x0x1x2x3
x0xbx2 xcxdxexfx10x11xcxdx12x13x14
x1 x2 3–1
1,x1ax6xbx44x2dx1dx1ex74x26x40x41x42:
(1)
8
>>>>
<
>>>>
:
2x1? x2 + x3 +3x4 =?4
x1 +3x2? x3 +4x4 = 6
2x1?2x2 +3x3? x4 = 3
3x1 +2x3 +2x4 = 3
(2)
8>
>>>
<
>>>
>:
x1?2x2 +3x3?x4 =?6
x1 + x2? x3 +x4 = 7
2x1? x2 + x3 = 1
x2 + x3 +x4 = 3
x5,(1) x1 =?5,x2 = 5,x3 = 8,x4 = 1.
(2) x1 = 4,x2 = 7,x3 = 0,x4 =?4.
2,x11x12x1ax5dx5ex15x5cx56xdx3dx4ax3ax1ex3dx4ax76x1dx1ex5dx5ex4cx22xdx79x70x5dx5ex3ax1ex79x70x5dx5e:
(1)
0
BB
B@
3 2 1 0 4
2 1 4 4?3
2 0 3 1?2
2 3?1 2 5
1
CC
CA;
(2)
0
BB
BB
B@
1 5 2 0 1
3?5 4 2 7
1?5 1 1 3
0 2 1?1 2
1?1 3?1 7
1
CC
CC
CA.
x5,(1)
0
BB
B@
3 2 1 0 4
0? 13 103 4? 173
0 0?11?15 18
0 0 0 1711? 1611
1
CC
CAx42
0
BB
B@
3 0 0 0 0
2 4 0 0 0
2 1 32 0 0
2 2? 103? 1718 0
1
CC
CA.
(2)
0
BB
BB
B@
1 5 2 0 1
0 2 1?1 2
0 0 4?4 12
0 0 0 0 0
0 0 0 0 0
1
CC
CC
CAx42
0
BB
BB
B@
1 0 0 0 0
3?2 0 0 0
1?1 0 0 0
0 1?8 0 0
1 1?16 0 0
1
CC
CC
CA.
3,x53x54:x74x26x40x41x42x15x3dx62x6b,x3dx34x6bx5cx56x3dx4ax4ex74x26x40x41x42x4cx2ax42x38x43x2dx15x74x26x40x41x42.
x3x4,(x69)
4,x53x54x5ex231.4.
x3x4,x2x5dx5eATx2cx1ax5ex231.3,x4aATx3ex24x6ax4ex48x6ax1ex5cx56xdx3dx4ax4cx2ax6bx4cxdx79x70x5dx5e,x76x79x53x3d
x4ax6cxdx3cx5dx5eAx15x1ex79,x6fx76Ax4cx2ax6bx4cx1ex79x70x5dx5e.
￠ 1 ￠
5,x6dxbx61:
(1)x74x26x40x41x42x15x2dxdx3ex24x68x2fx13x70x71x15x48x66x17xd,x20x30,x74x26x70x71x16x0x48x17xdx13x29x48x1dx13x6ex66x74
x26x40x41x42x15x2dxdx54x6f? x8x1fx13x77x6e,x20x30,x70xd,x14x17xdf(0;0;￠￠￠ ;0)gx42x37x17xdf(0;0;￠￠￠ ;0);(1;1;￠￠￠ ;1)g
x11x12x13x70x6ex15x74x26x40x41x42x15x2dxdx54x6f? x8x1fx55x13x77x6e,x20x30,x70x6ex15x17xdx71x13x6ex66x74x26x40x41x42x15x2dxdx54
x6f?
(2)x74x26x40x41x42x15x5cx56x3dx4ax4ex74x26x40x41x42x3dx2ax43x2dx15x74x26x40x41x42.x20x30,x37x66x43x2dx15x74x26x40x41x42x13
x29x48x1dx3ex24x72x4ex5cx56x3dx4ax64x4cx6f?
x5,(1)x11x72x70xdx42x14x17xdx30,x74x26x40x41x42x15x2dxdx54x48x1dx13x2cx73xd,x70xdx13x34x35x40x41x42x15x2dxd,x14
x17xdf(0;0;￠￠￠ ;0)gx3ex24x13x24x1dx40x41x42 8
>>>
>><
>>>>
>:
x1 = 0
x2 = 0
...
xn = 0
x15x2dxd,x74x26x40x41x42x15x2dxdx54x13x48x66x74x26x74x36,x2dxdx54x15x26x27x3ex5ex68x2,x6,x7x15x55x23.
(2)x6bx75x76x54x17x44x77x7xcx78x40x41“0 = 0”x15x4fx79x1d,x4fx22x23x13x72x64x15.
x1 x2 3–2
1,x1ax6xbx44x2dx1dx1ex74x26x40x41x42:
(1)
8>
>>>
<
>>>
>:
x1?2x2 +3x3? x4? x5 =2
x1 + x2? x3 + x4?2x5 =1
2x1? x2 + x3?2x5 =2
2x1 +2x2?5x3 +2x4? x5 =5
(2)
8
>>>>
<
>>>>
:
x1?2x2 +3x3? x4? x5 = 4
x1 + x2? x3 + x4?2x5 = 1
2x1? x2 + x3?2x5 = 3
2x1 +2x2?5x3 +2x4? x5 =?4
(3)
8
>><
>>:
x1?2x2 +x3 + x4 = 1
x1?2x2 +x3? x4 =?1
x1?2x2 +x3 +5x4 = 5
(4)
8
>>>>
<
>>>>
:
2x1 + x2 + x3 = 2
x1 +2x2 + x3 = 3
x1 + x2 +5x3 =?7
2x1 +2x2?3x3 = 12
(5)
8>
>>><
>>>>
:
2x1? 2x2 + x3? x4 + x5 =0
x1? 4x2 +2x3?2x4 +3x5 =0
4x1?10x2 +5x3?5x4 +7x5 =0
x1 + 2x2? x3 + x4?2x5 =0
x5,(1)x2cx2d.
(2) x3 = 1?x1 +2x2,x4 =?4x1 +5x2,x5 =?1?x1 +2x2,x1;x2x22x67x4ex7axbx20.
(3) x1 = 2x2?x3,x4 = 1,x2;x3x22x67x4ex7axbx20.
￠ 2 ￠
(4) x1 = 1,x2 = 2,x3 =?2.
(5) x1 = 13 x5,x2 = 16 (3x3?3x4 +5x5),x3;x4;x5x22x67x4ex7axbx20.
2,x7bx7c?,x27x40x41x42 8
>><
>>:
2x1? x2 + x3 + x4 = 1
x1 +2x2? x3 + 4x4 = 2
x1 +7x2?4x3 +11x4 =?
x47x2d,x57x73x38x15x48x7dx2d.
x5,x63x62? = 5x52x47x2cx7ex9x2d,x3cx48x7dx2dx22x1 = 15 (4?x3?6x4),x2 = 15 (3+3x3?7x4),x3;x4x22
x67x4ex7axbx20.
3,a,bx7ax8x4x52,x74x26x40x41x428
>>>>
<
>>>>
:
x1 + x2 + x3 + x4 + x5 = 1
3x1 +2x2 + x3 + x4?3x5 =a
x2 +2x3 +2x4 +6x5 = 3
5x1 +4x2 +3x3 +3x4? x5 = b
x47x2d? x6bx47x2dx15xfx21x1d,x73x48x7dx2d.
x5,x63x62a = 0;b = 2x52x47x2d,x3cx48x7dx2dx22x1 =?2 + x3 + x4 + 5x5,x2 = 3? 2x3? 2x4? 6x5,
x3;x4;x5x22x67x4ex7axbx20.
4,x53x54x40x41x42 8
>>>
>>>>
<
>>>
>>>>
:
x1?x2 = a1
x2?x3 = a2
x3?x4 = a3
x4?x5 = a4
x5?x1 = a5
x47x2dx15x30x11x40x26x31x32x13
a1 +a2 +a3 +a4 +a5 = 0:
x6bx47x2dx15xfx21x1d,x73x38x15x48x7dx2d.
x3x4,())x8x74x26x40x41x42x47x2d,x18(c1;c2;c3;c4;c5)x22x3cx48x66x2d,x76x38x51x52x4bx40x41x42,x57x76x28x29x65x17,
x4x50a1 +a2 +a3 +a4 +a5 = 0.
(() x8a1 + a2 + a3 + a4 + a5 = 0,x4ax4ex6x7x48x66x40x41x50x5 = x1 + a5,x47x48x51x52x4fx48x66x40x41,x50
x4 = a4 +a5 +x1,x3 = a3 +a4 +a5 +x1,x2 = a2 +a3 +a4 +a5 +x1,x76x2;x3;x4;x5x51x52x3dx48x66x40x41,
x50
x1?(a2 +a3 +a4 +a5 +x1) =?a2?a3?a4?a5 = a1:
x23x24x4bx40x41x42x15x48x7dx2dx228
>>>
><
>>>
>:
x2 = a2 +a3 +a4 +a5 +x1
x3 = a3 +a4 +a5 +x1
x4 = a4 +a5 +x1
x5 = a5 +x1
x1x22x67x4ex7axbx20.
5,x73x48x9x3ax29f(x) = a0x3+a1x2+a2x+a3,x27f(1) =?3,f(?1) =?7,f(2) =?1,f(?2) =?21.
x5,f(x) = x3?2x2 +x?3.
￠ 3 ￠
x1 x2 3–3
1,x18fi1 = (2;5;1;3),fi2 = (10;1;5;10),fi3 = (4;1;?1;1),x73x48x1fx20fi,x273(fi1?fi)+2(fi2 +fi) =
5(fi3 +fi).
x5,fi = (1;2;3;4).
2,xaxb3fi+4fl = (2;1;1;2),2fi?3fl = (?1;2;3;1),x73fix42fl.
x5,fi = 117 (2;11;15;10),fl = 117 (7;?4;?7;1).
3,x4ex1fx20flx1bx2ax1fx20fi1,fi2,fi3x15x74x26x42x54:
(1) fi1 = (1;1;1),fi2 = (1;1;?1),fi3 = (1;?1;?1),fl = (1;2;1);
(2) fi1 = (1;3;5),fi2 = (6;3;?2),fi3 = (3;1;0),fl = (5;8;8);
x5,(1) fl = fi1 + 12 fi2? 12 fi3.
(2) fl = 2fi1 +fi2?fi3.
4,x7cx12x1dx1ex1fx20x42x13x29x74x26x65x2a:
(1) fi1 = (1;1;1),fi2 = (1;2;3),fi3 = (1;3;6);
(2) fi1 = (3;2;?5;4),fi2 = (2;1;?3;?5),fi3 = (3;5;?13;11),fi4 = (4;5;?14;?3);
(3) fi1 = (1;?1;2;4),fi2 = (0;3;1;2),fi3 = (1;7;8;9),fi4 = (3;2;1;2);
(4) fi1 = (1;2;?1;4),fi2 = (9;1;2;?3),fi3 = (3;5;0;2),fi4 = (3;2;2;1),fi5 = (1;3;3;2).
x5,(1)x29; (2)x13; (3)x29; (4)x13.
5,x18a1;a2;￠￠￠ ;anx13x64x55x65x43x15xe,x49
fi1 = (1;a1;a21;￠￠￠ ;an?11 );
fi2 = (1;a2;a22;￠￠￠ ;an?12 );
:::::::::::::::::::::::::
fin = (1;an;a2n;￠￠￠ ;an?1n ):
x53x54:x0x48nx46x1fx20x6dx3ex24x4ex1fx20x42fi1;fi2;￠￠￠ ;finx74x26x1bx1c.
x3x4,x1fx20x42fi1;fi2;￠￠￠ ;finx75x2ax15xdx1ex29
jAj =
flfl
flfl
flfl
flfl
fl
1 a1 ￠￠￠ an?11
1 a2 ￠￠￠ an?12
￠￠￠ ￠￠￠ ￠￠￠ ￠￠￠
1 an ￠￠￠ an?1n
flfl
flfl
flfl
flfl
fl
=
Y
16j<i6n
(ai?aj) 6= 0;
x23x24fi1;fi2;￠￠￠ ;finx74x26x2cx2a.
x51x2x0x1x15nx46x1fx20fl,x1fx20x42fl;fi1;￠￠￠ ;finx74x26x65x2a,x43x25x1fx20flx3ex4ex1fx20x42fi1;fi2;￠￠￠ ;finx74
x26x1bx1c.
6,x18x1fx20x42fi1,fi2,fi3x74x26x2cx2a,x53x54:x1fx20x42fi1 +fi2,fi2 +fi3,fi3 +fi1x67x74x26x2cx2a.
x3x4,x18
k1(fi1 +fi2)+k2(fi2 +fi3)+k3(fi3 +fi1) = 0;
x4a
(k1 +k3)fi1 +(k1 +k2)fi2 +(k2 +k3)fi3 = 0:
￠ 4 ￠
x21x22fi1,fi2,fi3x74x26x2cx2a,x23x24 8
><
>:
k1 +k3 = 0
k1 +k2 = 0
k2 +k3 = 0
x2dx50k1 = k2 = k3 = 0,x23x24fi1 +fi2,fi2 +fi3,fi3 +fi1x74x26x2cx2a.
7,x53x54,fi1;fi2;￠￠￠ ;fis (x3cx16fi1 6= 0) x74x26x65x2ax15x30x26x31x32x13x45x46x47x48x66fii (1 < i 6 s) x3ex49
fi1;fi2;￠￠￠ ;fii?1x74x26x1bx1c.
x3x4,x21x22fi1;fi2;￠￠￠ ;fisx74x26x65x2a,x23x24x31x6bx55x33x22x6fx15xek1;k2;￠￠￠ ;ks,x27
k1fi1 +k2fi2 +￠￠￠+ksfis = 0:
x18k1;k2;￠￠￠ ;ksx16x6x7x48x66x55x22x6fx15x22ki,x4ai 6= 1(x29x4afi1 = 0x42x31x18x34x35),x43x25i > 1,x21
k1fi1 +￠￠￠+ki?1fii?1 =?kifii;
fii =? k1k
i
fi1? k2k
i
fi2?￠￠￠? ki?1k
i
fii?1:
8,x53x54:x8x1fx1fx20x42x15x48x66x3ax7fx42x74x26x65x2a,x4ax4fx1fx20x42x67x74x26x65x2a.
x3x4,x18x1fx20x42(II)x22(I)x15x3ax7fx42,x8x1fx20x42(I)x74x26x2cx2a,x4ax4ex6a3.9xb,(II)x67x74x26x2cx2a,x42xaxb
x34x35,x21x4fx1fx20x42x74x26x2cx2a.
9,x1dx1ex23x7dx13x29x2ax2b? x2x15,x17x24x53x54;x0x15,x1x25x4dx6a.
(1)x1efi1;fi2;￠￠￠ ;fisx74x26x65x2a,x4ax3cx16x73x48x66x6dx3ex4ex3cx24x1fx20x74x26x1bx1c;
(2)x1ex1fx20x42fi1;fi2;￠￠￠ ;firx74x26x2cx2a,x1fx20x42fl1;fl2;￠￠￠ ;flsx74x26x2cx2a,x4ax1fx20x42fi1;fi2;￠￠￠ ;fir;fl1;
fl2;￠￠￠ ;flsx67x74x26x2cx2a;
(3) x1ex1fx20x42fi1;fi2;￠￠￠ ;fis x74x26x2cx2a,x1fx20x42fl1;fl2;￠￠￠ ;fls x74x26x2cx2a,x4ax1fx20x42fi1 + fl1;fi2 +
fl2;￠￠￠ ;fis +flsx67x74x26x2cx2a;
(4)x1ex1fx20x42fi1;fi2;￠￠￠ ;firx74x26x65x2a,x4ax48x1dx31x6brx66x55x56x3cx6fx15xek1;k2;￠￠￠ ;kr,x27
k1fi1 +k2fi2 +￠￠￠+krfir = 0;
(5)x1ex1fx20x42fi1;fi2;￠￠￠ ;firx74x26x2cx2a,x4ax38x15x0x8x74x26x42x54x6dx55x56x3cx6f.
x5,(1)x0,x8fi1 = (0;0),fi2 = (1;1)x74x26x65x2a,x71fi2x55x3ex4efi1x74x26x1bx1c.
(2)x0,x8fi1 = (1;1),fi2 = (1;2)x74x26x2cx2a,fl1 = (2;2),fl2 = (0;1)x74x26x2cx2a,x71fi1;fi2;fl1;fl2x74x26
x65x2a.
(3) x0,x8 fi1 = (1;1),fi2 = (0;1),fl1 = (1;?1),fl2 = (1;2) x74x26x2cx2a,x71 fi1 + fl1 = (0;0),
fi2 +fl2 = (1;3)x74x26x65x2a.
(4)x0,x8fi1 = (0;0),fi2 = (0;1)x74x26x65x2a,x71x2x0x1x15k1 6= 0;k2 6= 0x6dx47k1fi1 +k2fi2 6= 0.
(5)x0,fi1;fi2;￠￠￠ ;firx15x6fx74x26x42x54x6fx56x3cx6f.
x1 x2 3–4
1,x6bx34x46x27x8x70x71R3x16,x1dx1exdx54Wx13x29x75x2aR3x15x74x26x0x70x71?
(1) W = f(a;b;c) 2R3 j (a;b;c)? (1;1;1)g;
(2) Wx13x2x17x6bx6ex2ex74x79x15x33x10x1fx20x23x75x2ax15xdx54;
(3) Wx13x42x70x71x16x6ex49x1dx6ex6fx1fx20(x0;y0;z0)x15x15x35x56x3cx1dx4x15x33x10x1fx20x23x75x2ax15xdx54.
x5,(1)x13; (2)x8x2ex74x4ex4bx17,x13;x29x4a,x55x13; (3)x15x35x56x3c …2,x13;x29x4a,x55x13.
￠ 5 ￠
2,x18V x22xex9Kx79nx46x1fx20x70x71,x7cx7dV x15x1dx1ex0xdWx13x29x75x2aV x15x74x26x0x70x71.
(1)x18fi1;fi2;￠￠￠ ;firx22V x16x1bx1dx15rx66x1fx20,
W = ffl 2 V j fi1;fi2;￠￠￠ ;fir;flx74x26x65x2ag;
(2)x18fi1;fi2;￠￠￠ ;firx22V x16x1bx1dx15rx66x1fx20,Wx13V x16x55x63x4efi1;fi2;￠￠￠ ;firx74x26x1bx1cx15x33x10x1fx20
x23x75x2ax15xdx54.
x5,(1)x13; (2)x55x13.
3,x18fi1;fi2;￠￠￠ ;firx22Knx16x1bx1dx15rx66x1fx20,x53x54:
W = f(c1;c2;￠￠￠ ;cr) j c1fi1 +c2fi2 +￠￠￠+crfir = 0g
x42x2aKrx15x0x70x71.
x3x4,x10x11W Krx3f(0;0;￠￠￠ ;0) 2 W,x43x25Wx6ex70.
x2x0x1x15(a1;￠￠￠ ;ar);(b1;￠￠￠ ;br) 2 Wx24x68k 2 K,x47
(a1 +b1)fi1 +￠￠￠+(ar +br)fir = a1fi1 +￠￠￠+arfir +b1fi1 +￠￠￠+brfir = 0+0 = 0:
x23x24
(a1;￠￠￠ ;ar)+(b1;￠￠￠ ;br) 2 W:
(ka1)fi1 +(ka2)fi2 +￠￠￠+(kar)fir = k(a1fi1 +a2fi2 +￠￠￠+arfir) = k￠0 = 0:
x23x24
k(a1;￠￠￠ ;ar) 2 W:
Wx2ax22Krx15x0x70x71.
4,x18W1;W2;￠￠￠ ;Wsx22Knx15sx66x74x26x0x70x71,W = W1 [W2 [￠￠￠[Ws,x53x54,Wx22Knx15x74x26
x0x70x71x15x30x11x40x26x31x32x13,x31x6bi(1 6 i 6 s),x27W = Wi.
x3x4,x30x11x26x13x10x11x15,x1dxfx53x40x26x26,x2sx1ax2x50x44.x62s = 1x52x22x23x10x11x2ax2b,x31x1dx22x23x2s?1
x2ax2b,xbx3W = W1 [W2 [￠￠￠[Ws,x8x1fW 6= Ws,x4ax3ex7afl 2 W nWs,x2x3cx0x1x15fi 2 Ws,x40x47
fl+kfi 2 W nWs (x43fl+kfi 2 Wsx24x68fi 2 Wsx3ex24x5ex50fl 2 Ws,x34x35),x62k = 1;￠￠￠ ;sx52,sx66x1fx20x16x40
x47x37x66x1fx20x4x3cx43x48x66Wi (1 6 i 6 s?1),x77x37x66x1fx20x65x5x7x3ex50fi 2 Wi,x21x4fWs W1[￠￠￠[Ws?1,
x3cx13W = W1 [￠￠￠[Ws?1,x33x1ax2x50x31x18,x3ex50x48x66i,1 6 i 6 s?1x27x50W = Wi,x22x23x2ax2b.
x1 x2 3–5
1,x73x4ex1dx1ex1fx20x23x2ex2ax15x74x26x0x70x71x15x7ax42x46xe:
(1) fi1=(2;1;11;2);fi2=(1;0;4;?1);fi3=(1;4;16;15);fi4=(2;?1;5;?6),fi5 = (1;6;22;23);
(2) fi1 = (1;?4;15;5;?4),fi2 = (0;7;29;?8;7),fi3 = (2;?1;1;1;?3),fi4 = (1;?4;3;5;?4).
x5,(1)x46xe2,x7afi1;fi2.
(2)x46xe4,x7afi1;fi2;fi3;fi4.
2,x18Wx22x1fx20x70x71V x15x0x70x71,fi1;fi2;￠￠￠ ;firx22Wx15x48x66x7a,fli =
rP
j=1
aijfij,i = 1;2;￠￠￠ ;r.
x53x54,fl1;fl2;￠￠￠ ;flrx67x13Wx15x7ax15x30x11x40x26x31x32x13fl
flfl
flfl
flfl
flfl
a11 a12 ￠￠￠ a1r
a21 a22 ￠￠￠ a2r
:::::::::::::::::::
ar1 ar2 ￠￠￠ arr
flfl
flfl
flfl
flfl
fl
6= 0:
￠ 6 ￠
x3x4,()) x18fl1;fl2;￠￠￠ ;flr x67x13W x15x7a,x18x5dx5eA = (aij),(k1;k2;￠￠￠ ;kr)x13x3x48x74x26x40x41x42
XA = 0x15x48x66x2d,x4
(k1 k2 ￠￠￠ kr)A = 0:
x4a
k1fl1 +k2fl2 +￠￠￠+krflr = (k1 ￠￠￠kr)A
0
B@
fi1
...
fir
1
CA = 0:
x21x22fl1;fl2;￠￠￠ ;flrx74x26x2cx2a,x23x24k1 = k2 = ￠￠￠ = kr = 0,x4XA = 0x7bx47x6fx2d,x23x24jAj6= 0.
(()x18
k1fl1 +k2fl2 +￠￠￠+krflr = 0;
x4a
(k1 ￠￠￠kr)A
0
B@
fi1
...
fir
1
CA = 0;
x21x22fi1;fi2;￠￠￠ ;firx74x26x2cx2a,x23x24
(k1 k2 ￠￠￠ kr)A = 0:
x4ex3cjAj 6= 0,x21x3x48x74x26x40x41x42x7bx47x6fx2d,x4k1 = k2 = ￠￠￠ = kr = 0,x3cx13fl1;fl2;￠￠￠ ;flrx74x26x2cx2a,x51
x21dimW = r,x23x24fl1;fl2;￠￠￠ ;flrx13Wx15x7a.
3,x18V x22xex9Kx79x15nx46x1fx20x70x71,x53x54:x2x0x8x3bx3cnx15x67x11xem,x48x1dx31x6bx4eV x15mx66x1fx20
x42x2ax15x1fx20x42,x27x3cx16x0x8nx66x1fx20x6dx75x2aV x15x7a.
x3x4,x4ex60x61 3–4.4 x15x22x1fx3ex24xbx60,V x55x3ex63x1bx1cx2ax38x15x47x73x9x66x5x74x26x0x70x71x15x57xd,x2
m > nx6cxdxex4fx2x50x44,x62m = nx52x22x23x2ax2b,x31x18xax6ax6x8x2dx2ex31x32x15m?1 > nx66x1fx20x15x1fx20x42
fi1;￠￠￠ ;fim?1,x4ex3cx16x0x1n?1x15x1fx20x7x2ax15x74x26x0x70x71x1fx22Wi (i = 1;￠￠￠ ;s),x4ax21V 6= SWi,x31x6b
x1fx20fim 62SWi (i = 1;￠￠￠ ;s),x4ax1fx20x42fi1;￠￠￠ ;fimx67x2dx2ex31x32.
x1 x2 3–6
1,?x7ax8x4x52,x40x41x42 8
>><
>>:
(?+3)x1 + x2 + 2x3 =?
x1 +(1)x2 + x3 =2?
3(?+1)x1 +?x2 +(?+3)x3 =3?
x47x2d? x6bx47x2dx15xfx21x1d,x73x25x48x7dx2d.
x5,x6axexdx1ex29x56x3c?2(1),x62? 6= 0;1x52,x40x41x42x47x2cx48x2d:
8
>>><
>>>:
x1 =31
x2 =?+31
x3 = 31 ;
x62? = 0x52,x48x7dx2dx22,x1 =?x3,x2 = x3,x3x13x67x4ex7axbx20;
x62? = 1x52,x4bx40x41x42x2cx2d.
￠ 7 ￠
2,a,bx7ax8x4x52,x40x41x42 8
>><
>>:
ax1 + x2 +x3 =4
x1 + bx2 +x3 =3
x1 +2bx2 +x3 =4
x47x2d? x6bx47x2dx15xfx21x1d,x73x25x48x7dx2d.
x5,(a)x62a 6= 1x3fb 6= 0x52,x40x41x42x47x2cx48x2d:
8>
>><
>>>:
x1 = 2b?1b(a?1)
x2 = 1b
x3 = 2ab?4b+1b(a?1) ;
(b)x62b = 0x52,x44x62a = 1,b 6= 12 x52,x4bx40x41x42x2cx2d;
(c)x62a = 1,b = 12 x52,x48x7dx2dx22,x1 = 2?x3,x2 = 2,x3x13x67x4ex7axbx20.
3,x73x1dx1ex3x48x74x26x40x41x42x15x7ax8x2dx6a:
(1)
8
>>>>
<
>>>>
:
x1 + x2 + x3 + x4 + x5 =0
3x1 +2x2 + x3 + x4? x5 =0
5x1 +4x2 +3x3 +3x4 + x5 =0
x2 +2x3 +2x4 +4x5 =0
(2)
8
>><
>>:
3x1 +2x2? 5x3 + 4x4 =0
3x1? x2 + 3x3? 3x4 =0
3x1 +5x2?13x3 +11x4 =0
(3)
8
>>>>
<
>>>>
:
x1 + x2 + x3 + x4 + x5 =0
2x1 +2x2 + x3 + x4?2x5 =0
5x1 +4x2?3x3 +4x4 + x5 =0
x2 +6x3? x4?4x5 =0
(4)
8>
>>><
>>>>
:
x1?2x2 +3x3?4x4 =0
x2? x3 + x4 =0
x1 +3x2?3x4 =0
x1?4x2 +3x3?2x4 =0
x5,(1) (1;?2;1;0;0),(1;?2;0;1;0),(3;?4;0;0;1).
(2) (?1;24;9;0),(2;?21;0;9).
(3) (?7;7;?1;1;0),(?25;28;?4;0;1).
(4) (0;1;2;1).
4,x53x54:x8x1fx3x48x74x26x40x41x42
8>
>>><
>>>>
:
a11x1 +a12x2 +￠￠￠+a1nxn = 0
a21x1 +a22x2 +￠￠￠+a2nxn = 0
::::::::::::::::::::::::::::::::
an1x1 +an2x2 +￠￠￠+annxn = 0
x15x6axex5dx5eAx15xdx1ex29jAj = 0,x40x41x42x15x9x13n? 1,x57x3fx5dx5eAx16aklx15x51xex24x0x29Akl 6= 0,x20x30
(Ak1;Ak2;￠￠￠ ;Akn)x13x4fx3x48x74x26x40x41x42x15x48x66x7ax8x2dx6a.
￠ 8 ￠
x3x4,x4ex3c
ak1Ak1 +ak2Ak2 +￠￠￠+aknAkn = jAj = 0;
ai1Ak1 +ai2Ak2 +￠￠￠+ainAkn = 0; x62i 6= kx52,
x21x4f(Ak1;Ak2;￠￠￠ ;Akn)x13x61x18x3x48x74x26x40x41x42x15x2d,x51x21Akl 6= 0,x77x13x48x66x6ex6fx2d,x4ex31x18xbx60x40x41
x42x15x9x13n?1,x23x24x4fx3x48x74x26x40x41x42x15x7ax8x2dx6ax4ex48x66x6ex6fx2dx75x2a,x21x4f(Ak1;Ak2;￠￠￠ ;Akn)x13x4f
x3x48x74x26x40x41x42x15x48x66x7ax8x2dx6a.
5,x18x3x48x74x26x40x41x42 8
>>>>
<
>>>>
:
a11x1 +a12x2 +￠￠￠+a1nxn = 0
a21x1 +a22x2 +￠￠￠+a2nxn = 0
::::::::::::::::::::::::::::::::
an?1;1x1 +an?1;2x2 +￠￠￠+an?1;nxn = 0
x15x6axex5dx5ex22A,Mix13x5dx5eAx16xax7x3dix1ex23x50x15(n?1)￡(n?1)x5dx5ex15xdx1ex29,x53x54:
(1) (M1;?M2;￠￠￠ ;(?1)n?1Mn)x13x40x41x42x15x48x66x2d;
(2)x8x1fx77x66x74x26x40x41x42x15x9x22n?1,x6ex66Mi 6= 0,x53x54x40x41x42x15x2dx33x13(M1;?M2;￠￠￠ ;(?1)(n?1)Mn)
x15x18xe.
x3x4,(1)x2fx3x48x74x26x40x41x42
8
>>>>
<
>>>>
:
a11x1 +a12x2 +￠￠￠+a1nxn = 0
::::::::::::::::::::::::::::::::
an?1;1x1 +an?1;2x2 +￠￠￠+an?1;nxn = 0
an1x1 +an2x2 +￠￠￠+annxn = 0
x3cx16an1 = an2 = ￠￠￠ = ann = 0,x4ax4fx74x26x40x41x42x42x4bx40x41x42x43x2d,x3fx6axex5dx5ex56x3c0,x21x4ex79x61,x6x7x48
xdx15x51xex24x0x29(An1;An2;￠￠￠ ;Ann)x22x4bx40x41x42x15x2d,x51Ani = (?1)n+iMi,x23x24(M1;?M2;￠￠￠ ;(?1)n?1Mn) =
(?1)n+1(An1;An2;￠￠￠ ;Ann)x22x4bx40x41x42x15x2d.
(2)x21x4bx40x41x42x15x9x22n?1,x3fx47x48x66Mi 6= 0,x21x4fx4bx40x41x42x15x7ax8x2dx6ax4ex48x66x6ex6fx2dx1fx20x75x2a.
x43x25x6ex6fx2d(M1;?M2;￠￠￠ ;(?1)n?1Mn)x75x2ax4bx74x26x40x41x42x15x48x66x7ax8x2dx6a,x21x4bx40x41x42x15x73x48x66x2dx6d
x13(M1;?M2;￠￠￠ ;(?1)n?1Mn)x15x18xe.
6,x1bx25xcxfx793x66x17(x1;y1);(x2;y2);(x3;y3)x28x74x15x30x11x40x26x31x32.
x5,x1ex17(x1;y1);(x2;y2);(x3;y3)x28x74,x55x3fx18x4fx2ex74x15x40x41x22Ax+By +C = 0,x4a
8>
<
>:
Ax1 +By1 +C = 0
Ax2 +By2 +C = 0
Ax3 +By3 +C = 0
,x3x48x74x26x40x41x42 8
><
>:
x1t1 +y1t2 +t3 = 0
x2t1 +y2t2 +t3 = 0
x3t1 +y3t2 +t3 = 0
x47x6ex6fx2d(A;B;C)
,x3cx6axex5dx5e
A =
0
@
x1 y1 1
x2 y2 1
x3 y3 1
1
A = 0:
￠ 9 ￠
7,x1bx25xcxfx793x31x2ex74
a1x+b1y +c1 =0;
a2x+b2y +c2 =0;
a3x+b3y +c3 = 0
x28x17x15x30x11x40x26x31x32.
x5,x4f3x31x2ex74x28x17x15x30x11x40x26x31x32x13x65x2cx15x3x48x74x26x40x41x42
a1x+b1y +c1z =0;
a2x+b2y +c2z =0;
a3x+b3y +c3z = 0
x47x6ex6fx2d,x62x3fx63x62x6axex5dx5ex56x3c0,x4
flfl
flfl
flfl
a1 b1 c1
a2 b2 c2
a3 b3 c3
flfl
flfl
flfl = 0:
8,x7ex25x72x4ex34x17(1;2),(1;?2),(0;?1)x15xbx40x41.
x5,(x?2)2 +y2 = 5.
9,x73x60x613–4.3x16x23x1dx4dx15x74x26x0x70x71x15x46xe.
x5,x18
fii =
0
BB
B@
a1i
a2i
...
ani
1
CC
CA;
A = (fi1;fi2;￠￠￠ ;fir):
x4a
(c1;￠￠￠ ;cr) 2 W,
X
cifii = 0
,A
0
B@
c1
...
cr
1
CA = 0,Wx22AX = 0x15x2dx70x71
x23x24
dimW = r?rankA = r?rankffi1;￠￠￠ ;firg:
x1 x2 3–7
1,x73x1dx1ex74x26x40x41x42x15x33x7cx2d:
(1)
8
>>>>
<
>>>>
:
2x1? x2 +5x3 + 7x4 =0
4x1?2x2 +7x3 + 5x4 =0
2x1? x2 + x3? 5x4 =0
2x1? x2 +6x3 +10x4 =0
(2)
8
>><
>>:
2x1 + x2? x3 +x4 =1
x1 +2x2 + x3?x4 =2
x1 + x2 +2x3 +x4 =3
￠ 10 ￠
(3)
8
>>>>
<
>>>>
:
x1 + x2?3x4? x5 =2
x1? x2 +2x3? x4 =1
4x1?2x2 +6x3 +3x4?4x5 =8
2x1 +4x2?2x3 +4x4?7x5 =9
(4)
8>
>>>
<
>>>
>:
3x1 +4x2 +2x3 + x4 +6x5 =?2
2x1 +5x2 +5x3 +3x4 + x5 = 4
x1? x2? x3?2x4 +5x5 =?6
x1 +6x2 +8x3 +5x4?3x5 = 9
x5,(1) k1(4;0;?3;1)+k2(0;8;3;?1).
(2) (1;0;1;0)+k(?3;3;?1;2).
(3) (1;0;0;0;?1)+k1(?1;1;1;0;0)+k2(7;5;0;2;6).
(4) (1;0;0;1;?1)+k(?1;1;0;?1;0).
2,x18·1;·2;￠￠￠ ;·rx13x6ex3x48x74x26x40x41x42
8>
>>><
>>>>
:
a11x1 + a12x2 +￠￠￠ + a1nxn = b1
a21x1 + a22x2 +￠￠￠ + a2nxn = b2
::::::::::::::::::::::::::::::::
am1x1 +am2x2 +￠￠￠ +amnxn =bm
x15x0x1rx66x2d,· =
rP
i=1
ki·i (ki 2 K;i = 1;2;￠￠￠ ;r),x53x54:x62x3fx63x62
rP
i=1
ki = 1x52,·x67x13x77x66x6ex3x48x74x26
x40x41x42x15x2d.
x3x4,x18A = (Aij),B =
0
B@
b1
...
bn
1
CA,x4aB 6= 0.
·x22x79x53x40x41x42x15x2d,A· = B,
nX
i=1
kiB = B,
nX
i=1
ki = 1:
3,x18 0x13x6ex3x48x74x26x40x41x42
8
>>>>
<
>>>>
:
a11x1 +a12x2 +￠￠￠+a1nxn = b1
a21x1 +a22x2 +￠￠￠+a2nxn = b2
::::::::::::::::::::::::::::::::
am1x1 +am2x2 +￠￠￠+amnxn = bm
x15x48x66x2d,·1;·2;￠￠￠ ;·rx13x38x15xcx25x42x15x7ax8x2dx6a.
x53x54,(1) 0;·1;·2;￠￠￠ ;·rx74x26x2cx2a;
(2) 0; 0 +·1; 0 +·2;￠￠￠ ; 0 +·rx67x74x26x2cx2a;
(3)x8 x13x77x66x6ex3x48x74x26x40x41x42x15x0x1x2d,x4a ; 0; 0 +·1;￠￠￠ ; 0 +·rx74x26x65x2a;
(4) Knx16x1fx20 x13x77x66x6ex3x48x74x26x40x41x42x15x2dx15x30x11x40x26x31x32x13x31x6br+1x66xeki(i = 0;1;￠￠￠ ;r),
rP
i=0
ki = 1,x27x50
= k0 0 +k1( 0 +·1)+k2( 0 +·2)+￠￠￠+kr( 0 +·r):
x3x4,(1)x18
fi = k0 0 +k1·1 +￠￠￠+kr·r = 0;
￠ 11 ￠
x4a
0 = Afi = k0A 0 +
rX
i=1
kiA·i = k0B:
x23x24k0 = 0,k1·1 + ￠￠￠ + kr·r = 0,x4ex3c·1;￠￠￠ ;·r x74x26x2cx2a,x3ex50k1 = k2 = ￠￠￠ = kr = 0,x21x4f
0;·1;·2;￠￠￠ ;·rx74x26x2cx2a.
(2)x18
k0 0 +k1( 0 +·1)+￠￠￠+kr( 0 +·r) = 0;
x4a
(k0 +k1 +￠￠￠+kr) 0 +k1·1 +￠￠￠+kr·r = 0;
x4e(1),0;·1;·2;￠￠￠ ;·rx74x26x2cx2a,x23x24
k1 = k2 = ￠￠￠ = kr = 0; k0 +k1 +￠￠￠+kr = 0;
xdx25k0 = k1 = ￠￠￠ = kr = 0,x21x4f 0; 0 +·1; 0 +·2;￠￠￠ ; 0 +·rx74x26x2cx2a.
(3) x8 x13x77x66x6ex3x48x74x26x40x41x42x15x2d,x4a? 0 x13x38x15xcx25x42x15x2d,x23x24x31x6b ki 2 K,x27
0 = Pki·i,x3cx13
= 0 +
X
ki·i =
1?
rX
i=1
ki
!
0 +
rX
i=1
ki( 0 +·i);
x43x25 ; 0; 0 +·1;￠￠￠ ; 0 +·rx74x26x65x2a.
(4) ()) x22x6ex3x48x74x26x40x41x42x15x2d,x4ax4e(3)x15x53x54x3ex50
=
1?
rX
i=1
ki
!
0 +
rX
i=1
ki( 0 +·i);
x43x25x4fx74x26x1bx1cx29x15x6axex39x3ax56x3c?
1?
rX
i=1
ki
!
+
rX
i=1
ki = 1:
(()x8
rP
i=0
ki = 1,x4ax4ex79x61x15x22x23x3exb x13x48x66x2d.
4,x53x54,Knx15x0x48x74x26x0x70x71x40x13Kx79x6ex66nxbx3x48x74x26x40x41x42x15x2dx70x71.
x3x4,x18W x22Knx15x0x48x0x70x71,fi1;fi2;￠￠￠ ;fir (x1ex1fx20)x22W x15x7a,x49A = (fi1;fi2;￠￠￠ ;fir),x4a
rankA = r,x18c1;￠￠￠ ;cn?r 2 Krx13ATX = 0x15x7ax8x2dx6a,x49C = (c1;￠￠￠ ;cn?r)T,x4arankC = n?r.
x3cx13x3x48x74x26x40x41x42CX = 0x15x2dx70x71x15x46xex56x3c r.
x4eATci = 0,x3ex50AT(c1;￠￠￠ ;cn?r) = 0,x4CA = 0,x77x1bx54fiix13CX = 0x15x2d.
x4ex3cfi1;￠￠￠ ;firx74x26x2cx2ax24x68CX = 0x15x2dx70x71x15x46xex56x3c r,x21fi1;￠￠￠ ;firx22CX = 0x15x7ax8x2d
x6a,x43x25Wx22CX = 0x15x2dx70x71.
5,x53x54,Knx15x48x66x6ex70x0xdY x13x48x66x74x26x74x36x15x30x11x40x26x31x32x13,x38x13x6ex48x74x26x40x41x42x15x2dxd
x54.
x3x4,())x18Y x13x48x66x74x26x74x36,x4aY = fi0 +W,x3cx16Wx13x48x66x74x26x0x70x71,fi0 =
0
B@
a1
...
an
1
CA2 Kn.
x4aWx22x6ex3x48x74x26x40x41x42AX = 0x15x2dx70x71,x49B = Afi0 =
0
B@
b1
...
bm
1
CA,xbx3AX = B,xexb,Y x22AX = B
x15x2dxd.
￠ 12 ￠
(()x6fx13x1dx57.4x15x22x23.
6,x18Y1,Y2 x22x1fx20x70x71V x15x37x66x74x26x74x36,x1dx1exdx54x13x29x75x2aV x15x74x26x74x36?
(1) Y1 [Y2;
(2) Y1 +Y2 = ffi1 +fi2 j fi1 2 Y1;fi2 2 Y2g.
x5,(1)x55x48x1d,x8x7afi;flx74x26x2cx2a,x49
Y1 = L(fi); Y2 = L(fl);
x4aY1;Y2x6dx13x74x26x74x36,x71fi+fl 62 Y1 [Y2.
(2)x13,x8
Y1 = fi1 +W1; Y2 = fi2 +W2;
x3cx16W1;W2x22x0x70x71,x4a
Y1 +Y2 = (fi1 +fi2)+(W1 +W2);
x3exbY1 +Y2x67x13x74x26x74x36.
7,x18fi0;fi1;￠￠￠ ;firx22V x15r +1x66x1fx20,x53x54:
Y =
( rX
i=0
kifii
flfl
flfl
fl
rX
i=0
ki = 1;ki 2 K;i = 0;1;￠￠￠r
)
x75x2aV x15x48x66x74x26x74x36.
x3x4,x18
rX
i=0
kifii 2 Y;
rX
i=0
ki = 1;
rX
i=0
lifii 2 Y;
rX
i=0
li = 1;
x4ax2x0x1x15k;l 2 K,k +l = 1,x47
k
rX
i=0
kifii
!
+l
rX
i=0
lifii
!
=
rX
i=0
(kki +lli)fii;
x25
rX
i=0
kki +
rX
i=0
lli = k
rX
i=0
ki +l
rX
i=0
li = k +l = 1;
x3cx13
k
rX
i=0
kifii
!
+l
rX
i=0
lifii
!
= 0;
x43x25Y x13x74x26x74x36.
8,x18Y x22x1fx20x70x71V x15x48x66x74x26x74x36,x53x54:x31x6bY x16x15r +1x66x1fx20fi0;fi1;￠￠￠ ;fir,x27
Y =
( rX
i=0
kifii
flfl
flfl
fl
rX
i=0
ki = 1;ki 2 K;i = 0;1;￠￠￠r
)
:
x3x4,x18Y = fi0 +W,x3cx16Wx13x0x70x71,x18Wx15x7ax22·1;￠￠￠ ;·r,x49
fi0 = fi0;fi1 = fi0 +·1;￠￠￠ ;fir = fi0 +·r;
x4afii 2 Y,x3fx2x0x1x15ki 2 K,
rP
i=0
ki = 1,x47
rX
i=0
kifii =
rX
i=0
kifi0 +
rX
i=0
ki·i 2 fi0 +W = Y:
￠ 13 ￠
x4dx39,x2x0x1x15fi = fi0 +· 2 Y = fi0 +W,x31x6bki,x27· =
rP
i=1
ki·i,x43x25
fi =
1?
rX
i=1
ki
!
fi0 +
rX
i=1
ki(fi0 +·i);
x3cx16?
1?
rX
i=1
ki
!
+k1 +k2 +￠￠￠+kr = 0:
x77x53x54x72
Y =
( rX
i=0
kifii
flfl
flfl
fl
rX
i=0
ki = 1;ki 2 K;i = 0;1;￠￠￠r
)
:
x1 x2 3–8
1,x6bx1bx1dx15x5cx5dx57x55x6ax16,x73x1dx1excxfx15x48x7dx40x41x3ax5exex40x41:
(1)x4e(?1;2;0);(?2;?1;4);(3;1;?5)x34x17x15xcxf;
(2)x4ex17(3;1;2)x3a(1;0;?2),xcxdx3cx1fx20?!v = (1;?2;?3)x15xcxf.
x5,(1)x4e3x17x15xcxfx34x17x29x40x41x13:
flfl
flfl
flfl
x+1?1 4
y?2?3?1
z 4?5
flfl
flfl
flfl = 0;
x65x66x7x50
19x+11y +13z?3 = 0:
x38x15x5exex40x41x22,8
><
>:
x =?1?u+4v
y = 2?3u?v
z = 4u?5v:
(2) x4exaxbx31x32,xcxfx72x4e(3;1;2),x38x15x40x1fx1fx20x13?1 =?!v = (1;?2;?3)x24x68?2 = (1? 3;0?
1;?2?2) = (?2;?1;?4),x21x4fxcxfx15x5exex40x41x22
8>
<
>:
x = 3+u?2v
y = 1?2u?v
z = 2?3u?4v;
xcxfx15x48x7dx40x41x22 fl
flfl
flfl
fl
x?3 1?2
y?1?2?1
z?2?3?4
flfl
flfl
flfl = 0;
x65x66x7x50x+2y?z?3 = 0.
2,x6bx1bx1dx15x5cx5dx57x55x6ax16,x73x1dx1excxfx15x48x7dx40x41:
(1)x4ex17(1;2;?4)x3axx45x15xcxf;
(2)x4ex17(2;1;2)x24x68xcxfΠ1, x+y?z = 0,Π2, 2x?3z?1 = 0x15x3x74x15xcxf;
(3)x4ex17(0;4;?3)x3a(1;?2;6),x3fxcxdx3cxx45x15xcxf;
(4)x4ex17(3;1;?2)x3fxcxdx3cxcxfx?2y?2z +1 = 0x15xcxf.
x5,(1)x18xcxfx15x48x7dx40x41x13Ax+By +Cz +D = 0,x21x22x38x4exx45,x23x24A = D = 0;x51x21x38x4ex17
(1;2;?4),x23x24B = 2C,x21xcxfx15x40x41x222y +z = 0.
￠ 14 ￠
(2)x2dx74x26x40x41x42 (
x+y?z = 0
2x?3z?1 = 0
x73x50xcxfx3x74x79x15x37x66x17
1
2 ;?
1
2 ;0
·
x24x68
0;? 13 ;? 13
·
,x50x8x23x73xcxfx15x34x17x29x40x41:
flfl
flfl
flfl
fl
x?2? 32?2
y?1? 32? 43
z?2?2? 73
flfl
flfl
flfl
fl
= 0;
x65x66x7x505x+3y?6z?1 = 0.
(3)x23x73xcxfx15x48x66x40x1fx1fx20x13(1;0;0),x21x4fx23x73xcxfx15x40x41x22:
flfl
flfl
flfl
x 1 1
y?4 0?6
z +3 0 9
flfl
flfl
flfl = 0;
x65x66x7x503y +2z?6 = 0.
(4)x4ex37xcxfxcxdx15x26x27x3exb,x23x73xcxfx15x40x41x2cx22x?2y?2z+D = 0,x21x1cxcxfx4e(3;1;?2)x17,
x23x243?2+4+D = 0,x4D =?5,x21x23x73x40x41x22x?2y?2z?5 = 0.
3,xaxbx48xcxfx72x4eP0(x0;y0;z0)(z0 6= 0),x3fx6bxx45x3ayx45x79x15xfx32x11x12x13ax3ab,x73x38x15x40x41.
x5,x18xcxfx6bzx45x79x15xfx32x22c,x4ax1cxcxfx15x40x41x22
x
a +
y
b +
z
c = 1:
x51x21P0x17x6bxcxfx79,x23x24 x0a + y0b + z0c = 1,x50 1c = 1z
0
1? x0a? y0b
·
,x23x24xcxfx40x41x22
x
a +
y
b +
1? x0a? y0b
· z
z0 = 1:
4,x73x4eP0(x0;y0;z0),x3fxcxdx3cxcxfAx+By +Cz +D = 0x15xcxfx15x40x41.
x5,x4ex37xcxfxcxdx15x26x27x3exb,x23x73xcxfx15x40x41x2cx22Ax + By + Cz + D1 = 0,x21x1cxcxfx4eP0x17,
x23x24D1 =?(Ax0 +By0 +Cz0),x21x23x73xcxfx40x41x22A(x?x0)+B(y?y0)+C(z?z0) = 0.
5,x53x54x37x618.2.
x3x4,x21x22x1fx20x42x74x26x74x36xcxdx1dx4dx22x77x66x1fx20x6bx74x26x0x70x71W x7b,x21x4f?!v x42xcxfxcxdx15x30x11
x40x26x31x32x13?!v 2 W,x25x74x26x0x70x71Wx51x13x4excx25x40x41Ax+By +Cz = 0x1dx4dx15,x23x24?!v 2 Wx15x30x11
x40x26x31x32x13?!v x15x11x20(X;Y;Z)x2dx2excx25x40x41,x4AX +BY +CZ = 0.
6,x7cx7dx1dx1ex28xcxfx15x65x2ax2fx59:
(1) 2x+y?z = 0x42 12 x+ 14 y? 14 z +2 = 0;
(2) x?2y +z?2 = 0x423x+y?2z?1 = 0;
(3) A1x+B1y +C1z +D1 = 0x42A2x+B2y +C2z +D2 = 0,x3cx16
flfl
flflA1 B1
A2 B2
flfl
flfl6= 0.
6,(1)x21x22
2
1
2
= 11
4
=?1
14
6= 02 ;
x23x24x37xcxfxcxd.
(2)x21x22 13 6=?21,x23x24x37xcxfx65x3.
(3)x21x22A1, B1 6= A2, B2,x23x24x37xcxfx65x3.
7,xaxbx37x66xcxfΠ1, x?2y +pz?1 = 0,Π2, 2x?4y +5z +q = 0,x1x62p;qx7ax8x4x52:
￠ 15 ￠
(1) Π1x42Π2x65x3; (2) Π1x42Π2xcxd; (3) Π1x42Π2x24x54.
x5,(1)x62 12 =?2?4 6= p5,x4p 6= 52,qx7ax0x1x32xex52x37xcxfx65x3.
(2)x62 12 =?2?4 = p5 6=?1q x52,x4p = 52,q 6=?2x52x37xcxfxcxd.
(3)x62 12 =?2?4 = p5 =?1q x52,x4p = 52,q =?2x52x37xcxfx24x54.
8,xaxbx17A(3;10;?5)x3axcxfΠ, 7x?4y?z?1 = 0,x73zx45x79x15x17Bx15x57x55,x27ABxcxdx3cΠ.
x5,x18 B x17x15x57x55x22 (0;0;k),x4a?!AB = (?3;?10;k + 5),x3dx3ex37x61 8.2,?!AB x15x11x20x40x41x2dx2e
7(?3)?4(?10)?(k +5) = 0,x4k = 14,x21Bx17x15x57x55x22(0;0;14).
9,x53x54x34x66xcxf
a1x+b1y +c1z +d1 = 0;
a2x+b2y +c2z +d2 = 0;
k(a1x+b1y +c1z)+l(a2x+b2y +c2z)+m = 0
x62m 6= kd1 +ld2x52,x10x47x66x28x17.
x3x4,x34x66xcxfx47x66x28x17x62x3fx63x62x74x26x40x41x42
8>
<
>:
a1x+b1y +c1z =?d1
a2x+b2y +c2z =?d2
k(a1x+b1y +c1z)+l(a2x+b2y +c2z) =?m
x47x2d,x2x77x66x40x41x42x2fx5cx56x3dx4a,x43x3d3x66x40x41x5x7x3d1x66x40x41x15k x18x3ax3d2x66x40x41x15lx18x7x50x8
0 = kd1 +ld2?m,x62m 6= kd1 +ld2x52,x77x13x66x34x35x40x41,x21x4fx10x47x66x28x17.
10,x53x54x0x8x48x66x6ax4ex65x3x15x37xcxf
Π1, A1x+B1y +C1z +D1 = 0;
Π2, A2x+B2y +C2z +D2 = 0
x15x65x3x2ex74Lx15xcxfx40x41x63x7ex2a
fi(A1x+B1y +C1z +D1)+fl(A2x+B2y +C2z +D2) = 0;
x3cx16,fi;flx13x55x33x22x6fx15x32xe.
x3x4,Lx16x15x17x15x57x55x2dx2eΠ1x42Π2x15x40x41,x43x25x67x2dx2ex40x41fi(A1x+B1y+C1z+D1)+fl(A2x+
B2y +C2z +D2) = 0,x21x4fLx6bx4fxcxfx79.
x4dx39,x8x1fxcxfΠ0x72x4ex2ex74L,x14x15x3ex6bΠ0x79x7ax48x66x55xax3cLx15x17M(x0;y0;z0),x49
fi0 = A2x0 +B2y0 +C2z0 +D2; fl0 = A1x0 +B1y0 +C1z0 +D1;
x4ex3cM 62 L,x23x24fi0;beta0x55x33x220,M0x15x57x55x10x11x2dx2ex24x1dx40x41:
fi0(A1x+B1y +C1z +D1)?fl0(A2x+B2y +C2z +D2) = 0;
x4ex4fxfx15x55x23xbx79x53x40x41x64x1dx15xcxfx48x1dx72x4ex3x74L,x4ex72x4ex48x31x2ex74x68x74x30x48x17x15xcxfx15x2cx48x26,x3e
x5fx79x53x40x41x1dx4dx15xcxfx6fx13Π0.
11,x18xcxfΠ, Ax+By+Cz+D = 0x42x4cx11x37x17M1(x1;y1;z1)x42M2(x2;y2;z2)x15x2ex74x65x3x3cx17
M,x25x3f!M1M = k!MM2,x53x54:
k =? Ax1 +By1 +Cz1 +DAx
2 +By2 +Cz2 +D
:
￠ 16 ￠
x3x4,x18x17Mx15x57x55x13(x0;y0;z0),x4ax4ex1dx65x11x17x66x29xb8
>>>
<
>>>:
x0 = x1 +kx21+k
y0 = y1 +ky21+k
z0 = z1 +kz21+k
(k 6=?1):
x71x4ex3cM0 2 Π,x23x24
A x1 +kx21+k +B y1 +ky21+k +C z1 +kz21+k +D = 0;
x4cx6bx7x4x50
k =? Ax1 +By1 +Cz1 +DAx
2 +By2 +Cz2 +D
:
12,x48xcxfx42x70x71x6cx38x36ABCDx15x38AB;BC;CD;DAx11x12x3x3cP,Q,R,S,x4a
AP
PB ￠
BQ
QC ￠
CR
RD ￠
DS
SA = 1:
x19x53x39.
x75x75x75x75x75x75x75x75x75x72x40x8x12x13x12
x12x12x13
x12x12x13
x12x12x12
x13x12x12
x13x12x12
x12x13x12
x12x13x12
x39x15x2x8x40x72x75x75x75x75x75x75x75x75x75
x2x15x39x12x13
x12x12x13
x12x12x12
x13x12x12
x13x12x12
x12x13x12
x12x13x12
x12x12x13
x12
x7x4dx3bx3bx3ax3bx3bx3ax3bx3bx3ax3bx6bx6bx6b
x6cx6bx6b
x6bx6bxd
x50x50x50x58
x5x18x42x22x23
x22x23x22
x23x22x23
x75x75x75x75x75x75x75x50x5
x5
x7x1dx34x8x20x40
x62x2dx72
x7x1fx3ax9x21x46x4dx1f
x4dx1f x4dx1f x4dx1f x4dx1f x4dx1f x4dx1f x7x58xa x58x24 x58xa
x66x2a x65x2b
x66x2b x66x2a
x65x2b x66xdx4 x42x18 x41x19 x41x19
x41x19 x41x19
x41
B
P A S
D
Q
C
R
x012x1x2
x3x4,x8x9,x14x15x24Bx17x22x4bx17,x24?!e1 =?!BP,?!e2 =?!BQ,?!e3 =?!BDx22x7ax1fx20x75x2fx48x66x5cx5dx57x55x6a
[B;?!BP;?!BQ;?!BD],x49
!QC = a?!BQ;?!CR = b?!RD;?!DS = c?!SA;?!PA = d?!BP:
x18
!BR = k?!BQ+l?!BD = k?!e
2 +l?!e3;
!BS = m?!BP +n?!BD = m?!e
1 +n?!e3;
x4a
!CR =?!BR!BC = (k?1?a)?!e
2 +l?!e3;
!RD =?!BD!BR =?k?!e
2 +(1?l)?!e3:
x4e?!CR = b?!RDx3ex50,(
k?1?a =?bk
l = b(1?l); x4
( a = k +l?1
1?l
b = l1?l,
x51x21
!DS =?!BS!BD = m?!e
1 +(n?1)?!e3;
!SA =?!BA!BS = (1+d?m)?!e
1?n?!e3:
x4e?!DS = c?!SAx3ex50,(
m = c(1+d?m)
n?1 =?cn; x4
( c = 1?n
n
d = m+n?11?n,
￠ 17 ￠
x23x24
AP
PB ￠
BQ
QC ￠
CR
RD ￠
DS
SA =
dbc
a =
l(m+n?1)
n(k +l?1), (*)
x51,
!QR =?!BR!BQ = (k?1)?!e
2 +l?!e3;
!QP =?!BP!BQ =?!e
1!e2;
!QS =?!BS!BQ = m?!e
1!e2 +n?!e3;
x21x22x773x66x1fx20x28xf,x23x24 fl
flfl
flfl
fl
0 1 m
k?1?1?1
l 0 n
flfl
flfl
flfl = l(m?1)?n(k?1) = 0:
x43x25
l(m+n?1) = ln+l(m?1) = ln+n(k?1) = n(k +l?1):
x76x79x29x51x52(*)x4x50:
AP
PB ￠
BQ
QC ￠
CR
RD ￠
DS
SA = 1:
x1 x2 3–9
1,x6bx1bx1dx15x5cx5dx57x55x6ax16,x73x1dx1ex2ex74x15x40x41:
(1)x4ex17P(3;1;?1)x3fxcxdx3cx1fx20?!v (4;7;?8);
(2)x4ex17P0(?3;0;1)x3aP1(2;5;1);
(3)xaxbx34x35x36x15x34x66x7ex17x13Ai(xi;yi;zi)(i = 1;2;3),x73x34x31x16x74x15x40x41.
x5,(1) x?34 = y?17 = z +1?8,
(2)x1ax37x17x29x40x41x3ex50 x+35 = y5 = z?10,
(3) x?xi
xi? xi+1 +xi+22
= y?yi
yi? yi+1 +yi+22
= z?zi
zi? zi+1 +zi+22
,x62i+1,i+2x3bx3c3x52x4x5x73.
2,x6bx2ex35x57x55x6ax16,x73x4ex17P(1;6;3)x3fxcxdx3cxcxf3x+y?2z?5 = 0x15x2ex74x15x40x41.
x5,x18x2ex74x15x40x1fx1fx20x22? = (A;B;C),x4a?x42xcxf3x+y?2z?5 = 0xcxd,x213A+B?2C = 0,
x25x2ex74x40x41x22 x?1A = y?6B = z?3C,
3,x73x4ex17A(0;?2;1)x3fxcxdx3cx2ex74
( x+6y?4z +2 = 0
x+y +z?3 = 0 x15x2ex74x40x41.
x5,xax76x2ex74x40x41x4cx22x55x12x40x41 x?22 = y?1 = z?1?1,x3cx40x1fx1fx20x22?(2;?1;?1),x21x23x73x2ex74
x40x41x22 x2 = y +2?1 = z?1?1,
4,x73x2ex74L, x3 = y?2?2 = z +42 x42xcxfΠ, 2x?3y +2z?2 = 0x15x3x17x57x55.
x5,x4ex2ex74x7ex2ax5exex40x41,8
><
>:
x = 3t
y = 2?2t
z =?4+2t:
x11x7x51x52xcxfx40x41x50
2(3t)?3(2?2t)+2(?4+2t)?2 = 0;
x2dx50t = 1,x23x24x3x17x22(3;0;?2).
￠ 18 ￠
5,x73x4ex17A(3;1;2)x68x2ex74L, x1 = y?2 = z +2?3 x15xcxfx15x40x41.
x5,x2ex74x79x47x48x17(0;0;?2),x21x4fxcxfx15x40x1fx1fx20x13?1 = (1;?2;?3),?2 = (?3;?1;?4),x21xcxf
x40x41x22 fl
flfl
flfl
fl
x?3 1?3
y?1?2?1
z?2?3?4
flfl
flfl
flfl = 0;
x78x67x505x+13y?7z?14 = 0.
6,xaxbx2ex74L, x?1m = y?a?2 = z +23,xcxfΠ, x?2y?4z +1 = 0,x1x62a;mx7ax2fx30x4x52
(1) Lx42Πx65x3; (2) Lxcxdx3cΠ; (3) Lx6bΠx7b.
x5,(1)x2ex74x15x40x1fx1fx20x13? = (m;?2;3),x1eLx42Πx65x3,x4a?x55x42Πxcxd,x21m + 4?12 6= 0,x4
m 6= 8,ax13x0x1x32xe.
(2)?x40x41x42Πxcxd,x4m = 8,x43x52Lx79x15x17(1;a;?2)x55x6bΠx79,x41?2a+8+1 6= 0,a 6= 5.
(3) m = 8,a = 5.
7,x73x72x4ex17(2;2;2)x3fx42x37x2ex74
x
1 =
y
2 =
z
3 x3a
x?1
2 =
y?2
1 =
z?3
4
x6dx65x3x15x2ex74x15x40x41.
x5,x18x23x73x2ex74x15x40x1fx1fx20x13? = (X;Y;Z),xaxbx3dx48x31x2ex74x79x15x17M1(0;0;0),x40x1fx1fx20?1 =
(1;2;3),x3dx62x31x2ex74x79x15x17M2(1;2;3),x40x1fx1fx20?2 = (2;1;4),x22x27x23x73x2ex74x42x3dx48x31x2ex74x65x3,x40x41
x27 fl
flfl
flfl
fl
2 1 X
2 2 Y
2 3 Z
flfl
flfl
flfl = 2X?4Y +2Z = 0:
x43x5,x22x27x23x73x2ex74x42x3dx62x31x2ex74x65x3,x40x41x27fl
flfl
flfl
fl
1 2 X
0 1 Y
1 4 Z
flfl
flfl
flfl = X?6Y +Z = 0:
x2dx4fx74x26x40x41x42x50X, Y, Z = 1, 0,?1,x21x4fx23x73x2ex74x15x40x41x13
x?2
1 =
y?2
0 =
z?2
1,
8,x76x1dx1ex2ex74x15x48x7dx29x40x41x4cx2ax55x12x40x41.
(1)
( 2x?3y +4z?12 = 0
x+4y?2z?10 = 0; (2)
( 3x+2y?4z?5 = 0
6x?y?2z?4 = 0;
(3)
( 3x?y +2 = 0
4y +3z +1 = 0:
x5,(1)x4fx2ex74x15x40x1fx1fx20x13 fl
flfl
fl
3 4
4?2
flfl
flfl;?
flfl
flfl2 4
1?2
flfl
flfl;
flfl
flfl2?3
1 4
flfl
flfl
= (?10;8;11):
x22x73x2ex74x79x15x17,x2dx4bx74x26x40x41x42,x50x8x48x66x2d(8;0;?1),x21x4fx2ex74x15x55x12x40x41x13
x?8
10 =
y
8 =
z +1
11,
(2)x4fx2ex74x15x40x1fx1fx20x13 fl
flfl
fl
2?4
1?2
flfl
flfl;?
flfl
flfl3?4
6?2
flfl
flfl;
flfl
flfl3 2
6?1
flfl
flfl
= (?8;?18;?15):
￠ 19 ￠
x22x73x2ex74x79x15x17,x2dx4bx74x26x40x41x42,x50x8x48x66x2d
1
3 ;0;?1
·
,x21x4fx2ex74x15x55x12x40x41x13
x? 13
8 =
y
18 =
z +1
15,
(3)x4fx2ex74x15x40x1fx1fx20x13 fl
flfl
fl
1 0
4 3
flfl
flfl;?
flfl
flfl3 0
0 3
flfl
flfl;
flfl
flfl3?1
0 4
flfl
flfl
= (?3;?9;12):
x4fx1fx20x3ex4cx6bx22(1;3;?4),x22x73x2ex74x79x15x17,x2dx4bx74x26x40x41x42,x50x8x48x66x2d(0;2;?3),x21x4fx2ex74x15x55x12
x40x41x13
x
1 =
y?2
3 =
z +3
4,
9,x73x2ex74x42xcxfx15x3x17.
(1) x+1?2 = y +13 = z?34 x423x+2y +z = 0;
(2)
( 2x+3y +z?1 = 0
x+2y?z +2 = 0 x42xOyxcxf.
x5,(1)x4ex2ex74x7ex2ax5exex40x41,8
><
>:
x =?1?2t
y =?1+3t
z = 3+4t:
x11x7x51x52xcxfx40x41x50
3(?1?2t)+2(?1+3t)+(3+4t) = 0;
x2dx50t = 12,x23x24x3x17x22
2; 12 ;5
·
.
(2)x3x17x15x57x55x13(x;y;0),x2dx40x41x42
( 2x+3y?1 = 0
x+2y +2 = 0 x50x = 8,y =?5,x21x3x17x22(8;?5;0).
10,x73x25x4ex17P0(x0;y0;z0)x57x3fx42x65x3xcxf
Πi, Aix+Biy +Ciz +Di = 0; i = 1;2
x6dxcxdx15x2ex74x15x40x41.
x5,x3dx3ex37x619.1,xcxfΠ1x42Π2x15x3x74x15x40x1fx1fx20x13 fl
flfl
fl
B1 C1
B2 C2
flfl
flfl;?
flfl
flflA1 C1
A2 C2
flfl
flfl;
flfl
flflA1 B1
A2 B2
flfl
flfl
:
x21x23x73x2ex74x40x41x42x3x74xcxd,x21x4fx77x67x13x23x73x2ex74x15x40x1fx1fx20,x21x23x73x2ex74x15x40x41x22
x?x0fl
flfl
fl
B1 C1
B2 C2
flfl
flfl
= y?y0flfl
flflC1 A1
C2 A2
flfl
flfl
= z?z0flfl
flflA1 B1
A2 B2
flfl
flfl
:
11,x2ex74x40x41 (
A1x+B1y +C1z +D1 = 0
A2x+B2y +C2z +D2 = 0
x15x6axex2cx2dx2ex2fx30x31x32x71x63x27x1cx2ex74x13x6bxOzx57x55xcxfx7b.
x5,xcxfxOzx15x40x41x13y = 0,x21x4fx2ex74x13x6bxcxfxOzx7bx15x30x11x40x26x31x32x13x40x41x42
8>
<
>:
A1x+B1y +C1z +D1 = 0
A2x+B2y +C2z +D2 = 0
y = 0
￠ 20 ￠
x47x2cx7ex2d,x25x79x53x40x41x42x42 8
><
>:
A1x+C1z +D1 = 0
A2x+C2z +D2 = 0
y = 0
(*)
x43x2d,x25x40x41x42(*)x47x2cx7ex9x2dx15x30x11x40x26x31x32x13x4fx37x66x40x41x15x6axex2ax65x6a,x4
A1
A2 =
C1
C2 =
D1
D2 6=
B1
B2 ;
x6x7x48x66x55x56x29x13x2ex74x40x41x15x40x26x31x32.
x1 x2 3–10
1,x73x72x4excxf4x?y +3z?1 = 0x3ax+5y?z +2 = 0x15x3x74x3fx2dx2ex1dx1ex31x32x39x48x15xcxf:
(1)x72x4ex4bx17;
(2)x42yx45xcxd;
(3)x72x4e(0;0;1)x17;
(4)x42xOyxcxfx15x3x74xcxdx3cx40x1f(4;5;0).
x5,(1)x23x73x40x41x22
k(4x?y +3z?1)+m(x+5y?z +2) = 0;
x1a(0;0;0)x51x52x50k = 2m,x23x24xcxfx40x41x22
9x+3y +5z = 0:
(2)x23x73x40x41x22
k(4x?y +3z?1)+m(x+5y?z +2) = 0;
yx45x15x40x1fx1fx20x13? = (0;1;0),xcxfx42?xcxdx15x31x32x13?k +5m = 0,x4k = 5m,x23x24xcxfx40x41x22
21x+14z?3 = 0:
(3)x23x73x40x41x22
k(4x?y +3z?1)+m(x+5y?z +2) = 0;
x1a(0;0;1)x51x52x502k +m = 0,x4m =?2k,x23x24xcxfx40x41x22
2x?11y +5z?5 = 0:
(4)x23x73x40x41x22
k(4x?y +3z?1)+m(x+5y?z +2) = 0;
x4fxcxfx42x1fx20(4;5;0)xcxdx15x31x32x134(4k +m)+5(?k +5m) = 0,x411k =?29m,x23x24xcxfx40x41x22
105x+84y?98z +51 = 0:
2,x73x42xcxfx?2y?z +2 = 0xcxdx3fx6bxx45x79x15xfx32x224x15xcxf.
x5,x3dx3excxdxcxfx14x26x27,x23x73xcxfx15x40x41x22x?2y?z +D = 0,x4cx22xfx32x29:
x
D +
y
D
2
+ zD = 1:
x3ex5fD =?4,x21x23x73x40x41x22x?2y?z?4 = 0.
3,x2ex74x40x41 (
A1x+B1y +C1z +D1 = 0
A2x+B2y +C2z +D2 = 0
￠ 21 ￠
x15x6axex2cx2dx2ex2fx30x31x32x71x63x27x1cx2ex74x13x6bxOyxcxfx7b.
x5,x4exaxbx2ex74x15xcxfx14x15x40x41x22
k(A1x+B1y +C1z +D1)+m(A2x+B2y +C2z +D2) = 0:
xcxfz = 0x2cx1cx6bx4fxcxfx14x7b,x23x24x47x24x1dx2ax6ax29:8
>>>
><
>>>
>:
kA1 +mA2 = 0
kB1 +mB2 = 0
kC1 +mC2 6= 0
kD1 +mD2 = 0:
x22x23x13
A1
A2 =
B1
B2 =
D1
D2 6=
C1
C2,
4,x42x55x28xfx15x2ex74
L1,
( A
1x+B1y +C1z +D1 = 0
A2x+B2y +C2z +D2 = 0
x3ax2ex74
L2,
( A
3x+B3y +C3z +D3 = 0
A4x+B4y +C4z +D4 = 0
x6dx65x3x15x2ex74Lx15x40x41x22(
k(A1x+B1y +C1z +D1)+m(A2x+B2y +C2z +D2) = 0
k0(A3x+B3y +C3z +D3)+m0(A4x+B4y +C4z +D4) = 0;
x3cx16,k;mx13x55x33x22x6fx15x32xe,k0;m0x67x13x55x33x22x6fx15x32xe.
x5,x4ex65x3x2ex74Lx42L1x64x1dx15xcxfx40x41x48x1dx47x24x1dx36x29:
k(A1x+B1y +C1z +D1)+m(A2x+B2y +C2z +D2) = 0;
x3cx16k;mx55x43x52x220,x43x5,x4eLx42L2x64x1dx15xcxfx40x41x22
k0(A3x+B3y +C3z +D3)+m0(A4x+B4y +C4z +D4) = 0;
x3cx16k0;m0x55x43x52x220,x4ex3cL1;L2x55x28xf,x79x53x37x66xcxfx55x24x54,x38x15x15x3x74x6fx13L.
￠ 22 ￠
x0xbx2 xcxdxexfx10x11xcxdx12x13x14
x1 x2 3–1
1,x1ax6xbx44x2dx1dx1ex74x26x40x41x42:
(1)
8
>>>>
<
>>>>
:
2x1? x2 + x3 +3x4 =?4
x1 +3x2? x3 +4x4 = 6
2x1?2x2 +3x3? x4 = 3
3x1 +2x3 +2x4 = 3
(2)
8>
>>>
<
>>>
>:
x1?2x2 +3x3?x4 =?6
x1 + x2? x3 +x4 = 7
2x1? x2 + x3 = 1
x2 + x3 +x4 = 3
x5,(1) x1 =?5,x2 = 5,x3 = 8,x4 = 1.
(2) x1 = 4,x2 = 7,x3 = 0,x4 =?4.
2,x11x12x1ax5dx5ex15x5cx56xdx3dx4ax3ax1ex3dx4ax76x1dx1ex5dx5ex4cx22xdx79x70x5dx5ex3ax1ex79x70x5dx5e:
(1)
0
BB
B@
3 2 1 0 4
2 1 4 4?3
2 0 3 1?2
2 3?1 2 5
1
CC
CA;
(2)
0
BB
BB
B@
1 5 2 0 1
3?5 4 2 7
1?5 1 1 3
0 2 1?1 2
1?1 3?1 7
1
CC
CC
CA.
x5,(1)
0
BB
B@
3 2 1 0 4
0? 13 103 4? 173
0 0?11?15 18
0 0 0 1711? 1611
1
CC
CAx42
0
BB
B@
3 0 0 0 0
2 4 0 0 0
2 1 32 0 0
2 2? 103? 1718 0
1
CC
CA.
(2)
0
BB
BB
B@
1 5 2 0 1
0 2 1?1 2
0 0 4?4 12
0 0 0 0 0
0 0 0 0 0
1
CC
CC
CAx42
0
BB
BB
B@
1 0 0 0 0
3?2 0 0 0
1?1 0 0 0
0 1?8 0 0
1 1?16 0 0
1
CC
CC
CA.
3,x53x54:x74x26x40x41x42x15x3dx62x6b,x3dx34x6bx5cx56x3dx4ax4ex74x26x40x41x42x4cx2ax42x38x43x2dx15x74x26x40x41x42.
x3x4,(x69)
4,x53x54x5ex231.4.
x3x4,x2x5dx5eATx2cx1ax5ex231.3,x4aATx3ex24x6ax4ex48x6ax1ex5cx56xdx3dx4ax4cx2ax6bx4cxdx79x70x5dx5e,x76x79x53x3d
x4ax6cxdx3cx5dx5eAx15x1ex79,x6fx76Ax4cx2ax6bx4cx1ex79x70x5dx5e.
￠ 1 ￠
5,x6dxbx61:
(1)x74x26x40x41x42x15x2dxdx3ex24x68x2fx13x70x71x15x48x66x17xd,x20x30,x74x26x70x71x16x0x48x17xdx13x29x48x1dx13x6ex66x74
x26x40x41x42x15x2dxdx54x6f? x8x1fx13x77x6e,x20x30,x70xd,x14x17xdf(0;0;￠￠￠ ;0)gx42x37x17xdf(0;0;￠￠￠ ;0);(1;1;￠￠￠ ;1)g
x11x12x13x70x6ex15x74x26x40x41x42x15x2dxdx54x6f? x8x1fx55x13x77x6e,x20x30,x70x6ex15x17xdx71x13x6ex66x74x26x40x41x42x15x2dxdx54
x6f?
(2)x74x26x40x41x42x15x5cx56x3dx4ax4ex74x26x40x41x42x3dx2ax43x2dx15x74x26x40x41x42.x20x30,x37x66x43x2dx15x74x26x40x41x42x13
x29x48x1dx3ex24x72x4ex5cx56x3dx4ax64x4cx6f?
x5,(1)x11x72x70xdx42x14x17xdx30,x74x26x40x41x42x15x2dxdx54x48x1dx13x2cx73xd,x70xdx13x34x35x40x41x42x15x2dxd,x14
x17xdf(0;0;￠￠￠ ;0)gx3ex24x13x24x1dx40x41x42 8
>>>
>><
>>>>
>:
x1 = 0
x2 = 0
...
xn = 0
x15x2dxd,x74x26x40x41x42x15x2dxdx54x13x48x66x74x26x74x36,x2dxdx54x15x26x27x3ex5ex68x2,x6,x7x15x55x23.
(2)x6bx75x76x54x17x44x77x7xcx78x40x41“0 = 0”x15x4fx79x1d,x4fx22x23x13x72x64x15.
x1 x2 3–2
1,x1ax6xbx44x2dx1dx1ex74x26x40x41x42:
(1)
8>
>>>
<
>>>
>:
x1?2x2 +3x3? x4? x5 =2
x1 + x2? x3 + x4?2x5 =1
2x1? x2 + x3?2x5 =2
2x1 +2x2?5x3 +2x4? x5 =5
(2)
8
>>>>
<
>>>>
:
x1?2x2 +3x3? x4? x5 = 4
x1 + x2? x3 + x4?2x5 = 1
2x1? x2 + x3?2x5 = 3
2x1 +2x2?5x3 +2x4? x5 =?4
(3)
8
>><
>>:
x1?2x2 +x3 + x4 = 1
x1?2x2 +x3? x4 =?1
x1?2x2 +x3 +5x4 = 5
(4)
8
>>>>
<
>>>>
:
2x1 + x2 + x3 = 2
x1 +2x2 + x3 = 3
x1 + x2 +5x3 =?7
2x1 +2x2?3x3 = 12
(5)
8>
>>><
>>>>
:
2x1? 2x2 + x3? x4 + x5 =0
x1? 4x2 +2x3?2x4 +3x5 =0
4x1?10x2 +5x3?5x4 +7x5 =0
x1 + 2x2? x3 + x4?2x5 =0
x5,(1)x2cx2d.
(2) x3 = 1?x1 +2x2,x4 =?4x1 +5x2,x5 =?1?x1 +2x2,x1;x2x22x67x4ex7axbx20.
(3) x1 = 2x2?x3,x4 = 1,x2;x3x22x67x4ex7axbx20.
￠ 2 ￠
(4) x1 = 1,x2 = 2,x3 =?2.
(5) x1 = 13 x5,x2 = 16 (3x3?3x4 +5x5),x3;x4;x5x22x67x4ex7axbx20.
2,x7bx7c?,x27x40x41x42 8
>><
>>:
2x1? x2 + x3 + x4 = 1
x1 +2x2? x3 + 4x4 = 2
x1 +7x2?4x3 +11x4 =?
x47x2d,x57x73x38x15x48x7dx2d.
x5,x63x62? = 5x52x47x2cx7ex9x2d,x3cx48x7dx2dx22x1 = 15 (4?x3?6x4),x2 = 15 (3+3x3?7x4),x3;x4x22
x67x4ex7axbx20.
3,a,bx7ax8x4x52,x74x26x40x41x428
>>>>
<
>>>>
:
x1 + x2 + x3 + x4 + x5 = 1
3x1 +2x2 + x3 + x4?3x5 =a
x2 +2x3 +2x4 +6x5 = 3
5x1 +4x2 +3x3 +3x4? x5 = b
x47x2d? x6bx47x2dx15xfx21x1d,x73x48x7dx2d.
x5,x63x62a = 0;b = 2x52x47x2d,x3cx48x7dx2dx22x1 =?2 + x3 + x4 + 5x5,x2 = 3? 2x3? 2x4? 6x5,
x3;x4;x5x22x67x4ex7axbx20.
4,x53x54x40x41x42 8
>>>
>>>>
<
>>>
>>>>
:
x1?x2 = a1
x2?x3 = a2
x3?x4 = a3
x4?x5 = a4
x5?x1 = a5
x47x2dx15x30x11x40x26x31x32x13
a1 +a2 +a3 +a4 +a5 = 0:
x6bx47x2dx15xfx21x1d,x73x38x15x48x7dx2d.
x3x4,())x8x74x26x40x41x42x47x2d,x18(c1;c2;c3;c4;c5)x22x3cx48x66x2d,x76x38x51x52x4bx40x41x42,x57x76x28x29x65x17,
x4x50a1 +a2 +a3 +a4 +a5 = 0.
(() x8a1 + a2 + a3 + a4 + a5 = 0,x4ax4ex6x7x48x66x40x41x50x5 = x1 + a5,x47x48x51x52x4fx48x66x40x41,x50
x4 = a4 +a5 +x1,x3 = a3 +a4 +a5 +x1,x2 = a2 +a3 +a4 +a5 +x1,x76x2;x3;x4;x5x51x52x3dx48x66x40x41,
x50
x1?(a2 +a3 +a4 +a5 +x1) =?a2?a3?a4?a5 = a1:
x23x24x4bx40x41x42x15x48x7dx2dx228
>>>
><
>>>
>:
x2 = a2 +a3 +a4 +a5 +x1
x3 = a3 +a4 +a5 +x1
x4 = a4 +a5 +x1
x5 = a5 +x1
x1x22x67x4ex7axbx20.
5,x73x48x9x3ax29f(x) = a0x3+a1x2+a2x+a3,x27f(1) =?3,f(?1) =?7,f(2) =?1,f(?2) =?21.
x5,f(x) = x3?2x2 +x?3.
￠ 3 ￠
x1 x2 3–3
1,x18fi1 = (2;5;1;3),fi2 = (10;1;5;10),fi3 = (4;1;?1;1),x73x48x1fx20fi,x273(fi1?fi)+2(fi2 +fi) =
5(fi3 +fi).
x5,fi = (1;2;3;4).
2,xaxb3fi+4fl = (2;1;1;2),2fi?3fl = (?1;2;3;1),x73fix42fl.
x5,fi = 117 (2;11;15;10),fl = 117 (7;?4;?7;1).
3,x4ex1fx20flx1bx2ax1fx20fi1,fi2,fi3x15x74x26x42x54:
(1) fi1 = (1;1;1),fi2 = (1;1;?1),fi3 = (1;?1;?1),fl = (1;2;1);
(2) fi1 = (1;3;5),fi2 = (6;3;?2),fi3 = (3;1;0),fl = (5;8;8);
x5,(1) fl = fi1 + 12 fi2? 12 fi3.
(2) fl = 2fi1 +fi2?fi3.
4,x7cx12x1dx1ex1fx20x42x13x29x74x26x65x2a:
(1) fi1 = (1;1;1),fi2 = (1;2;3),fi3 = (1;3;6);
(2) fi1 = (3;2;?5;4),fi2 = (2;1;?3;?5),fi3 = (3;5;?13;11),fi4 = (4;5;?14;?3);
(3) fi1 = (1;?1;2;4),fi2 = (0;3;1;2),fi3 = (1;7;8;9),fi4 = (3;2;1;2);
(4) fi1 = (1;2;?1;4),fi2 = (9;1;2;?3),fi3 = (3;5;0;2),fi4 = (3;2;2;1),fi5 = (1;3;3;2).
x5,(1)x29; (2)x13; (3)x29; (4)x13.
5,x18a1;a2;￠￠￠ ;anx13x64x55x65x43x15xe,x49
fi1 = (1;a1;a21;￠￠￠ ;an?11 );
fi2 = (1;a2;a22;￠￠￠ ;an?12 );
:::::::::::::::::::::::::
fin = (1;an;a2n;￠￠￠ ;an?1n ):
x53x54:x0x48nx46x1fx20x6dx3ex24x4ex1fx20x42fi1;fi2;￠￠￠ ;finx74x26x1bx1c.
x3x4,x1fx20x42fi1;fi2;￠￠￠ ;finx75x2ax15xdx1ex29
jAj =
flfl
flfl
flfl
flfl
fl
1 a1 ￠￠￠ an?11
1 a2 ￠￠￠ an?12
￠￠￠ ￠￠￠ ￠￠￠ ￠￠￠
1 an ￠￠￠ an?1n
flfl
flfl
flfl
flfl
fl
=
Y
16j<i6n
(ai?aj) 6= 0;
x23x24fi1;fi2;￠￠￠ ;finx74x26x2cx2a.
x51x2x0x1x15nx46x1fx20fl,x1fx20x42fl;fi1;￠￠￠ ;finx74x26x65x2a,x43x25x1fx20flx3ex4ex1fx20x42fi1;fi2;￠￠￠ ;finx74
x26x1bx1c.
6,x18x1fx20x42fi1,fi2,fi3x74x26x2cx2a,x53x54:x1fx20x42fi1 +fi2,fi2 +fi3,fi3 +fi1x67x74x26x2cx2a.
x3x4,x18
k1(fi1 +fi2)+k2(fi2 +fi3)+k3(fi3 +fi1) = 0;
x4a
(k1 +k3)fi1 +(k1 +k2)fi2 +(k2 +k3)fi3 = 0:
￠ 4 ￠
x21x22fi1,fi2,fi3x74x26x2cx2a,x23x24 8
><
>:
k1 +k3 = 0
k1 +k2 = 0
k2 +k3 = 0
x2dx50k1 = k2 = k3 = 0,x23x24fi1 +fi2,fi2 +fi3,fi3 +fi1x74x26x2cx2a.
7,x53x54,fi1;fi2;￠￠￠ ;fis (x3cx16fi1 6= 0) x74x26x65x2ax15x30x26x31x32x13x45x46x47x48x66fii (1 < i 6 s) x3ex49
fi1;fi2;￠￠￠ ;fii?1x74x26x1bx1c.
x3x4,x21x22fi1;fi2;￠￠￠ ;fisx74x26x65x2a,x23x24x31x6bx55x33x22x6fx15xek1;k2;￠￠￠ ;ks,x27
k1fi1 +k2fi2 +￠￠￠+ksfis = 0:
x18k1;k2;￠￠￠ ;ksx16x6x7x48x66x55x22x6fx15x22ki,x4ai 6= 1(x29x4afi1 = 0x42x31x18x34x35),x43x25i > 1,x21
k1fi1 +￠￠￠+ki?1fii?1 =?kifii;
fii =? k1k
i
fi1? k2k
i
fi2?￠￠￠? ki?1k
i
fii?1:
8,x53x54:x8x1fx1fx20x42x15x48x66x3ax7fx42x74x26x65x2a,x4ax4fx1fx20x42x67x74x26x65x2a.
x3x4,x18x1fx20x42(II)x22(I)x15x3ax7fx42,x8x1fx20x42(I)x74x26x2cx2a,x4ax4ex6a3.9xb,(II)x67x74x26x2cx2a,x42xaxb
x34x35,x21x4fx1fx20x42x74x26x2cx2a.
9,x1dx1ex23x7dx13x29x2ax2b? x2x15,x17x24x53x54;x0x15,x1x25x4dx6a.
(1)x1efi1;fi2;￠￠￠ ;fisx74x26x65x2a,x4ax3cx16x73x48x66x6dx3ex4ex3cx24x1fx20x74x26x1bx1c;
(2)x1ex1fx20x42fi1;fi2;￠￠￠ ;firx74x26x2cx2a,x1fx20x42fl1;fl2;￠￠￠ ;flsx74x26x2cx2a,x4ax1fx20x42fi1;fi2;￠￠￠ ;fir;fl1;
fl2;￠￠￠ ;flsx67x74x26x2cx2a;
(3) x1ex1fx20x42fi1;fi2;￠￠￠ ;fis x74x26x2cx2a,x1fx20x42fl1;fl2;￠￠￠ ;fls x74x26x2cx2a,x4ax1fx20x42fi1 + fl1;fi2 +
fl2;￠￠￠ ;fis +flsx67x74x26x2cx2a;
(4)x1ex1fx20x42fi1;fi2;￠￠￠ ;firx74x26x65x2a,x4ax48x1dx31x6brx66x55x56x3cx6fx15xek1;k2;￠￠￠ ;kr,x27
k1fi1 +k2fi2 +￠￠￠+krfir = 0;
(5)x1ex1fx20x42fi1;fi2;￠￠￠ ;firx74x26x2cx2a,x4ax38x15x0x8x74x26x42x54x6dx55x56x3cx6f.
x5,(1)x0,x8fi1 = (0;0),fi2 = (1;1)x74x26x65x2a,x71fi2x55x3ex4efi1x74x26x1bx1c.
(2)x0,x8fi1 = (1;1),fi2 = (1;2)x74x26x2cx2a,fl1 = (2;2),fl2 = (0;1)x74x26x2cx2a,x71fi1;fi2;fl1;fl2x74x26
x65x2a.
(3) x0,x8 fi1 = (1;1),fi2 = (0;1),fl1 = (1;?1),fl2 = (1;2) x74x26x2cx2a,x71 fi1 + fl1 = (0;0),
fi2 +fl2 = (1;3)x74x26x65x2a.
(4)x0,x8fi1 = (0;0),fi2 = (0;1)x74x26x65x2a,x71x2x0x1x15k1 6= 0;k2 6= 0x6dx47k1fi1 +k2fi2 6= 0.
(5)x0,fi1;fi2;￠￠￠ ;firx15x6fx74x26x42x54x6fx56x3cx6f.
x1 x2 3–4
1,x6bx34x46x27x8x70x71R3x16,x1dx1exdx54Wx13x29x75x2aR3x15x74x26x0x70x71?
(1) W = f(a;b;c) 2R3 j (a;b;c)? (1;1;1)g;
(2) Wx13x2x17x6bx6ex2ex74x79x15x33x10x1fx20x23x75x2ax15xdx54;
(3) Wx13x42x70x71x16x6ex49x1dx6ex6fx1fx20(x0;y0;z0)x15x15x35x56x3cx1dx4x15x33x10x1fx20x23x75x2ax15xdx54.
x5,(1)x13; (2)x8x2ex74x4ex4bx17,x13;x29x4a,x55x13; (3)x15x35x56x3c …2,x13;x29x4a,x55x13.
￠ 5 ￠
2,x18V x22xex9Kx79nx46x1fx20x70x71,x7cx7dV x15x1dx1ex0xdWx13x29x75x2aV x15x74x26x0x70x71.
(1)x18fi1;fi2;￠￠￠ ;firx22V x16x1bx1dx15rx66x1fx20,
W = ffl 2 V j fi1;fi2;￠￠￠ ;fir;flx74x26x65x2ag;
(2)x18fi1;fi2;￠￠￠ ;firx22V x16x1bx1dx15rx66x1fx20,Wx13V x16x55x63x4efi1;fi2;￠￠￠ ;firx74x26x1bx1cx15x33x10x1fx20
x23x75x2ax15xdx54.
x5,(1)x13; (2)x55x13.
3,x18fi1;fi2;￠￠￠ ;firx22Knx16x1bx1dx15rx66x1fx20,x53x54:
W = f(c1;c2;￠￠￠ ;cr) j c1fi1 +c2fi2 +￠￠￠+crfir = 0g
x42x2aKrx15x0x70x71.
x3x4,x10x11W Krx3f(0;0;￠￠￠ ;0) 2 W,x43x25Wx6ex70.
x2x0x1x15(a1;￠￠￠ ;ar);(b1;￠￠￠ ;br) 2 Wx24x68k 2 K,x47
(a1 +b1)fi1 +￠￠￠+(ar +br)fir = a1fi1 +￠￠￠+arfir +b1fi1 +￠￠￠+brfir = 0+0 = 0:
x23x24
(a1;￠￠￠ ;ar)+(b1;￠￠￠ ;br) 2 W:
(ka1)fi1 +(ka2)fi2 +￠￠￠+(kar)fir = k(a1fi1 +a2fi2 +￠￠￠+arfir) = k￠0 = 0:
x23x24
k(a1;￠￠￠ ;ar) 2 W:
Wx2ax22Krx15x0x70x71.
4,x18W1;W2;￠￠￠ ;Wsx22Knx15sx66x74x26x0x70x71,W = W1 [W2 [￠￠￠[Ws,x53x54,Wx22Knx15x74x26
x0x70x71x15x30x11x40x26x31x32x13,x31x6bi(1 6 i 6 s),x27W = Wi.
x3x4,x30x11x26x13x10x11x15,x1dxfx53x40x26x26,x2sx1ax2x50x44.x62s = 1x52x22x23x10x11x2ax2b,x31x1dx22x23x2s?1
x2ax2b,xbx3W = W1 [W2 [￠￠￠[Ws,x8x1fW 6= Ws,x4ax3ex7afl 2 W nWs,x2x3cx0x1x15fi 2 Ws,x40x47
fl+kfi 2 W nWs (x43fl+kfi 2 Wsx24x68fi 2 Wsx3ex24x5ex50fl 2 Ws,x34x35),x62k = 1;￠￠￠ ;sx52,sx66x1fx20x16x40
x47x37x66x1fx20x4x3cx43x48x66Wi (1 6 i 6 s?1),x77x37x66x1fx20x65x5x7x3ex50fi 2 Wi,x21x4fWs W1[￠￠￠[Ws?1,
x3cx13W = W1 [￠￠￠[Ws?1,x33x1ax2x50x31x18,x3ex50x48x66i,1 6 i 6 s?1x27x50W = Wi,x22x23x2ax2b.
x1 x2 3–5
1,x73x4ex1dx1ex1fx20x23x2ex2ax15x74x26x0x70x71x15x7ax42x46xe:
(1) fi1=(2;1;11;2);fi2=(1;0;4;?1);fi3=(1;4;16;15);fi4=(2;?1;5;?6),fi5 = (1;6;22;23);
(2) fi1 = (1;?4;15;5;?4),fi2 = (0;7;29;?8;7),fi3 = (2;?1;1;1;?3),fi4 = (1;?4;3;5;?4).
x5,(1)x46xe2,x7afi1;fi2.
(2)x46xe4,x7afi1;fi2;fi3;fi4.
2,x18Wx22x1fx20x70x71V x15x0x70x71,fi1;fi2;￠￠￠ ;firx22Wx15x48x66x7a,fli =
rP
j=1
aijfij,i = 1;2;￠￠￠ ;r.
x53x54,fl1;fl2;￠￠￠ ;flrx67x13Wx15x7ax15x30x11x40x26x31x32x13fl
flfl
flfl
flfl
flfl
a11 a12 ￠￠￠ a1r
a21 a22 ￠￠￠ a2r
:::::::::::::::::::
ar1 ar2 ￠￠￠ arr
flfl
flfl
flfl
flfl
fl
6= 0:
￠ 6 ￠
x3x4,()) x18fl1;fl2;￠￠￠ ;flr x67x13W x15x7a,x18x5dx5eA = (aij),(k1;k2;￠￠￠ ;kr)x13x3x48x74x26x40x41x42
XA = 0x15x48x66x2d,x4
(k1 k2 ￠￠￠ kr)A = 0:
x4a
k1fl1 +k2fl2 +￠￠￠+krflr = (k1 ￠￠￠kr)A
0
B@
fi1
...
fir
1
CA = 0:
x21x22fl1;fl2;￠￠￠ ;flrx74x26x2cx2a,x23x24k1 = k2 = ￠￠￠ = kr = 0,x4XA = 0x7bx47x6fx2d,x23x24jAj6= 0.
(()x18
k1fl1 +k2fl2 +￠￠￠+krflr = 0;
x4a
(k1 ￠￠￠kr)A
0
B@
fi1
...
fir
1
CA = 0;
x21x22fi1;fi2;￠￠￠ ;firx74x26x2cx2a,x23x24
(k1 k2 ￠￠￠ kr)A = 0:
x4ex3cjAj 6= 0,x21x3x48x74x26x40x41x42x7bx47x6fx2d,x4k1 = k2 = ￠￠￠ = kr = 0,x3cx13fl1;fl2;￠￠￠ ;flrx74x26x2cx2a,x51
x21dimW = r,x23x24fl1;fl2;￠￠￠ ;flrx13Wx15x7a.
3,x18V x22xex9Kx79x15nx46x1fx20x70x71,x53x54:x2x0x8x3bx3cnx15x67x11xem,x48x1dx31x6bx4eV x15mx66x1fx20
x42x2ax15x1fx20x42,x27x3cx16x0x8nx66x1fx20x6dx75x2aV x15x7a.
x3x4,x4ex60x61 3–4.4 x15x22x1fx3ex24xbx60,V x55x3ex63x1bx1cx2ax38x15x47x73x9x66x5x74x26x0x70x71x15x57xd,x2
m > nx6cxdxex4fx2x50x44,x62m = nx52x22x23x2ax2b,x31x18xax6ax6x8x2dx2ex31x32x15m?1 > nx66x1fx20x15x1fx20x42
fi1;￠￠￠ ;fim?1,x4ex3cx16x0x1n?1x15x1fx20x7x2ax15x74x26x0x70x71x1fx22Wi (i = 1;￠￠￠ ;s),x4ax21V 6= SWi,x31x6b
x1fx20fim 62SWi (i = 1;￠￠￠ ;s),x4ax1fx20x42fi1;￠￠￠ ;fimx67x2dx2ex31x32.
x1 x2 3–6
1,?x7ax8x4x52,x40x41x42 8
>><
>>:
(?+3)x1 + x2 + 2x3 =?
x1 +(1)x2 + x3 =2?
3(?+1)x1 +?x2 +(?+3)x3 =3?
x47x2d? x6bx47x2dx15xfx21x1d,x73x25x48x7dx2d.
x5,x6axexdx1ex29x56x3c?2(1),x62? 6= 0;1x52,x40x41x42x47x2cx48x2d:
8
>>><
>>>:
x1 =31
x2 =?+31
x3 = 31 ;
x62? = 0x52,x48x7dx2dx22,x1 =?x3,x2 = x3,x3x13x67x4ex7axbx20;
x62? = 1x52,x4bx40x41x42x2cx2d.
￠ 7 ￠
2,a,bx7ax8x4x52,x40x41x42 8
>><
>>:
ax1 + x2 +x3 =4
x1 + bx2 +x3 =3
x1 +2bx2 +x3 =4
x47x2d? x6bx47x2dx15xfx21x1d,x73x25x48x7dx2d.
x5,(a)x62a 6= 1x3fb 6= 0x52,x40x41x42x47x2cx48x2d:
8>
>><
>>>:
x1 = 2b?1b(a?1)
x2 = 1b
x3 = 2ab?4b+1b(a?1) ;
(b)x62b = 0x52,x44x62a = 1,b 6= 12 x52,x4bx40x41x42x2cx2d;
(c)x62a = 1,b = 12 x52,x48x7dx2dx22,x1 = 2?x3,x2 = 2,x3x13x67x4ex7axbx20.
3,x73x1dx1ex3x48x74x26x40x41x42x15x7ax8x2dx6a:
(1)
8
>>>>
<
>>>>
:
x1 + x2 + x3 + x4 + x5 =0
3x1 +2x2 + x3 + x4? x5 =0
5x1 +4x2 +3x3 +3x4 + x5 =0
x2 +2x3 +2x4 +4x5 =0
(2)
8
>><
>>:
3x1 +2x2? 5x3 + 4x4 =0
3x1? x2 + 3x3? 3x4 =0
3x1 +5x2?13x3 +11x4 =0
(3)
8
>>>>
<
>>>>
:
x1 + x2 + x3 + x4 + x5 =0
2x1 +2x2 + x3 + x4?2x5 =0
5x1 +4x2?3x3 +4x4 + x5 =0
x2 +6x3? x4?4x5 =0
(4)
8>
>>><
>>>>
:
x1?2x2 +3x3?4x4 =0
x2? x3 + x4 =0
x1 +3x2?3x4 =0
x1?4x2 +3x3?2x4 =0
x5,(1) (1;?2;1;0;0),(1;?2;0;1;0),(3;?4;0;0;1).
(2) (?1;24;9;0),(2;?21;0;9).
(3) (?7;7;?1;1;0),(?25;28;?4;0;1).
(4) (0;1;2;1).
4,x53x54:x8x1fx3x48x74x26x40x41x42
8>
>>><
>>>>
:
a11x1 +a12x2 +￠￠￠+a1nxn = 0
a21x1 +a22x2 +￠￠￠+a2nxn = 0
::::::::::::::::::::::::::::::::
an1x1 +an2x2 +￠￠￠+annxn = 0
x15x6axex5dx5eAx15xdx1ex29jAj = 0,x40x41x42x15x9x13n? 1,x57x3fx5dx5eAx16aklx15x51xex24x0x29Akl 6= 0,x20x30
(Ak1;Ak2;￠￠￠ ;Akn)x13x4fx3x48x74x26x40x41x42x15x48x66x7ax8x2dx6a.
￠ 8 ￠
x3x4,x4ex3c
ak1Ak1 +ak2Ak2 +￠￠￠+aknAkn = jAj = 0;
ai1Ak1 +ai2Ak2 +￠￠￠+ainAkn = 0; x62i 6= kx52,
x21x4f(Ak1;Ak2;￠￠￠ ;Akn)x13x61x18x3x48x74x26x40x41x42x15x2d,x51x21Akl 6= 0,x77x13x48x66x6ex6fx2d,x4ex31x18xbx60x40x41
x42x15x9x13n?1,x23x24x4fx3x48x74x26x40x41x42x15x7ax8x2dx6ax4ex48x66x6ex6fx2dx75x2a,x21x4f(Ak1;Ak2;￠￠￠ ;Akn)x13x4f
x3x48x74x26x40x41x42x15x48x66x7ax8x2dx6a.
5,x18x3x48x74x26x40x41x42 8
>>>>
<
>>>>
:
a11x1 +a12x2 +￠￠￠+a1nxn = 0
a21x1 +a22x2 +￠￠￠+a2nxn = 0
::::::::::::::::::::::::::::::::
an?1;1x1 +an?1;2x2 +￠￠￠+an?1;nxn = 0
x15x6axex5dx5ex22A,Mix13x5dx5eAx16xax7x3dix1ex23x50x15(n?1)￡(n?1)x5dx5ex15xdx1ex29,x53x54:
(1) (M1;?M2;￠￠￠ ;(?1)n?1Mn)x13x40x41x42x15x48x66x2d;
(2)x8x1fx77x66x74x26x40x41x42x15x9x22n?1,x6ex66Mi 6= 0,x53x54x40x41x42x15x2dx33x13(M1;?M2;￠￠￠ ;(?1)(n?1)Mn)
x15x18xe.
x3x4,(1)x2fx3x48x74x26x40x41x42
8
>>>>
<
>>>>
:
a11x1 +a12x2 +￠￠￠+a1nxn = 0
::::::::::::::::::::::::::::::::
an?1;1x1 +an?1;2x2 +￠￠￠+an?1;nxn = 0
an1x1 +an2x2 +￠￠￠+annxn = 0
x3cx16an1 = an2 = ￠￠￠ = ann = 0,x4ax4fx74x26x40x41x42x42x4bx40x41x42x43x2d,x3fx6axex5dx5ex56x3c0,x21x4ex79x61,x6x7x48
xdx15x51xex24x0x29(An1;An2;￠￠￠ ;Ann)x22x4bx40x41x42x15x2d,x51Ani = (?1)n+iMi,x23x24(M1;?M2;￠￠￠ ;(?1)n?1Mn) =
(?1)n+1(An1;An2;￠￠￠ ;Ann)x22x4bx40x41x42x15x2d.
(2)x21x4bx40x41x42x15x9x22n?1,x3fx47x48x66Mi 6= 0,x21x4fx4bx40x41x42x15x7ax8x2dx6ax4ex48x66x6ex6fx2dx1fx20x75x2a.
x43x25x6ex6fx2d(M1;?M2;￠￠￠ ;(?1)n?1Mn)x75x2ax4bx74x26x40x41x42x15x48x66x7ax8x2dx6a,x21x4bx40x41x42x15x73x48x66x2dx6d
x13(M1;?M2;￠￠￠ ;(?1)n?1Mn)x15x18xe.
6,x1bx25xcxfx793x66x17(x1;y1);(x2;y2);(x3;y3)x28x74x15x30x11x40x26x31x32.
x5,x1ex17(x1;y1);(x2;y2);(x3;y3)x28x74,x55x3fx18x4fx2ex74x15x40x41x22Ax+By +C = 0,x4a
8>
<
>:
Ax1 +By1 +C = 0
Ax2 +By2 +C = 0
Ax3 +By3 +C = 0
,x3x48x74x26x40x41x42 8
><
>:
x1t1 +y1t2 +t3 = 0
x2t1 +y2t2 +t3 = 0
x3t1 +y3t2 +t3 = 0
x47x6ex6fx2d(A;B;C)
,x3cx6axex5dx5e
A =
0
@
x1 y1 1
x2 y2 1
x3 y3 1
1
A = 0:
￠ 9 ￠
7,x1bx25xcxfx793x31x2ex74
a1x+b1y +c1 =0;
a2x+b2y +c2 =0;
a3x+b3y +c3 = 0
x28x17x15x30x11x40x26x31x32.
x5,x4f3x31x2ex74x28x17x15x30x11x40x26x31x32x13x65x2cx15x3x48x74x26x40x41x42
a1x+b1y +c1z =0;
a2x+b2y +c2z =0;
a3x+b3y +c3z = 0
x47x6ex6fx2d,x62x3fx63x62x6axex5dx5ex56x3c0,x4
flfl
flfl
flfl
a1 b1 c1
a2 b2 c2
a3 b3 c3
flfl
flfl
flfl = 0:
8,x7ex25x72x4ex34x17(1;2),(1;?2),(0;?1)x15xbx40x41.
x5,(x?2)2 +y2 = 5.
9,x73x60x613–4.3x16x23x1dx4dx15x74x26x0x70x71x15x46xe.
x5,x18
fii =
0
BB
B@
a1i
a2i
...
ani
1
CC
CA;
A = (fi1;fi2;￠￠￠ ;fir):
x4a
(c1;￠￠￠ ;cr) 2 W,
X
cifii = 0
,A
0
B@
c1
...
cr
1
CA = 0,Wx22AX = 0x15x2dx70x71
x23x24
dimW = r?rankA = r?rankffi1;￠￠￠ ;firg:
x1 x2 3–7
1,x73x1dx1ex74x26x40x41x42x15x33x7cx2d:
(1)
8
>>>>
<
>>>>
:
2x1? x2 +5x3 + 7x4 =0
4x1?2x2 +7x3 + 5x4 =0
2x1? x2 + x3? 5x4 =0
2x1? x2 +6x3 +10x4 =0
(2)
8
>><
>>:
2x1 + x2? x3 +x4 =1
x1 +2x2 + x3?x4 =2
x1 + x2 +2x3 +x4 =3
￠ 10 ￠
(3)
8
>>>>
<
>>>>
:
x1 + x2?3x4? x5 =2
x1? x2 +2x3? x4 =1
4x1?2x2 +6x3 +3x4?4x5 =8
2x1 +4x2?2x3 +4x4?7x5 =9
(4)
8>
>>>
<
>>>
>:
3x1 +4x2 +2x3 + x4 +6x5 =?2
2x1 +5x2 +5x3 +3x4 + x5 = 4
x1? x2? x3?2x4 +5x5 =?6
x1 +6x2 +8x3 +5x4?3x5 = 9
x5,(1) k1(4;0;?3;1)+k2(0;8;3;?1).
(2) (1;0;1;0)+k(?3;3;?1;2).
(3) (1;0;0;0;?1)+k1(?1;1;1;0;0)+k2(7;5;0;2;6).
(4) (1;0;0;1;?1)+k(?1;1;0;?1;0).
2,x18·1;·2;￠￠￠ ;·rx13x6ex3x48x74x26x40x41x42
8>
>>><
>>>>
:
a11x1 + a12x2 +￠￠￠ + a1nxn = b1
a21x1 + a22x2 +￠￠￠ + a2nxn = b2
::::::::::::::::::::::::::::::::
am1x1 +am2x2 +￠￠￠ +amnxn =bm
x15x0x1rx66x2d,· =
rP
i=1
ki·i (ki 2 K;i = 1;2;￠￠￠ ;r),x53x54:x62x3fx63x62
rP
i=1
ki = 1x52,·x67x13x77x66x6ex3x48x74x26
x40x41x42x15x2d.
x3x4,x18A = (Aij),B =
0
B@
b1
...
bn
1
CA,x4aB 6= 0.
·x22x79x53x40x41x42x15x2d,A· = B,
nX
i=1
kiB = B,
nX
i=1
ki = 1:
3,x18 0x13x6ex3x48x74x26x40x41x42
8
>>>>
<
>>>>
:
a11x1 +a12x2 +￠￠￠+a1nxn = b1
a21x1 +a22x2 +￠￠￠+a2nxn = b2
::::::::::::::::::::::::::::::::
am1x1 +am2x2 +￠￠￠+amnxn = bm
x15x48x66x2d,·1;·2;￠￠￠ ;·rx13x38x15xcx25x42x15x7ax8x2dx6a.
x53x54,(1) 0;·1;·2;￠￠￠ ;·rx74x26x2cx2a;
(2) 0; 0 +·1; 0 +·2;￠￠￠ ; 0 +·rx67x74x26x2cx2a;
(3)x8 x13x77x66x6ex3x48x74x26x40x41x42x15x0x1x2d,x4a ; 0; 0 +·1;￠￠￠ ; 0 +·rx74x26x65x2a;
(4) Knx16x1fx20 x13x77x66x6ex3x48x74x26x40x41x42x15x2dx15x30x11x40x26x31x32x13x31x6br+1x66xeki(i = 0;1;￠￠￠ ;r),
rP
i=0
ki = 1,x27x50
= k0 0 +k1( 0 +·1)+k2( 0 +·2)+￠￠￠+kr( 0 +·r):
x3x4,(1)x18
fi = k0 0 +k1·1 +￠￠￠+kr·r = 0;
￠ 11 ￠
x4a
0 = Afi = k0A 0 +
rX
i=1
kiA·i = k0B:
x23x24k0 = 0,k1·1 + ￠￠￠ + kr·r = 0,x4ex3c·1;￠￠￠ ;·r x74x26x2cx2a,x3ex50k1 = k2 = ￠￠￠ = kr = 0,x21x4f
0;·1;·2;￠￠￠ ;·rx74x26x2cx2a.
(2)x18
k0 0 +k1( 0 +·1)+￠￠￠+kr( 0 +·r) = 0;
x4a
(k0 +k1 +￠￠￠+kr) 0 +k1·1 +￠￠￠+kr·r = 0;
x4e(1),0;·1;·2;￠￠￠ ;·rx74x26x2cx2a,x23x24
k1 = k2 = ￠￠￠ = kr = 0; k0 +k1 +￠￠￠+kr = 0;
xdx25k0 = k1 = ￠￠￠ = kr = 0,x21x4f 0; 0 +·1; 0 +·2;￠￠￠ ; 0 +·rx74x26x2cx2a.
(3) x8 x13x77x66x6ex3x48x74x26x40x41x42x15x2d,x4a? 0 x13x38x15xcx25x42x15x2d,x23x24x31x6b ki 2 K,x27
0 = Pki·i,x3cx13
= 0 +
X
ki·i =
1?
rX
i=1
ki
!
0 +
rX
i=1
ki( 0 +·i);
x43x25 ; 0; 0 +·1;￠￠￠ ; 0 +·rx74x26x65x2a.
(4) ()) x22x6ex3x48x74x26x40x41x42x15x2d,x4ax4e(3)x15x53x54x3ex50
=
1?
rX
i=1
ki
!
0 +
rX
i=1
ki( 0 +·i);
x43x25x4fx74x26x1bx1cx29x15x6axex39x3ax56x3c?
1?
rX
i=1
ki
!
+
rX
i=1
ki = 1:
(()x8
rP
i=0
ki = 1,x4ax4ex79x61x15x22x23x3exb x13x48x66x2d.
4,x53x54,Knx15x0x48x74x26x0x70x71x40x13Kx79x6ex66nxbx3x48x74x26x40x41x42x15x2dx70x71.
x3x4,x18W x22Knx15x0x48x0x70x71,fi1;fi2;￠￠￠ ;fir (x1ex1fx20)x22W x15x7a,x49A = (fi1;fi2;￠￠￠ ;fir),x4a
rankA = r,x18c1;￠￠￠ ;cn?r 2 Krx13ATX = 0x15x7ax8x2dx6a,x49C = (c1;￠￠￠ ;cn?r)T,x4arankC = n?r.
x3cx13x3x48x74x26x40x41x42CX = 0x15x2dx70x71x15x46xex56x3c r.
x4eATci = 0,x3ex50AT(c1;￠￠￠ ;cn?r) = 0,x4CA = 0,x77x1bx54fiix13CX = 0x15x2d.
x4ex3cfi1;￠￠￠ ;firx74x26x2cx2ax24x68CX = 0x15x2dx70x71x15x46xex56x3c r,x21fi1;￠￠￠ ;firx22CX = 0x15x7ax8x2d
x6a,x43x25Wx22CX = 0x15x2dx70x71.
5,x53x54,Knx15x48x66x6ex70x0xdY x13x48x66x74x26x74x36x15x30x11x40x26x31x32x13,x38x13x6ex48x74x26x40x41x42x15x2dxd
x54.
x3x4,())x18Y x13x48x66x74x26x74x36,x4aY = fi0 +W,x3cx16Wx13x48x66x74x26x0x70x71,fi0 =
0
B@
a1
...
an
1
CA2 Kn.
x4aWx22x6ex3x48x74x26x40x41x42AX = 0x15x2dx70x71,x49B = Afi0 =
0
B@
b1
...
bm
1
CA,xbx3AX = B,xexb,Y x22AX = B
x15x2dxd.
￠ 12 ￠
(()x6fx13x1dx57.4x15x22x23.
6,x18Y1,Y2 x22x1fx20x70x71V x15x37x66x74x26x74x36,x1dx1exdx54x13x29x75x2aV x15x74x26x74x36?
(1) Y1 [Y2;
(2) Y1 +Y2 = ffi1 +fi2 j fi1 2 Y1;fi2 2 Y2g.
x5,(1)x55x48x1d,x8x7afi;flx74x26x2cx2a,x49
Y1 = L(fi); Y2 = L(fl);
x4aY1;Y2x6dx13x74x26x74x36,x71fi+fl 62 Y1 [Y2.
(2)x13,x8
Y1 = fi1 +W1; Y2 = fi2 +W2;
x3cx16W1;W2x22x0x70x71,x4a
Y1 +Y2 = (fi1 +fi2)+(W1 +W2);
x3exbY1 +Y2x67x13x74x26x74x36.
7,x18fi0;fi1;￠￠￠ ;firx22V x15r +1x66x1fx20,x53x54:
Y =
( rX
i=0
kifii
flfl
flfl
fl
rX
i=0
ki = 1;ki 2 K;i = 0;1;￠￠￠r
)
x75x2aV x15x48x66x74x26x74x36.
x3x4,x18
rX
i=0
kifii 2 Y;
rX
i=0
ki = 1;
rX
i=0
lifii 2 Y;
rX
i=0
li = 1;
x4ax2x0x1x15k;l 2 K,k +l = 1,x47
k
rX
i=0
kifii
!
+l
rX
i=0
lifii
!
=
rX
i=0
(kki +lli)fii;
x25
rX
i=0
kki +
rX
i=0
lli = k
rX
i=0
ki +l
rX
i=0
li = k +l = 1;
x3cx13
k
rX
i=0
kifii
!
+l
rX
i=0
lifii
!
= 0;
x43x25Y x13x74x26x74x36.
8,x18Y x22x1fx20x70x71V x15x48x66x74x26x74x36,x53x54:x31x6bY x16x15r +1x66x1fx20fi0;fi1;￠￠￠ ;fir,x27
Y =
( rX
i=0
kifii
flfl
flfl
fl
rX
i=0
ki = 1;ki 2 K;i = 0;1;￠￠￠r
)
:
x3x4,x18Y = fi0 +W,x3cx16Wx13x0x70x71,x18Wx15x7ax22·1;￠￠￠ ;·r,x49
fi0 = fi0;fi1 = fi0 +·1;￠￠￠ ;fir = fi0 +·r;
x4afii 2 Y,x3fx2x0x1x15ki 2 K,
rP
i=0
ki = 1,x47
rX
i=0
kifii =
rX
i=0
kifi0 +
rX
i=0
ki·i 2 fi0 +W = Y:
￠ 13 ￠
x4dx39,x2x0x1x15fi = fi0 +· 2 Y = fi0 +W,x31x6bki,x27· =
rP
i=1
ki·i,x43x25
fi =
1?
rX
i=1
ki
!
fi0 +
rX
i=1
ki(fi0 +·i);
x3cx16?
1?
rX
i=1
ki
!
+k1 +k2 +￠￠￠+kr = 0:
x77x53x54x72
Y =
( rX
i=0
kifii
flfl
flfl
fl
rX
i=0
ki = 1;ki 2 K;i = 0;1;￠￠￠r
)
:
x1 x2 3–8
1,x6bx1bx1dx15x5cx5dx57x55x6ax16,x73x1dx1excxfx15x48x7dx40x41x3ax5exex40x41:
(1)x4e(?1;2;0);(?2;?1;4);(3;1;?5)x34x17x15xcxf;
(2)x4ex17(3;1;2)x3a(1;0;?2),xcxdx3cx1fx20?!v = (1;?2;?3)x15xcxf.
x5,(1)x4e3x17x15xcxfx34x17x29x40x41x13:
flfl
flfl
flfl
x+1?1 4
y?2?3?1
z 4?5
flfl
flfl
flfl = 0;
x65x66x7x50
19x+11y +13z?3 = 0:
x38x15x5exex40x41x22,8
><
>:
x =?1?u+4v
y = 2?3u?v
z = 4u?5v:
(2) x4exaxbx31x32,xcxfx72x4e(3;1;2),x38x15x40x1fx1fx20x13?1 =?!v = (1;?2;?3)x24x68?2 = (1? 3;0?
1;?2?2) = (?2;?1;?4),x21x4fxcxfx15x5exex40x41x22
8>
<
>:
x = 3+u?2v
y = 1?2u?v
z = 2?3u?4v;
xcxfx15x48x7dx40x41x22 fl
flfl
flfl
fl
x?3 1?2
y?1?2?1
z?2?3?4
flfl
flfl
flfl = 0;
x65x66x7x50x+2y?z?3 = 0.
2,x6bx1bx1dx15x5cx5dx57x55x6ax16,x73x1dx1excxfx15x48x7dx40x41:
(1)x4ex17(1;2;?4)x3axx45x15xcxf;
(2)x4ex17(2;1;2)x24x68xcxfΠ1, x+y?z = 0,Π2, 2x?3z?1 = 0x15x3x74x15xcxf;
(3)x4ex17(0;4;?3)x3a(1;?2;6),x3fxcxdx3cxx45x15xcxf;
(4)x4ex17(3;1;?2)x3fxcxdx3cxcxfx?2y?2z +1 = 0x15xcxf.
x5,(1)x18xcxfx15x48x7dx40x41x13Ax+By +Cz +D = 0,x21x22x38x4exx45,x23x24A = D = 0;x51x21x38x4ex17
(1;2;?4),x23x24B = 2C,x21xcxfx15x40x41x222y +z = 0.
￠ 14 ￠
(2)x2dx74x26x40x41x42 (
x+y?z = 0
2x?3z?1 = 0
x73x50xcxfx3x74x79x15x37x66x17
1
2 ;?
1
2 ;0
·
x24x68
0;? 13 ;? 13
·
,x50x8x23x73xcxfx15x34x17x29x40x41:
flfl
flfl
flfl
fl
x?2? 32?2
y?1? 32? 43
z?2?2? 73
flfl
flfl
flfl
fl
= 0;
x65x66x7x505x+3y?6z?1 = 0.
(3)x23x73xcxfx15x48x66x40x1fx1fx20x13(1;0;0),x21x4fx23x73xcxfx15x40x41x22:
flfl
flfl
flfl
x 1 1
y?4 0?6
z +3 0 9
flfl
flfl
flfl = 0;
x65x66x7x503y +2z?6 = 0.
(4)x4ex37xcxfxcxdx15x26x27x3exb,x23x73xcxfx15x40x41x2cx22x?2y?2z+D = 0,x21x1cxcxfx4e(3;1;?2)x17,
x23x243?2+4+D = 0,x4D =?5,x21x23x73x40x41x22x?2y?2z?5 = 0.
3,xaxbx48xcxfx72x4eP0(x0;y0;z0)(z0 6= 0),x3fx6bxx45x3ayx45x79x15xfx32x11x12x13ax3ab,x73x38x15x40x41.
x5,x18xcxfx6bzx45x79x15xfx32x22c,x4ax1cxcxfx15x40x41x22
x
a +
y
b +
z
c = 1:
x51x21P0x17x6bxcxfx79,x23x24 x0a + y0b + z0c = 1,x50 1c = 1z
0
1? x0a? y0b
·
,x23x24xcxfx40x41x22
x
a +
y
b +
1? x0a? y0b
· z
z0 = 1:
4,x73x4eP0(x0;y0;z0),x3fxcxdx3cxcxfAx+By +Cz +D = 0x15xcxfx15x40x41.
x5,x4ex37xcxfxcxdx15x26x27x3exb,x23x73xcxfx15x40x41x2cx22Ax + By + Cz + D1 = 0,x21x1cxcxfx4eP0x17,
x23x24D1 =?(Ax0 +By0 +Cz0),x21x23x73xcxfx40x41x22A(x?x0)+B(y?y0)+C(z?z0) = 0.
5,x53x54x37x618.2.
x3x4,x21x22x1fx20x42x74x26x74x36xcxdx1dx4dx22x77x66x1fx20x6bx74x26x0x70x71W x7b,x21x4f?!v x42xcxfxcxdx15x30x11
x40x26x31x32x13?!v 2 W,x25x74x26x0x70x71Wx51x13x4excx25x40x41Ax+By +Cz = 0x1dx4dx15,x23x24?!v 2 Wx15x30x11
x40x26x31x32x13?!v x15x11x20(X;Y;Z)x2dx2excx25x40x41,x4AX +BY +CZ = 0.
6,x7cx7dx1dx1ex28xcxfx15x65x2ax2fx59:
(1) 2x+y?z = 0x42 12 x+ 14 y? 14 z +2 = 0;
(2) x?2y +z?2 = 0x423x+y?2z?1 = 0;
(3) A1x+B1y +C1z +D1 = 0x42A2x+B2y +C2z +D2 = 0,x3cx16
flfl
flflA1 B1
A2 B2
flfl
flfl6= 0.
6,(1)x21x22
2
1
2
= 11
4
=?1
14
6= 02 ;
x23x24x37xcxfxcxd.
(2)x21x22 13 6=?21,x23x24x37xcxfx65x3.
(3)x21x22A1, B1 6= A2, B2,x23x24x37xcxfx65x3.
7,xaxbx37x66xcxfΠ1, x?2y +pz?1 = 0,Π2, 2x?4y +5z +q = 0,x1x62p;qx7ax8x4x52:
￠ 15 ￠
(1) Π1x42Π2x65x3; (2) Π1x42Π2xcxd; (3) Π1x42Π2x24x54.
x5,(1)x62 12 =?2?4 6= p5,x4p 6= 52,qx7ax0x1x32xex52x37xcxfx65x3.
(2)x62 12 =?2?4 = p5 6=?1q x52,x4p = 52,q 6=?2x52x37xcxfxcxd.
(3)x62 12 =?2?4 = p5 =?1q x52,x4p = 52,q =?2x52x37xcxfx24x54.
8,xaxbx17A(3;10;?5)x3axcxfΠ, 7x?4y?z?1 = 0,x73zx45x79x15x17Bx15x57x55,x27ABxcxdx3cΠ.
x5,x18 B x17x15x57x55x22 (0;0;k),x4a?!AB = (?3;?10;k + 5),x3dx3ex37x61 8.2,?!AB x15x11x20x40x41x2dx2e
7(?3)?4(?10)?(k +5) = 0,x4k = 14,x21Bx17x15x57x55x22(0;0;14).
9,x53x54x34x66xcxf
a1x+b1y +c1z +d1 = 0;
a2x+b2y +c2z +d2 = 0;
k(a1x+b1y +c1z)+l(a2x+b2y +c2z)+m = 0
x62m 6= kd1 +ld2x52,x10x47x66x28x17.
x3x4,x34x66xcxfx47x66x28x17x62x3fx63x62x74x26x40x41x42
8>
<
>:
a1x+b1y +c1z =?d1
a2x+b2y +c2z =?d2
k(a1x+b1y +c1z)+l(a2x+b2y +c2z) =?m
x47x2d,x2x77x66x40x41x42x2fx5cx56x3dx4a,x43x3d3x66x40x41x5x7x3d1x66x40x41x15k x18x3ax3d2x66x40x41x15lx18x7x50x8
0 = kd1 +ld2?m,x62m 6= kd1 +ld2x52,x77x13x66x34x35x40x41,x21x4fx10x47x66x28x17.
10,x53x54x0x8x48x66x6ax4ex65x3x15x37xcxf
Π1, A1x+B1y +C1z +D1 = 0;
Π2, A2x+B2y +C2z +D2 = 0
x15x65x3x2ex74Lx15xcxfx40x41x63x7ex2a
fi(A1x+B1y +C1z +D1)+fl(A2x+B2y +C2z +D2) = 0;
x3cx16,fi;flx13x55x33x22x6fx15x32xe.
x3x4,Lx16x15x17x15x57x55x2dx2eΠ1x42Π2x15x40x41,x43x25x67x2dx2ex40x41fi(A1x+B1y+C1z+D1)+fl(A2x+
B2y +C2z +D2) = 0,x21x4fLx6bx4fxcxfx79.
x4dx39,x8x1fxcxfΠ0x72x4ex2ex74L,x14x15x3ex6bΠ0x79x7ax48x66x55xax3cLx15x17M(x0;y0;z0),x49
fi0 = A2x0 +B2y0 +C2z0 +D2; fl0 = A1x0 +B1y0 +C1z0 +D1;
x4ex3cM 62 L,x23x24fi0;beta0x55x33x220,M0x15x57x55x10x11x2dx2ex24x1dx40x41:
fi0(A1x+B1y +C1z +D1)?fl0(A2x+B2y +C2z +D2) = 0;
x4ex4fxfx15x55x23xbx79x53x40x41x64x1dx15xcxfx48x1dx72x4ex3x74L,x4ex72x4ex48x31x2ex74x68x74x30x48x17x15xcxfx15x2cx48x26,x3e
x5fx79x53x40x41x1dx4dx15xcxfx6fx13Π0.
11,x18xcxfΠ, Ax+By+Cz+D = 0x42x4cx11x37x17M1(x1;y1;z1)x42M2(x2;y2;z2)x15x2ex74x65x3x3cx17
M,x25x3f!M1M = k!MM2,x53x54:
k =? Ax1 +By1 +Cz1 +DAx
2 +By2 +Cz2 +D
:
￠ 16 ￠
x3x4,x18x17Mx15x57x55x13(x0;y0;z0),x4ax4ex1dx65x11x17x66x29xb8
>>>
<
>>>:
x0 = x1 +kx21+k
y0 = y1 +ky21+k
z0 = z1 +kz21+k
(k 6=?1):
x71x4ex3cM0 2 Π,x23x24
A x1 +kx21+k +B y1 +ky21+k +C z1 +kz21+k +D = 0;
x4cx6bx7x4x50
k =? Ax1 +By1 +Cz1 +DAx
2 +By2 +Cz2 +D
:
12,x48xcxfx42x70x71x6cx38x36ABCDx15x38AB;BC;CD;DAx11x12x3x3cP,Q,R,S,x4a
AP
PB ￠
BQ
QC ￠
CR
RD ￠
DS
SA = 1:
x19x53x39.
x75x75x75x75x75x75x75x75x75x72x40x8x12x13x12
x12x12x13
x12x12x13
x12x12x12
x13x12x12
x13x12x12
x12x13x12
x12x13x12
x39x15x2x8x40x72x75x75x75x75x75x75x75x75x75
x2x15x39x12x13
x12x12x13
x12x12x12
x13x12x12
x13x12x12
x12x13x12
x12x13x12
x12x12x13
x12
x7x4dx3bx3bx3ax3bx3bx3ax3bx3bx3ax3bx6bx6bx6b
x6cx6bx6b
x6bx6bxd
x50x50x50x58
x5x18x42x22x23
x22x23x22
x23x22x23
x75x75x75x75x75x75x75x50x5
x5
x7x1dx34x8x20x40
x62x2dx72
x7x1fx3ax9x21x46x4dx1f
x4dx1f x4dx1f x4dx1f x4dx1f x4dx1f x4dx1f x7x58xa x58x24 x58xa
x66x2a x65x2b
x66x2b x66x2a
x65x2b x66xdx4 x42x18 x41x19 x41x19
x41x19 x41x19
x41
B
P A S
D
Q
C
R
x012x1x2
x3x4,x8x9,x14x15x24Bx17x22x4bx17,x24?!e1 =?!BP,?!e2 =?!BQ,?!e3 =?!BDx22x7ax1fx20x75x2fx48x66x5cx5dx57x55x6a
[B;?!BP;?!BQ;?!BD],x49
!QC = a?!BQ;?!CR = b?!RD;?!DS = c?!SA;?!PA = d?!BP:
x18
!BR = k?!BQ+l?!BD = k?!e
2 +l?!e3;
!BS = m?!BP +n?!BD = m?!e
1 +n?!e3;
x4a
!CR =?!BR!BC = (k?1?a)?!e
2 +l?!e3;
!RD =?!BD!BR =?k?!e
2 +(1?l)?!e3:
x4e?!CR = b?!RDx3ex50,(
k?1?a =?bk
l = b(1?l); x4
( a = k +l?1
1?l
b = l1?l,
x51x21
!DS =?!BS!BD = m?!e
1 +(n?1)?!e3;
!SA =?!BA!BS = (1+d?m)?!e
1?n?!e3:
x4e?!DS = c?!SAx3ex50,(
m = c(1+d?m)
n?1 =?cn; x4
( c = 1?n
n
d = m+n?11?n,
￠ 17 ￠
x23x24
AP
PB ￠
BQ
QC ￠
CR
RD ￠
DS
SA =
dbc
a =
l(m+n?1)
n(k +l?1), (*)
x51,
!QR =?!BR!BQ = (k?1)?!e
2 +l?!e3;
!QP =?!BP!BQ =?!e
1!e2;
!QS =?!BS!BQ = m?!e
1!e2 +n?!e3;
x21x22x773x66x1fx20x28xf,x23x24 fl
flfl
flfl
fl
0 1 m
k?1?1?1
l 0 n
flfl
flfl
flfl = l(m?1)?n(k?1) = 0:
x43x25
l(m+n?1) = ln+l(m?1) = ln+n(k?1) = n(k +l?1):
x76x79x29x51x52(*)x4x50:
AP
PB ￠
BQ
QC ￠
CR
RD ￠
DS
SA = 1:
x1 x2 3–9
1,x6bx1bx1dx15x5cx5dx57x55x6ax16,x73x1dx1ex2ex74x15x40x41:
(1)x4ex17P(3;1;?1)x3fxcxdx3cx1fx20?!v (4;7;?8);
(2)x4ex17P0(?3;0;1)x3aP1(2;5;1);
(3)xaxbx34x35x36x15x34x66x7ex17x13Ai(xi;yi;zi)(i = 1;2;3),x73x34x31x16x74x15x40x41.
x5,(1) x?34 = y?17 = z +1?8,
(2)x1ax37x17x29x40x41x3ex50 x+35 = y5 = z?10,
(3) x?xi
xi? xi+1 +xi+22
= y?yi
yi? yi+1 +yi+22
= z?zi
zi? zi+1 +zi+22
,x62i+1,i+2x3bx3c3x52x4x5x73.
2,x6bx2ex35x57x55x6ax16,x73x4ex17P(1;6;3)x3fxcxdx3cxcxf3x+y?2z?5 = 0x15x2ex74x15x40x41.
x5,x18x2ex74x15x40x1fx1fx20x22? = (A;B;C),x4a?x42xcxf3x+y?2z?5 = 0xcxd,x213A+B?2C = 0,
x25x2ex74x40x41x22 x?1A = y?6B = z?3C,
3,x73x4ex17A(0;?2;1)x3fxcxdx3cx2ex74
( x+6y?4z +2 = 0
x+y +z?3 = 0 x15x2ex74x40x41.
x5,xax76x2ex74x40x41x4cx22x55x12x40x41 x?22 = y?1 = z?1?1,x3cx40x1fx1fx20x22?(2;?1;?1),x21x23x73x2ex74
x40x41x22 x2 = y +2?1 = z?1?1,
4,x73x2ex74L, x3 = y?2?2 = z +42 x42xcxfΠ, 2x?3y +2z?2 = 0x15x3x17x57x55.
x5,x4ex2ex74x7ex2ax5exex40x41,8
><
>:
x = 3t
y = 2?2t
z =?4+2t:
x11x7x51x52xcxfx40x41x50
2(3t)?3(2?2t)+2(?4+2t)?2 = 0;
x2dx50t = 1,x23x24x3x17x22(3;0;?2).
￠ 18 ￠
5,x73x4ex17A(3;1;2)x68x2ex74L, x1 = y?2 = z +2?3 x15xcxfx15x40x41.
x5,x2ex74x79x47x48x17(0;0;?2),x21x4fxcxfx15x40x1fx1fx20x13?1 = (1;?2;?3),?2 = (?3;?1;?4),x21xcxf
x40x41x22 fl
flfl
flfl
fl
x?3 1?3
y?1?2?1
z?2?3?4
flfl
flfl
flfl = 0;
x78x67x505x+13y?7z?14 = 0.
6,xaxbx2ex74L, x?1m = y?a?2 = z +23,xcxfΠ, x?2y?4z +1 = 0,x1x62a;mx7ax2fx30x4x52
(1) Lx42Πx65x3; (2) Lxcxdx3cΠ; (3) Lx6bΠx7b.
x5,(1)x2ex74x15x40x1fx1fx20x13? = (m;?2;3),x1eLx42Πx65x3,x4a?x55x42Πxcxd,x21m + 4?12 6= 0,x4
m 6= 8,ax13x0x1x32xe.
(2)?x40x41x42Πxcxd,x4m = 8,x43x52Lx79x15x17(1;a;?2)x55x6bΠx79,x41?2a+8+1 6= 0,a 6= 5.
(3) m = 8,a = 5.
7,x73x72x4ex17(2;2;2)x3fx42x37x2ex74
x
1 =
y
2 =
z
3 x3a
x?1
2 =
y?2
1 =
z?3
4
x6dx65x3x15x2ex74x15x40x41.
x5,x18x23x73x2ex74x15x40x1fx1fx20x13? = (X;Y;Z),xaxbx3dx48x31x2ex74x79x15x17M1(0;0;0),x40x1fx1fx20?1 =
(1;2;3),x3dx62x31x2ex74x79x15x17M2(1;2;3),x40x1fx1fx20?2 = (2;1;4),x22x27x23x73x2ex74x42x3dx48x31x2ex74x65x3,x40x41
x27 fl
flfl
flfl
fl
2 1 X
2 2 Y
2 3 Z
flfl
flfl
flfl = 2X?4Y +2Z = 0:
x43x5,x22x27x23x73x2ex74x42x3dx62x31x2ex74x65x3,x40x41x27fl
flfl
flfl
fl
1 2 X
0 1 Y
1 4 Z
flfl
flfl
flfl = X?6Y +Z = 0:
x2dx4fx74x26x40x41x42x50X, Y, Z = 1, 0,?1,x21x4fx23x73x2ex74x15x40x41x13
x?2
1 =
y?2
0 =
z?2
1,
8,x76x1dx1ex2ex74x15x48x7dx29x40x41x4cx2ax55x12x40x41.
(1)
( 2x?3y +4z?12 = 0
x+4y?2z?10 = 0; (2)
( 3x+2y?4z?5 = 0
6x?y?2z?4 = 0;
(3)
( 3x?y +2 = 0
4y +3z +1 = 0:
x5,(1)x4fx2ex74x15x40x1fx1fx20x13 fl
flfl
fl
3 4
4?2
flfl
flfl;?
flfl
flfl2 4
1?2
flfl
flfl;
flfl
flfl2?3
1 4
flfl
flfl
= (?10;8;11):
x22x73x2ex74x79x15x17,x2dx4bx74x26x40x41x42,x50x8x48x66x2d(8;0;?1),x21x4fx2ex74x15x55x12x40x41x13
x?8
10 =
y
8 =
z +1
11,
(2)x4fx2ex74x15x40x1fx1fx20x13 fl
flfl
fl
2?4
1?2
flfl
flfl;?
flfl
flfl3?4
6?2
flfl
flfl;
flfl
flfl3 2
6?1
flfl
flfl
= (?8;?18;?15):
￠ 19 ￠
x22x73x2ex74x79x15x17,x2dx4bx74x26x40x41x42,x50x8x48x66x2d
1
3 ;0;?1
·
,x21x4fx2ex74x15x55x12x40x41x13
x? 13
8 =
y
18 =
z +1
15,
(3)x4fx2ex74x15x40x1fx1fx20x13 fl
flfl
fl
1 0
4 3
flfl
flfl;?
flfl
flfl3 0
0 3
flfl
flfl;
flfl
flfl3?1
0 4
flfl
flfl
= (?3;?9;12):
x4fx1fx20x3ex4cx6bx22(1;3;?4),x22x73x2ex74x79x15x17,x2dx4bx74x26x40x41x42,x50x8x48x66x2d(0;2;?3),x21x4fx2ex74x15x55x12
x40x41x13
x
1 =
y?2
3 =
z +3
4,
9,x73x2ex74x42xcxfx15x3x17.
(1) x+1?2 = y +13 = z?34 x423x+2y +z = 0;
(2)
( 2x+3y +z?1 = 0
x+2y?z +2 = 0 x42xOyxcxf.
x5,(1)x4ex2ex74x7ex2ax5exex40x41,8
><
>:
x =?1?2t
y =?1+3t
z = 3+4t:
x11x7x51x52xcxfx40x41x50
3(?1?2t)+2(?1+3t)+(3+4t) = 0;
x2dx50t = 12,x23x24x3x17x22
2; 12 ;5
·
.
(2)x3x17x15x57x55x13(x;y;0),x2dx40x41x42
( 2x+3y?1 = 0
x+2y +2 = 0 x50x = 8,y =?5,x21x3x17x22(8;?5;0).
10,x73x25x4ex17P0(x0;y0;z0)x57x3fx42x65x3xcxf
Πi, Aix+Biy +Ciz +Di = 0; i = 1;2
x6dxcxdx15x2ex74x15x40x41.
x5,x3dx3ex37x619.1,xcxfΠ1x42Π2x15x3x74x15x40x1fx1fx20x13 fl
flfl
fl
B1 C1
B2 C2
flfl
flfl;?
flfl
flflA1 C1
A2 C2
flfl
flfl;
flfl
flflA1 B1
A2 B2
flfl
flfl
:
x21x23x73x2ex74x40x41x42x3x74xcxd,x21x4fx77x67x13x23x73x2ex74x15x40x1fx1fx20,x21x23x73x2ex74x15x40x41x22
x?x0fl
flfl
fl
B1 C1
B2 C2
flfl
flfl
= y?y0flfl
flflC1 A1
C2 A2
flfl
flfl
= z?z0flfl
flflA1 B1
A2 B2
flfl
flfl
:
11,x2ex74x40x41 (
A1x+B1y +C1z +D1 = 0
A2x+B2y +C2z +D2 = 0
x15x6axex2cx2dx2ex2fx30x31x32x71x63x27x1cx2ex74x13x6bxOzx57x55xcxfx7b.
x5,xcxfxOzx15x40x41x13y = 0,x21x4fx2ex74x13x6bxcxfxOzx7bx15x30x11x40x26x31x32x13x40x41x42
8>
<
>:
A1x+B1y +C1z +D1 = 0
A2x+B2y +C2z +D2 = 0
y = 0
￠ 20 ￠
x47x2cx7ex2d,x25x79x53x40x41x42x42 8
><
>:
A1x+C1z +D1 = 0
A2x+C2z +D2 = 0
y = 0
(*)
x43x2d,x25x40x41x42(*)x47x2cx7ex9x2dx15x30x11x40x26x31x32x13x4fx37x66x40x41x15x6axex2ax65x6a,x4
A1
A2 =
C1
C2 =
D1
D2 6=
B1
B2 ;
x6x7x48x66x55x56x29x13x2ex74x40x41x15x40x26x31x32.
x1 x2 3–10
1,x73x72x4excxf4x?y +3z?1 = 0x3ax+5y?z +2 = 0x15x3x74x3fx2dx2ex1dx1ex31x32x39x48x15xcxf:
(1)x72x4ex4bx17;
(2)x42yx45xcxd;
(3)x72x4e(0;0;1)x17;
(4)x42xOyxcxfx15x3x74xcxdx3cx40x1f(4;5;0).
x5,(1)x23x73x40x41x22
k(4x?y +3z?1)+m(x+5y?z +2) = 0;
x1a(0;0;0)x51x52x50k = 2m,x23x24xcxfx40x41x22
9x+3y +5z = 0:
(2)x23x73x40x41x22
k(4x?y +3z?1)+m(x+5y?z +2) = 0;
yx45x15x40x1fx1fx20x13? = (0;1;0),xcxfx42?xcxdx15x31x32x13?k +5m = 0,x4k = 5m,x23x24xcxfx40x41x22
21x+14z?3 = 0:
(3)x23x73x40x41x22
k(4x?y +3z?1)+m(x+5y?z +2) = 0;
x1a(0;0;1)x51x52x502k +m = 0,x4m =?2k,x23x24xcxfx40x41x22
2x?11y +5z?5 = 0:
(4)x23x73x40x41x22
k(4x?y +3z?1)+m(x+5y?z +2) = 0;
x4fxcxfx42x1fx20(4;5;0)xcxdx15x31x32x134(4k +m)+5(?k +5m) = 0,x411k =?29m,x23x24xcxfx40x41x22
105x+84y?98z +51 = 0:
2,x73x42xcxfx?2y?z +2 = 0xcxdx3fx6bxx45x79x15xfx32x224x15xcxf.
x5,x3dx3excxdxcxfx14x26x27,x23x73xcxfx15x40x41x22x?2y?z +D = 0,x4cx22xfx32x29:
x
D +
y
D
2
+ zD = 1:
x3ex5fD =?4,x21x23x73x40x41x22x?2y?z?4 = 0.
3,x2ex74x40x41 (
A1x+B1y +C1z +D1 = 0
A2x+B2y +C2z +D2 = 0
￠ 21 ￠
x15x6axex2cx2dx2ex2fx30x31x32x71x63x27x1cx2ex74x13x6bxOyxcxfx7b.
x5,x4exaxbx2ex74x15xcxfx14x15x40x41x22
k(A1x+B1y +C1z +D1)+m(A2x+B2y +C2z +D2) = 0:
xcxfz = 0x2cx1cx6bx4fxcxfx14x7b,x23x24x47x24x1dx2ax6ax29:8
>>>
><
>>>
>:
kA1 +mA2 = 0
kB1 +mB2 = 0
kC1 +mC2 6= 0
kD1 +mD2 = 0:
x22x23x13
A1
A2 =
B1
B2 =
D1
D2 6=
C1
C2,
4,x42x55x28xfx15x2ex74
L1,
( A
1x+B1y +C1z +D1 = 0
A2x+B2y +C2z +D2 = 0
x3ax2ex74
L2,
( A
3x+B3y +C3z +D3 = 0
A4x+B4y +C4z +D4 = 0
x6dx65x3x15x2ex74Lx15x40x41x22(
k(A1x+B1y +C1z +D1)+m(A2x+B2y +C2z +D2) = 0
k0(A3x+B3y +C3z +D3)+m0(A4x+B4y +C4z +D4) = 0;
x3cx16,k;mx13x55x33x22x6fx15x32xe,k0;m0x67x13x55x33x22x6fx15x32xe.
x5,x4ex65x3x2ex74Lx42L1x64x1dx15xcxfx40x41x48x1dx47x24x1dx36x29:
k(A1x+B1y +C1z +D1)+m(A2x+B2y +C2z +D2) = 0;
x3cx16k;mx55x43x52x220,x43x5,x4eLx42L2x64x1dx15xcxfx40x41x22
k0(A3x+B3y +C3z +D3)+m0(A4x+B4y +C4z +D4) = 0;
x3cx16k0;m0x55x43x52x220,x4ex3cL1;L2x55x28xf,x79x53x37x66xcxfx55x24x54,x38x15x15x3x74x6fx13L.
￠ 22 ￠
