66xcxfx15x3x74x15x40x1fx1fx20x13
(2;3;4)￡(0;0;1) = (3;?2;0),x21x4f?x42(3;?2;0)x22x2e,x43A?2B = 0,x2dx50A = 23 B,C =? 1312 B,x21
x23x73x2ex74x15x40x41x22
x?1
8 =
y?1
12 =
z?1
13,
10,x73x1dx1ex37x2ex74x71x15x32x33,x8x37x2ex74x47x28x22x74,x73x25x38x15x15x66x22x74x15x40x41.
￠ 15 ￠
(1)
‰ 2x+2y?z?10 = 0
x?y?z?22 = 0; x42
x+7
3 =
y?5
1 =
z?9
4 ;
(2)
8<
:
x = 3t?5
y = 2t?5
z =?2t+1;
x42
‰ x?y +z +5 = 0
x+2y?z?14 = 0:
x5,(1)x3dx48x31x2ex74x4cx2ax55x12x40x41x22
x?6
3 =
y +6
1 =
z +10
4 ;
x21x4fx37x2ex74xcxd,x32x33x22
d = j(13;?11;?19)￡(3;?1;4)jp26 =
p16250
p26 = 25:
(2)x2ex74L1x72x4ex17M1(?5;?5;1),x40x1fx1fx20x13?1 = (3;2;?2),x2ex74L2x72x4ex17M2(4;1;?8),x40x1fx1f
x20x13?2 = (?1;2;3),x21x4f
1 ￡?2 = (10;?7;8):
x3cx13L1x42L2x15x32x33x13
d = j(
!M
1M2;?1 ￡?2)j
j?1 ￡?2j =
24p
213 =
24p213
213,
x4eM1;?1;?1 ￡?2x64x1dx15xcxfΠ1x15x40x41x13
flfl
flfl
flfl
x+5 3 10
y +5 2?7
z?1?2 8
flfl
flfl
flfl = 2x?44y?41z?169 = 0:
x4eM2;?2;?1 ￡?2x64x1dx15xcxfΠ2x15x40x41x13
flfl
flfl
flfl
x?4?1 10
y?1 2?7
z +8 3 8
flfl
flfl
flfl = 37x+38y?13z?290 = 0:
x6bx4cx7x3ex50x66x22x74x40x41x22 (
2x?44y?41z?169 = 0
37x+38y?13z?290 = 0:
11,xaxbx48x17P(a;b;c) (abc 6= 0),x4ePx17x1fx28x66x57x55xfx2fx22x74,x22x2ex11x12x22L;M;N,x73x53,OPx42
x28x66xfOMN;ONL;OLMx15x3x35x65x56.
x3x4,x3dx3ex31x18,x22x2ex11x12x22L(a;b;0),M(a;0;c),N(0;b;c),x4axfOMN x15x44x1fx20x3ex7ax22”1 =
!OM ￡!ON = (?bc;?ac;ab),x6bx6cx6d,xfOLMx15x44x1fx20x3ex7ax22”
2 = (bc;?ac;?ab),xfONLx15x44x1fx20x3e
x7ax22”3 = (?bc;ac;?ab),x25?!OP = (a;b;c),x4ex3c
!OP ￠”
1 =
!OP ￠”
2 =
!OP ￠”
3 =?abc;
x3exbOPx42x773x66xfx15x3x35x65x56.
12,xaxbx37x31x5fxfx2ex74L1x3aL2,x73x53x4cx11L1x79x0x48x17x3aL2x79x0x48x17x15x74x78x15x16x17x42x43x13x66x22
x74x78x15x22x2excx11xf.
x3x4,x3bx62x7bx7cx57x55x6ax3ex27L1x22xx45,x40x41x22
( y = 0
z = 0; L2x15x40x41x4ax22
( kx?y = 0
z = a; (ak 6= 0),x18
P1(x1;0;0)x22L1x79x0x1x48x66x17,P2(x2;kx2;a)x13L2x79x0x1x17,x4aP1P2x15x16x17x57x55x22
x
1 +x22 ; kx22 ; a2
·
.
￠ 16 ￠
x4x4fx42x43x2dx2ex5exex40x41 8
>><
>>:
x = 12 u+ 12 v
y = k2 u
z = a2 ;
u;vx22x5exe.
x10x11x13xcxfz = a2, x77x67x13x37x31x5fxfx2ex74x15x66x22x74x78x15x22x2excx11xf.
13,xaxbx2ex74Lx72x4ex17(1;1;0)x3fx42x2ex74
L1, x?14 = y?2 = z1 ; L2, x4 = y +3?3 = z?1?2
x22x2e,x73x2ex74Lx6bx28x66x57x55xfx79x15x5dx17x15x40x41.
x5,x21x22Lx42L1;L2x6dx22x2e,x23x24x3ex7aLx15x40x1fx1fx20x22? = (4;?2;1)￡(4;?3;?2) = (7;12;?4),x4a
Lx15x40x41x22
x?1
7 =
y?1
12 =
z
4,
x38x6bxOy xcxfz = 0x79x15x16x17x40x41x22 x?17 = y?112 = z0,x6byOz xcxfx = 0x79x15x16x17x40x41x22
x
0 =
y?1
12 =
z
4,x6bxOzxcxfy = 0x79x15x16x17x40x41x22
x?1
7 =
y
0 =
z
4,
14,x73x4ex17(2;?3;?1)x3fx42x2ex74
x?1
2 =
y +1
1 =
z
1
x22x2ex65x3x15x2ex74.
x5,x18x23x73x2ex74x15x40x1fx1fx20x22? = (A;B;C),x21x38x40x41x42xaxbx2ex74x22x2e,x21x47?2A?B + C = 0.
x12x7ex25x23x73x2ex74x15x5exex40x41
8
><
>:
x = 2+At
y =?3+Bt
z =?1+Ct;
x51x52xaxbx2ex74x15x40x41,x50
1+At
2 =
2+Bt
1 =
1+Ct
1,
x4ex79x29x3ex50(3A?B+5C)t = 0,x4ex3ct = 0x10x11x55x13x2d,x25x37x2ex74x65x3x58x54tx48x1dx47x2d,x21x4f3A?B+5C =
0,x6x7x2dx50A, B, C = 4,?13,?5,x43x25x2ex74x40x41x22
x?2
4 =
y +3
13 =
z +1
5,
x1 x2 5–6
1,x6bx55x12x53x27x1ex50x70x71R4x16,x73x1fx20flx6bx4ex1fx20fi1,fi2,fi3x7x2ax15x0x70x71Wx79x15x72x3x16x17,x18
(1) fi1 = (2;2;?3;1),fi2 = (?2;1;?2;3),fi3 = (1;2;?3;2),fl = (1;1;?2;1);
(2) fi1 = (?1;2;?1;1),fi2 = (2;?1;1;0),fi3 = (0;1;?1;2),fl = (1;2;?1;0).
x5,(1) x18fl = x1fi1 + x2fi2 + x3fi3 + fl2,x3cx16fl2 = fl? x1fi1? x2fi2? x3fi3 2 W?,x4ex56x29
(fl2;fii) = 0,i = 1;2;3,x3ex24xcx25x24x1dx3x48x74x26x40x41x42:
8>
<
>:
18x1 +7x2 +17x3 = 11
7x1 +18x2 +12x3 = 6
17x1 +12x2 +18x3 = 11
x2dx50(x1;x2;x3) =
1
2 ;
1
12 ;
1
12
·
,x21x4fflx6bWx79x15x72x3x16x17x22:
1
2 fi1 +
1
12 fi2 +
1
12 fi3 =
11
12 ;
5
4 ;?
23
12 ;
11
12
:
￠ 17 ￠
(2)
1
2 ;2;0;
1
2
·
.
2,x18A 2 Mn(R),B 2Rn,x53x54:x32x6axex74x26x40x41x42AX = Bx47x2dx15x30x11x40x26x31x32x13Bx42x40x41x42
ATX = 0x15x2dx70x71x72x3.
x3x4,()) x1eAX = Bx47x2d,x4ax47C = (c1 c2 ￠￠￠ cn)Tx27x50B = AC,x3cx13x2ATX = 0x15x0x1x2d
D = (d1 d2 ￠￠￠ dn)T,x47
DTB = DTAC = (ATD)TC = 0;
x23x24Bx42ATX = 0x15x2dx70x71x72x3.
(()x18ATX = 0x15x2dx70x71x22W1,Ax15x1ex1fx20x42x2ex2ax15x0x70x71x22W2,x4aW1? W2,x51x21dimW1 =
n?rankA = n?dimW2,x23x24V = W1 'W2,x43x25W2 = W?1, xaxbB? W1,x3ex50B 2 W2,x4Bx3ex4e
Ax15x1ex1fx20x42x74x26x1bx1c,x3cx13x31x6bC 2Rnx27x50B = AC.
3,x18V1,V2x13x53x27x1ex50x70x71V x15x37x66x0x70x71,x3fV1x15x46xex5cx3cV2x15x46xe,x53x54,V2x16x40x47x48x6e
x6fx1fx20x72x3x3cV1x16x23x47x1fx20.
x3x4,x4ex37x616.2,V = V1 'V?1,dimV?1 = n?dimV1.
dim(V2 \V?1 ) = dimV2 +dimV?1?dim(V2 +V?1 )
> n?dimV1 +dimV2?n
= dimV2?dimV1 > 1:
x23x24V2 \V?1 6= 0,x31x6bx6ex6fx1fx20fi 2 V2 \V?1,x4fi 2 V2,fi? V1.
4,x18Ux22nx46x53x27x1ex50x70x71V x15x0x70x71,x53x54,?U?￠? = U.
x3x4,x21x22Ux15x1fx20x6dx42U?x72x3,x21x4fU (U?)?,x51x21
dim(U?)? = n?dimU? = n?(n?dimU) = dimU;
x21x4fU = (U?)?.
5,x18V1,V2x22nx46x53x27x1ex50x70x71V x15x37x66x0x70x71,x53x54:
(1) (V1 +V2)? = V?1 \V?2 ;
(2) (V1 \V2)? = V?1 +V?2,
x3x4,(1)x1efi 2 (V1 +V2)?,x4afi? V1x3ffi? V2,x43x25fi 2 V?1 \V?2, x23x24(V1 +V2)? V?1 \V?2,
x8x1ffi 2 V?1 \V?2,x4afi? V1x3ffi? V2,fi? V1 + V2,x23x24fi 2 (V1 + V2)?,x77x58x54V?1 \V?2
(V1 +V2)?.
x51x79x4x47(V1 +V2)? = V?1 \V?2,
(2) (V1 \V2)? = ￡(V?1 )? \(V?2 ) =
h?
V?1 +V?2 ￠?
i?
= V?1 +V?2,
6,x18Wx22x53x27x1ex50x70x71V x15x0x70x71,fix13V x15x48x66x1fx20,x1dx4dfix8Wx15x32x33
d(fi;W) = jfi?fi0j;
x3cx16,fi0x22fix6bWx79x15x72x3x16x17.
x53x54:x8x1ffi1;fi2;￠￠￠ ;fimx22Wx15x7a,x4a
d(fi;W) =
s
jG(fi1;fi2;￠￠￠ ;fim;fi)j
jG(fi1;fi2;￠￠￠ ;fim)j,
x77x1ex15G(￠￠￠)x13x1fx20x42x15x5ax60x5bx5dx5e(x5fx60x615–3.10).
￠ 18 ￠
x3x4,x18fi0 =
mP
i=1
xifii,x43
(fi?fi0;fij) =
fi?
mX
i=1
xifii;fij
!
= 0; j = 1;￠￠￠ ;m;
x3ex50 0
B@
(fi;fi1)
...
(fi;fim)
1
CA = G(fi
1;￠￠￠ ;fim)
0
B@
x1
...
xm
1
CA;
x4ex3cfi1;￠￠￠ ;fimx74x26x2cx2a,x23x24G = G(fi1;￠￠￠ ;fim)x3exc(x5ex5fx61x605–3.10),x21x4f
0
B@
x1
...
xm
1
CA = G?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA:
fi0 = (fi1 ￠￠￠ fim)
0
B@
x1
...
xm
1
CA = (fi
1 ￠￠￠ fim)G?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA:
x23x24
d(fi;W)2 = (fi?fi0;fi?fi0)
=
0
B@fi?(fi
1 ￠￠￠ fim)G?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA;fi?(fi
1 ￠￠￠ fim)G?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA
1
CA
= (fi;fi)?2((fi;fi1) ￠￠￠ (fi;fim))G?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA
+((fi;fi1) ￠￠￠ (fi;fim))G?TGG?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA
= (fi;fi)?((fi;fi1) ￠￠￠ (fi;fim))G?1
0
B@
(fi;fi1)
...
(fi;fim)
1
CA
= 1jGj
2
64(fi;fi)jGj?((fi;fi
1) ￠￠￠ (fi;fim))G?
0
B@
(fi;fi1)
...
(fi;fim)
1
CA
3
75
= 1jGj
flfl
flfl
flfl
flfl
fl
(fi1;fi1) ￠￠￠ (fi1;fim) (fi1;fi)
(fi2;fi1) ￠￠￠ (fi2;fim) (fi2;fi)
...,..,..,..
(fi;fi1) ￠￠￠ (fi;fim) (fi;fi)
flfl
flfl
flfl
flfl
fl
= jG(fi1;￠￠￠ ;fim;fi)jjGj,
) d(fi;W) =
s
jG(fi1;fi2;￠￠￠ ;fim;fi)j
jG(fi1;fi2;￠￠￠ ;fim)j,
7,x18V1,V2x22x53x27x1ex50x70x71V x15x37x66x0x70x71,x;y 2 V,x74x26x74x36L1 = x+V1,L2 = y +V2x39x71
￠ 19 ￠
x15x32x33x1dx4dx22
d(L1;L2) = minjfi?flj; 8fi 2 L1; fl 2 L2:
x53x54,d(L1;L2) = d(x?y;V1 +V2).
x3x4,x4eV = (V1+V2)'(V1+V2)?,x3ex50x?y = fl1?fi1+–,x3cx16fi1 2 V1,fl1 2 V2,– 2 (V1+V2)?.
x3cx13
d(x?y;V1 +V2) = j–j = j(x+fi1)?(y +fl1)j> d(L1;L2):
x4dx39,x2x0x1x15fi = x+fi1 2 L1,fl = y +fl1 2 L2,x49
fi?fl = (x?y)+(fi1?fl1) = +–;
x3cx16 2 V1 +V2,– 2 (V1 +V2)?,x4a
x?y = (?fi1 +fl1)+–:
x3cx13
jfi?flj2 = j +–j2 = j j2 +j–j2 >j–j2 = d(x?y;V1 +V2)2:
(x3cx16j +–j2 = j j2 +j–j2x13x21x22? –.) x23x24
d(L1;L2) = minjfi?flj> d(x?y;V1 +V2):
x6x2x3ex50d(L1;L2) = d(x?y;V1 +V2).
8,x73x37x66xcxfL1 = x+L(fi1;fi2)x42L2 = y +L(fl1;fl2)x39x71x15x32x33,x3cx16
fi1 = (1;?2;0;?3); fi2 = (2;?2;1;2); x = (4;5;3;2);
fl1 = (1;0;1;1); fl2 = (1;?2;0;?1); y = (1;?2;1;?3):
x5,W = L(fi1;fi2)+L(fl1;fl2) = L(fi1;fi2;fl1),x23x24
d(L1;L2) = d(x?y;W) =
s
jG(fi1;fi2;fl1;(x?y))j
jG(fi1;fi2;fl1)j =
r 324
36 = 3:
9,x73x1dx1ex40x41x15x6x5cx62xfx2d,8
>>>>
<
>>>>
:
3:4x?1:6y = 1
3:3x?1:7y = 1
3:2x?1:5y = 1
2:6x?1:1y = 1:
x5,A =
0
BB
B@
3:4?1:6
3:3?1:7
3:2?1:5
2:6?1:1
1
CC
CA,B =
0
BB
B@
1
1
1
1
1
CC
CA.
x4ax6x5cx62xfx2d(x;y)x22x74x26x40x41ATAX = ATBx15x2d,x2dx77x66x40x41,x50(
x … 0:69
y … 0:78
x1 x2 5–7
1,x18A = (aij) 2 Mn(R)x22x72x3x5dx5e,x3fjAj = 1.
x53x54,aij = Aij,x3cx16Aijx22aijx15x51xex24x0x29.
x3x4,x21x22jAj = 1,x23x24AA? = E,x43x25
A? = A?1 = AT;
￠ 20 ￠
x37x38x65x62x7x3ex50
aij = Aij; i;j = 1;￠￠￠ ;n:
2,x18x41 x13x53x27x1ex50x70x71V x15x48x66x3dx4a.
x53x54:x8x1fx41 x46x47x7bx10x55x3d,x4x2x23x47x15fi;fl 2 V,(x41fi;x41fl) = (fi;fl),x20x30x38x48x1dx13x74x26x15,x21x25
x13x72x3x3dx4a.
x3x4,x2x0x1x15fi;fl 2 V,x47
(x41(fi+fl)?x41fi?x41fl;x41(fi+fl)?x41fi?x41fl)
= (x41(fi+fl);x41(fi+fl))?2(x41(fi+fl);x41fi)?2(x41(fi+fl);x41fl)
+(x41fi;x41fi)+2(x41fi;x41fl)+(x41fl;x41fl)
= (fi+fl;fi+fl)?2(fi+fl;fi)?2(fi+fl;fl)+(fi;fi)+2(fi;fl)+(fl;fl)
= ((fi+fl)?fi?fl;(fi+fl)?fi?fl) = 0:
x23x24x41(fi+fl) = x41fi+ x41fl.
x6bx6cx6d,
(x41(kfi)?kx41fi;x41(kfi)?kx41fi)
= (x41(kfi);x41(kfi))?2k(x41(kfi);x41fi)+k2(x41fi;x41fi)
= (kfi;kfi)?2k(kfi;fi)+k2(fi;fi) = 0:
x23x24x41(kfi) = kx41fi.
x21x4fx41 x13x74x26x3dx4a.
3,x18"1;"2;￠￠￠ ;"nx42fi1;fi2;￠￠￠ ;finx13x53x27x1ex50x70x71x15x37x66x54x3dx72x3x7a,x53x54,x31x6bx72x3x3dx4ax41,
x27
x41("i) = fii (i = 1;2;￠￠￠ ;n):
x3x4,x4ex3c"1;"2;￠￠￠ ;"nx13x74x26x70x71x15x7a,x21x4fx2dx2ex61x18x31x32x15x74x26x3dx4ax41 x48x1dx31x6b,x2x3cx0x1
x15x37x66x1fx20fi =
nP
i=1
ai"i,fl =
nP
i=1
bi"i,x47x41(fi) =
nP
i=1
aix41("i) =
nP
i=1
aifii,x41(fl) =
nP
i=1
bifii,x21x4f
(x41(fi);x41(fl)) =
nX
i=1
aifii;
nX
i=1
bifii
!
=
nX
i=1
aibi = (fi;fl):
x23x24x41 x13x72x3x3dx4a.
4,x73x1dx1ex72x3x40x5ex15x53x60x35:
(1)
0
BB
@
1
2
p3
2 0
p3
2
1
2 00 0 1
1
CC
A; (2)
0
@
0 0 1
0?1 0
1 0 0
1
A; (3)
0
BB
B@
p2
4
p6
4
p2
2
p2
4?
p6
4
p2
2p
3
2?
1
2 0
1
CC
CA.
x5,(1) = 0,`+? = 5…3 ;
(2) = …2,` = …2,? = …2 ;
(3)x4er33 = cos = 0,2 [0;…],x3ex50 = …2, x12x4er31 = sin? =
p3
2 x24x68r32 = cos? =?
1
2 x3ex50
= 2…3, x6x7x4er13 = sin` =
p2
2 x24x68r23 =?cos` =
p2
2 x3ex50` =
3…
4,
5,x18x17Px15x57x55x22(1;1;0),x73x22x45?!OPx5bx1ex32x40x1fx1bx1c …
6 x15x72x3x3dx4a.
￠ 21 ￠
x5,x5ex68x6a7.5,x1bx1cx45x15x14x2fx1fx20x13? =
p
2
2 ;
p2
2 ;0
,x49· =!k =
p
2
2 ;
p2
2 ;?1
,x63·
x40x1fx15x64x5dx1fx22x53,x4ex3c8
>>>>
>><
>>>
>>>:
x53 (?!i ) =?!i?2 (
!i ;·)
(·;·) · =
1
2
!i? 1
2
!j + p2
2
!k
x53 (?!j ) =?!j?2 (
!j ;·)
(·;·) · =?
1
2
!i + 1
2
!j + p2
2
!k
x53 (?!k ) =?!k?2 (
!k ;·)
(·;·) · =
p2
2
!i + p2
2
!j
x21x4fx53 x6bx7a?!i ;?!j ;?!k x1dx15x5dx5ex22
S =
0
BB
B@
1
2?
1
2
p2
2
12 12
p2
2p
2
2
p2
2 0
1
CC
CA:
x16x48x40xfx1bx1cx52?!k ;?…
6
x15x5dx5ex13
R =
0
BB
@
p3
2
1
2 0
12
p3
2 00 0 1
1
CC
A:
x6x7x50x8x23x73x72x3x3dx4ax15x5dx5ex22
SRS =
0
BB
B@
p3
4 +
1
2?
p3
4 +
1
2
p2
4
p3
4 +
1
2
p3
4 +
1
2?
p2
4
p2
4
p2
4
p3
2
1
CC
CA:
6,x73x72x3x3dx4a
x41(X) =
0
BB
B@
p2
2
p2
2 0
1
2?
1
2
p2
2
1
2?
1
2?
p2
2
1
CC
CAX
x15x1bx1cx45x42x1bx1cx35.
x5,x73x1bx1cx45x65x62x3cx73AX = Xx15x2dx1fx20X 2R3,x2dx50x1bx1cx45x15x40x1fx1fx20x13? = (p2+1;1;p2?1).
x22x73x1bx1cx35,x7ax48x66x42?x72x3x15x1fx20fi = (1;?2;?1),x4ax1bx1cx35 = hfi;x41(fi)i.
cos = (fi;x41(fi))jfij2 =? 34,
x51x21x3ex54x10
(?;fi;x41(fi)) =? 212 < 0;
x23x24x1bx1cx35 = … +arccos 34,
￠ 22 ￠
