線形数学 I II 演習問題 - 公式ウェブサイトyamaguti/jugyo/linalg/senkei12e.pdf ·...
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線形数学 I・II 演習問題
Transcript of 線形数学 I II 演習問題 - 公式ウェブサイトyamaguti/jugyo/linalg/senkei12e.pdf ·...
III
I 1 1
I 2 5
I 3 12
I 4 1 20
I 5 1 31
I 6 59
I 7 68
I 8 96
I 9 105
I 10 118
II 11 123
II 12 131
II 13 135
II 14 148
II 15 1 157
II 16 1 190
II 17 210
II 18 233
II 19 252
II 20 283
I 1
1. , , , , .(1) f1 : R R, f1(x) = sinx (2) f2 :
[2 ,
2
] R, f2(x) = sinx
(3) f3 : R [1, 1], f3(x) = sinx (4) f4 :[2 ,
2
] [1, 1], f4(x) = sinx
(5) f5 : R (), f5(t) =
(1 + t
2 t
)(6) f6 : () (), f6
((x
y
))=
(2x y + 14x 2y 1
)
2. a, b, c ( a = 0), f : R R f(x) = ax2 + bx+ c 2.(1) (ff)(x) .(2) (ff)(x) x x , (ff)(x) x f(x) x = ax2 + (b 1)x+ c .
(3) (ff)(x) x f(x) x = ax2 + (b 1)x + c g(x) . 2 f(x) x = 0,g(x) = 0 D, D D a D
(4) 2 f(x) x = 0 g(x) = 0 , g(x) = 0
(5) 4 (ff)(x) x = 0 D .(6) 2 g(x) = 0 , , ff , (ff)(), (ff)() D
(7) 4 ff 3 D b
3. a, b , a, b : R R a(x) = ax, b(x) = x+ b . , .(1) ba, ab , x , .(2) ab = aba .(3) : R R (x) = x2 , a, b, c . b, a(b), c(a(b)) , x , .
(4) 0 , , f(x) = x2 + x+ 2 f : R R . , f = c(a(b)) a, b, c , , , .
4. p 2 A , f : () () f(x) = Ax+ p .A , f , f y A, y, p .
5. a, b, c, d, p, q, r, s R , ad bc ps qr 0. X, Y
X =
R c = 0{x R x = dc } c = 0 , Y =R r = 0{x R x = sr } r = 0
, f : X R, g : Y R f(x) = ax+ bcx+ d
, g(x) =px+ q
rx+ s.
(1) f(x) Y x X X Z , Z .(2) f : Z Y f(x) = f(x) . , gf : Z R , x Z , a, b, c, d, p, q, r, s x (gf)(x) .
6. () p 2 A , f : () () f(x) = Ax+ p. , A .
1
1
1. (1) f1(x) = sinx = 2 x , f1 . f1(0) = sin 0 = 0, f1() = sin = 0
f1(0) = f1() , f1 .
(2) f2(x) = sinx = 2 x , f2 . sinx [2 ,
2
]
, f2 .
(3) sinx 1 1 , f3 . f3(0) = sin 0 = 0, f3() = sin = 0 f3(0) = f3() , f3 .
(4) x 2
2, sinx , 1 1 , f4
.
(5) t , f5(t) =
(1 + t
2 t
),
x = 1 + ty = 2 t . , f5(t) =
(0
0
) t . f
. f5(s) = f5(t)
(1 + s
2 s
)=
(1 + t
2 t
), 1.
1 + s = 1 + t , s = t , f5 .
(6)
(p
q
), f6
((x
y
))=
(p
q
)
(x
y
), f6 ,(
2x y + 14x 2y 1
)=
(p
q
),
2x y + 1 = p (i)4x 2y 1 = q (ii) . x, y , (ii) (i) 2 3 = q 2p . ,
(p
q
), f6
((x
y
))=
(p
q
)
(x
y
), p, q q = 2p 3 .
(p
q
)=
(0
0
)
q = 2p 3 , f6
((x
y
))=
(0
0
)
(x
y
). f6
. f6
((x
y
))= f6
((x
y
))
(2x y + 14x 2y 1
)=
(2x y + 14x 2y 1
)
2x y = 2x y (i)4x 2y = 4x 2y (ii). (ii) (i) 2, (i) (ii), f6
((x
y
))= f6
((x
y
))
. f6
((x
y
))= f6
((x
y
)) (i). ,
x = y = 0, x = 1, y = 2 (i), f6
((0
0
))= f6
((1
2
)), f6
.
2. (1) (ff)(x) = f(f(x)) = f(ax2 + bx+ c) = a(ax2 + bx+ c)2 + b(ax2 + bx+ c) + c =a3x4 + 2a2bx3 + a(b2 + 2ac+ b)x2 + b(2ac+ b)x+ c(ac+ b+ 1)
(2) (ff)(x) x = a3x4 + 2a2bx3 + a(b2 + 2ac+ b)x2 + (2abc+ b2 1)x+ c(ac+ b+ 1) =(ax2 + (b 1)x+ c)(a2x2 + a(b+1)x+ ac+ b+1) (ff)(x) x f(x) x = ax2 + (b 1)x+ c a2x2 + a(b+ 1)x+ ac+ b+ 1 , 0.
(3) D = (b1)24ac, (2) g(x) = a2x2+a(b+1)x+ac+b+1D = a2(b+1)24a2(ac+b+1) =a2(b2 2b+ 1 4ac 4) = a2((b 1)2 4ac 4) = a2(D 4) .
2
(4) f(x)x = 0 g(x) = 0 ,
a2 + (b 1)+ c = 0 (i)a22 + a(b+ 1)+ ac+ b+ 1 = 0 (ii). (ii) (i) a 2a+ b+ 1 = 0 = b+ 1
2a.
(i) , 4a (b 1)2 4ac 4 = 0 . D 4 , (3) D = a2(D 4) = 0 , g(x) = 0 . g(x) = 0, (3) a2(D4) = D = 0, (b1)24ac4 = D4 = 0
. , = b+ 12a
f() = 0 g() = 0 , f(x) x = 0 g(x) = 0
= b+ 12a
.
(5) (ff)(x) x 2 f(x) x g(x) , f(x) x = 0 g(x) = 0 g(x) = 0 . f(x)x = 0 D < 0, D = 0, D > 0 0, 1, 2 , D = a2(D 4) , g(x) = 0 D < 4, D = 4, D > 4 0, 1, 2 . D = 4 g(x) = 0 f(x) x = 0 , (ff)(x) x = 0 , D < 0 0 , D = 0 1 , 0 < D 4 2 , D > 4 4 .(6) (ff)(x) = 4a3x3 + 6a2bx2 + a(b2 + 2ac+ b)x+ b(2ac+ b) (ff)(x) g(x) (ff)(x) =
(4ax+2b4)g(x)b2+2b+4ac+4 = (4ax+2b4)g(x)D+5, (ff)() = (ff)() = D+5.
(7) (ff)(x) = 12a3x2+12a2bx+a(b2+2ac+b) = 12a3(x+
b
2a
)2+a(2b2+2ac+b)2b2+2ac+b 0
(ff) . , (ff) 0 1, ff 3 . 2b2 + 2ac + b < 0 , (ff)(x) = 0 2 ,
+ = ba, =
b2 + 2ac+ b
12a2. (ff)(x) = (ff)(x)
(x
3+
b
6a
) D 1
3(2ax + b)
(ff)()(ff)() = (D 1)2
9(2a+ b)(2a+ b) =
(D 1)2
27(2b2 + 2ac+ b) . (ff)(), (ff)()
3 (ff) (ff)(x) = 0 3,. , (ff)(x) = 0 (ff) , ff 3. D = 1 2b2 + 2ac + b < 0 . , 2b2 + 2ac+ b = 1
2(D + 3b2 1) , D b 1 3b2 < D < 1
D > 1.
3. (1) (ba)(x) = b(a(x)) = b(ax) = ax+ b, (ab)(x) = a(b(x)) = a(x+ b) = a(x+ b).(2) , x (ab)(x) = a(x + b) = ax + ab, (aba)(x) = ax + ab
(ab)(x) = (aba)(x) . ab = aba .(3) (b)(x) = (b(x)) = (x + b) = (x + b)2, (a(b))(x) = a((b)(x)) = a((x + b)2) = a(x + b)2,
(c(a(b)))(x) = c(a(b))(x)) = c(a(x+ b)2) = a(x+ b)2 + c.
(4) f(x) =
(x+
2
)2+
4 2
4 (3) a = , b =
2, c =
4 2
4.
4. y , f(x) = y x , Ax+p = y Ax = yp ,A, x = A1(yp). x = A1(yp) f(x) = Ax+p = AA1(yp)+p =(y p) + p = y f . f(x) = f(x) Ax+ p = Ax + p Ax = Ax , A vx = x , f . f .
, y , x = A1(y p) x f(x) = y , f y A1(y p) .
5. (1) r = 0 Y = R , Z = X . r = 0 . ar + cs = 0 a = csr
ad bc = c(br + ds)r
, ad bc = 0 br + ds = 0 . ax+ bcx+ d
= sr
3
(ar+ cs)x = (br+ ds) , ar+ cs = 0 ax+ bcx+ d
= sr x X . r = 0
ar + cs = 0 Z = X . ar + cs = 0 ax+ bcx+ d
= sr x br + ds
ar + cs, r = 0
ad bc = 0 br + dsar + cs
= dc, Z =
{x R
x = br+dsar+cs ,dc } = X { br+dsar+cs} . , Z =
X r(ar + cs) = 0X { br+dsar+cs} r(ar + cs) = 0 .(2) (gf)(x) = g(f(x)) = g(f(x)) = g
(ax+ b
cx+ d
)=pax+bcx+d + q
r ax+bcx+d + s=
(ap+ cq)x+ bp+ dq
(ar + cs)x+ br + ds.
6. e1 =
(1
0
), e2 =
(0
1
), f , f(x1) = e1 + p, f(x2) = e2 + p x1,
x2 . f(x1) = Ax1 + p, f(x2) = Ax2 + p f(x1) = e1 + p, f(x2) = e2 + p Ax1 = e1, Ax2 = e2
. , A =
(a b
c d
), x1 =
(x
y
), x2 =
(z
w
) Ax1 = e1
ax+ by = 1 (i)cx+ dy = 0 (ii) , Ax2 = e2
az + bw = 0 (iii)cz + dw = 1 (iv) . (i) d (ii) b (adbc)x = d, (ii) a (i) c (adbc)y = c. (iii) d (iv) b (ad bc)z = b , (iv) a (iii) c (ad bc)w = a . , ad bc = 0 a = b = c = d = 0 A . f p , f . ad bc = 0 A .
4
I 2
1. (1) a =
(a1
a2
). b =
(b1
b2
) a a1b2 a2b1 = 0
.
(2) a =
a1a2a3
. b =b1b2b3
a a1b2 a2b1 =a1b3 a3b1 = a2b3 a2b1 = 0 .
2. (1) y = ax+ b
(x
y
)=
(p
q
)+ t
(u
v
).
(2) x = c
(x
y
)=
(p
q
)+ t
(u
v
).
(3)
(u
v
)=
(0
0
),
(x
y
)=
(p
q
)+ t
(u
v
).
3. 2 A, B .
(1) A(2, 1, 5), B(1, 3, 4) (2) A(1, 1, 3), B(2, 1,1) (3) A(1, 1, 5), B(1, 7, 5)
4. 3 A, B, C .
(1) A(3, 1, 1), B(2, 0,1), C(4, 1, 2) (2) A(1,1, 3), B(2,1, 4), C(3,1,1)(3) A(3, 4, 5), B(1, 4, 2), C(2, 0, 3)
5. .
(1) x+ 2y z = 3 (2) 3x z = 1 (3) x = 2 (4) x y 3z = 0
6. .
(1) x =
101
+ s120
+ t 011
(2) x =123
+ s311
+ t221
(3) x =123
+ s210
+ t130
7. 3 A,B,C .
(1) A(2, 1,2), B(1, 3,9), C(4, 2, 2) (2) A(2, 3, 4), B(0, 9, 12), C(5, 4, 6)(3) A(2, 1, 5), B(1, 2,4), C(0,1,1)
8. A =
1 2 11 3 31 4 6
, .(1) TA ,
x 32
=y 11
= z 5 .(2) x+ 3y+ 3z + 1 = 0 2, ,
.
(3) TA , x+ 3y + 3z + 1 = 0 .
(4) TA , x+ 3y + 3z + 1 = 0 .
9. 2 l, m l :x 32
=y 11
= z 5, m : x 2 = y + 12
=z + 1
3.
(1) l m .
(2) l m .
5
10. 2 l :x a2
= y =z 13
, m :x 15
=y b2
=z 24,
, a, b .
11. x y+2z = 1, 2x+ y z = 1 . .(1) 2.
(2) (1) l , l xy, l .
12. a , A =
(2a+ 3 a+ 1
a 10 a 3
). TA a
. TA 1 .
13.
(a
b
),
(c
d
), 2 A =
(ac bc
ad bd
).
(1) TA , .
(2) TA ,
.
14. () a, b , 0 . A =
(1 ab a2
b2 1 + ab
), TA
, .
6
2
1. (1) a1b2 a2b1 = 0 . a = 0 a1 a2 0 . a1 = 0 , t =b1a1
b1 = ta1 , a1b2 a2b1 = 0 , b2 = ta2 b = ta . a2 = 0 , t =b2a2
b = ta . b a . b = ta b1 = ta1 b2 = ta2 a1b2 a2b1 = a1ta2 a2ta1 = 0 .(2) a1b2 a2b1 = a1b3 a3b1 = a2b3 a2b1 = 0 . a = 0 a1, a2, a3 0
. a1 = 0 , t =b1a1 b1 = ta1 , a1b2 a2b1 = a1b3 a3b1 = 0 , b2 = ta2 b3 = ta3
b = ta . a2 = 0 t =b2a2, a3 = 0 t =
b3a3 b = ta
. b a . b = ta j = 1, 2, 3 bj = taj
aibj ajbi = aitaj ajtai = 0 (1 i < j 3) .
2. (1) x = t y = at+ b
(x
y
)=
(t
at+ b
)=
(0
b
)+ t
(1
a
).
(2) y = t x = c
(x
y
)=
(c
t
)=
(c
0
)+ t
(0
1
).
(3)
(x
y
)=
(p
q
)+ t
(u
v
)
x = p+ ut (i)y = q + vt (ii) (i) v (ii) u , vx uy = pv qu , vx uy pv + qu = 0 .
3. (1)
321
, x =215
+ t321
, x 23
=y 12
= z + 5.
(2)
104
, x =113
+ t 104
, x 1 = z 34
, y = 1.
(3)
010
, x =113
+ t010
, x = 1, z = 3.
4. (1)
112
,101
, x =311
+ s112
+ t101
.(2)
101
, 204
, x = 11
3
+ s101
+ t 204
.(3)
403
,142
, x =345
+ s403
+ t142
.
5. (1) (3, 0, 0),
210
,101
121
, x =300
+s 21
0
+t101
.
7
(2) (0, 0,1),
103
,010
301
, x= 001
+s 103
+t010
.(3) (2, 0, 0),
010
,001
100
, x =200
+s010
+t001
.(4) (0, 0, 0) ,
110
,301
113
, x = s110
+ t301
.
6. (1)
211
120
, 011
, (1, 0, 1) , 2x+ y + z = 1.(2)
114
311
,221
, (1, 2, 3) , x+ y 4z = 9.(3)
001
210
,130
, (1, 2, 3) , z = 3.
7. (1)
127
,214
, 321
, A, 3x 2y z = 6.(2)
268
,312
, 175
, A, x+ 7y 5z = 3.(3)
319
,226
, 301
, A, 3x z = 1.
8. (1)
211
, (3, 1, 5),xyz
=315
+ t 21
1
. A
315
=102137
, A 21
1
=124
, TA xyz
=102137
+ t124
. x 10 = y 21
2=z 37
4.
(2)
133
, .
xyz
133
x + 3y + 3z = 0 , 31
0
8
301
133
, . (1, 0, 0) , 31
0
301
xyz
=10
0
+ s 31
0
+ t 301
.(3) A
100
=111
, A 31
0
= 101
, A 301
= 203
, (2) TA
xyz
=111
+ s 101
+ t 203
.(4) TA H . H
uvw
, (3),
101
203
, u, v, w u w = 0 (i)2u 3w = 0 (ii) .
(i), (ii) u = w = 0 w ,
010
H . (3) H (1,1,1) , H y + 1 = 0 .
9. (1) l, m ,
211
,123
. , l 315
, , l, m H H
xyz
=315
+ s 21
1
+ t123
. m
211
, H , 211
=315
+ s 21
1
+ t123
s, t . ,
2s+ t = 1 (i)
s+ 2t = 1 (ii)
s+ 3t = 6 (iii)
. (i), (ii) s, t
, s = 15, t = 3
5, s+ 3t = 2 (iii).
211
=315
+ s 21
1
+ t123
s, t . l m .
(2) l, m ,
xyz
=315
+ s 21
1
,xyz
= 21
1
+ t123
9
, l, m , A, B,OA =
2s+ 3s+ 1s+ 5
, OB = t+ 22t 13t 1
.AB =
t 2s 12t+ s 23t s 6
l, m , 0 . l
211
AB 3t 6s 6 , m 123
AB 14t 3s 23 ,
3t 6s 6 = 014t 3s 23 = 0 . s = 15 , t = 85 , OA =
13565245
, OB =
185115195
.
AB =
111
, A AB , x 13
5= y 6
5= z + 24
5
10. l, m , H , H l m
. l, m ,
213
, 52
4
, H uvw
2u+ v + 3w = 0 (i)5u 2v + 4w = 0 (ii) . (i), (ii) v , 9u + 10w = 0 , u = 10,w = 9 u, w , (i) v = 2u 3w = 7 v (i) (ii), H
1079
, . H 10x+ 7y 9z = 0 . l, m (a, 0, 1), (1, b, 2) , H . 10a 9 = 0, 10 + 7b 18 = 0 , a =
9
10, b =
8
7.
11. (1) ,
x y + 2z = 1 (i)2x+ y z = 1 (ii) xyz
. (i), (ii) z 5x + y = 1 , (i), (ii) y 3x + z = 0 . , x = t , y = 5t 1, z = 3t xyz
= t5t 1
3t
= 01
0
+ t 153
.(2) l xy l , z 0 . (1) l 3t = 0,
t = 0 l (0,1, 0) l xy. (1) l
153
, , (0,1, 0) , x 5(y+1) 3z = 0, x 5y 3z 5 = 0 .
12. TA v Av = v , A 1
10
. A x 2 x2 ax a2 + 2a+ 1 = 0 , A 1 2 + a a2 = 0, a = 2 a = 1 . a = 2 , Av = v
1 15
(1
2
),
(15, 2
5
)
( 1
5, 2
5
). a = 1
, Av = v 1 110
(1
3
),
(110, 3
10
)
( 1
10, 3
10
).
13. (1) a =
(a
b
), b =
(c
d
). x =
(x
y
) R2 , TA(x) = Ax =
(c(ax+ by)
d(ax+ by)
)= (ax + by)b
, TA(x) b . TA b =
(c
d
)
, .
(2) x =
(x
y
), (1) TA(x) = (ax+ by)b , b , x TA
, ax+ by = 0 . , ax+ by = 0 x a
, ax+ by = 0 x a (
(b
a
))
. TA ,
(b
a
)
.
14. A t2 2t + 1 = 0 1 . 1 A v =
(x
y
)
,
(1 ab a2
b2 1 + ab
)(x
y
)=
(x
y
), x, y
a(bx+ ay) = 0 (i)b(bx+ ay) = 0 (ii) . a, b 0 , , 1 bx + ay = 0 . , x = a, y = b
, v =
(a
b
) 1 A . , TA
v . l , v , l x = p + tv
. w =
(ba
) v w , xv + yw
. p = v + w , Av = v, Aw =
(a(a2 + b2) bb(a2 + b2) + a
)= (a2 + b2)v + w
Ap = A(v + w) = Av + Aw = v + (a2 + b2)v + w = p+ (a2 + b2)v . l TA
x = p+(t+ (a2 + b2)
)v . v , p
, l .
(a
b
) TA
.
11
I 3
1. , , .
(1) (1 1). (2) 1 2.(3) 2 (2 1). (4) 1 3.(5) 4 (4 1). (6) 2 3.(7) 2. (8) 4 3.(9) 2.
A =
1 01 21 3
B = 12
1
C =3 2 11 1 20 1 0
D =2 1 0 12 1 4 10
0 1 2 0
E =3 11 20 1
1 1
F =
(1
1
)G =
(1 2
)H =
(1 3 0
2 1 3
)I =
(0 1 3 1
)2. . a.
(1)
123
(1 2 3) (2) (1 3 24 0 1
) 1 0 41 4 05 3 2
(3)1 42 53 6
(1 2 34 5 6
)(4)
0 1 0 0
0 0 1 0
0 0 0 1
a 0 0 0
4
(5)
1 1 12 0 33 1 2
1 30 21 4
(1 2 3 42 0 2 1
)(6)
1 1 12 0 33 1 2
1 30 21 4
(0 2 3 11 2 0 1
)
(7)
(1 1 43 1 0
) 0 2 11 3 11 2 0
1 30 21 4
(8) (1 1 32 1 0
) 0 2 11 2 11 2 0
1 30 2
1 4
3. (1) 3 A
1 1 12 1 22 1 1
, AX =1 23 11 1
X .(2) 3 B
1 1 12 1 22 1 1
, BY =1 22 11 1
Y .
4. J =
0 1 00 0 10 0 0
, .(1) Jn . (2) (aE3 + J)n . (3) (J + atJ2)3n . (4) (J2 + atJ)3n .
5. (1) e1, e2, e3, e4, e5 R5 , A =
(e2 e4 e5 e3 e1
), A5 .
(2) e1, e2, e3, e4 R4 , 4 5 A A =
(e2 e4 e1 e3 e2
), (A tA)5
.
6. A, B 3. A , AB = BA B A
.
7. 0 , A A2 = 2A 2En n .(1) Ak = akA+ bkEn , ak+1, bk+1 ak, bk .
12
(2) {ak}k=1, {bk}k=1 , Ak A, En, k, .(. {ak+1 ak}k=1 ,
{ akk
}k=1.)
8. . P =
( 1
0
).
(1) A =
(a b
c d
), AP = xA+ yP + zE2 x, y, z c = 0
.
(2) A =
(a b
0 d
), Q =
(p q
0 r
). a = d , Q = uA+ vP + wE2 u, v, w , , a, b, d, p,
q, r .
9. X 2. 2 A AX = XA, A = E2+X
, .
10. Nn , 1, 2 j n , j n ej1 n,X =
(xij)m n.
(1) N ln .
(2) X NmX = O .
(3) X XNn = O .
(4) X NmX = XNn .
(5) a = b , (aEm +Nm)X = X(bEn +Nn) X .
11. n1, n2, . . . , nk , a1, a2, . . . , ak , n = n1 + n2 + + nk .
A =
a1En1 0
a2En2. . .
0 akEnk
n X AX = XA , X .
X =
X1 0
X2. . .
0 Xk
(, Xi ni )
12. I, J , K 4.
I =
0 1 0 01 0 0 0
0 0 0 1
0 0 1 0
, J =0 0 1 00 0 0 11 0 0 0
0 1 0 0
, K =
0 0 0 1
0 0 1 00 1 0 0
1 0 0 0
(1) I2 = J2 = K2 = E4, IJ = JI = K, JK = KJ = I, KI = IK = J .(2) A = aE4 + bI + cJ + dK (a, b, c, d R) , A = aE4 bI cJ dK, A =
a2 + b2 + c2 + d2
, AA = AA = A2E4 .(3) H aE4 + bI + cJ + dK (a, b, c, d R) 4. A,B H AB H , A H A , A1 H .(4) X2 + E4 = O X H .
13
3
1. (1) G, F (2) I, E (3) H, B (4) G, H (5) E, F
(6) H, C (7) H, A (8) E, H (9) F , G
2. (1)
123
(1 2 3) =1 2 32 4 63 6 9
(2)
(1 3 24 0 1
) 1 0 41 4 05 3 2
= (12 6 01 3 14
)
(3)
1 42 53 6
(1 2 34 5 6
)=
17 22 2722 29 3627 36 45
(4)
0 1 0 0
0 0 1 0
0 0 0 1
a 0 0 0
4
=
0 1 0 0
0 0 1 0
0 0 0 1
a 0 0 0
2
0 1 0 0
0 0 1 0
0 0 0 1
a 0 0 0
2
=
0 0 1 0
0 0 0 1
a 0 0 0
0 a 0 0
2
=
a 0 0 0
0 a 0 0
0 0 a 0
0 0 0 a
= aE4
(5)
1 1 12 0 33 1 2
1 30 21 4
(1 2 3 42 0 2 1
)=
2 11 181 15
(1 2 3 42 0 2 1
)=
4 4 4 735 2 39 2231 2 27 11
(6)
1 1 12 0 33 1 2
1 30 21 4
(0 2 3 11 2 0 1
)=
2 11 181 15
(0 2 3 11 2 0 1
)=
1 2 6 118 38 3 1915 28 3 14
(7)
(1 1 43 1 0
) 0 2 11 3 11 2 0
1 30 21 4
= (5 7 21 9 2
) 1 30 21 4
= (7 371 23
)
(8)
(1 1 32 1 0
) 0 2 11 2 11 2 0
1 30 2
1 4
= (4 10 01 6 1
)1 30 21 4
= (4 82 13
)
3. (1) AX =
1 23 11 1
A1 , X = 1 1 12 1 22 1 1
1 23 11 1
=1 01 1
0 2
.(2) BY =
1 22 11 1
B1 , Y =1 1 12 1 22 1 1
1 22 11 1
= 2 42 5
3 6
.
4. (1) J2 =
0 0 10 0 00 0 0
, J3 = O n 3 Jn = O .(2) aE3 J ((aE3)J = J(aE3) = aJ) (x + y)n =
nk=0
nCkxnkyk x, y ,
aE3, J (aE3 + J)n =nk=0
nCk(aE3)nkJk . k 3 Jk = O
14
, (aE3 + J)n = anE3 + nan1J +n(n 1)
2an2J2 =
an nan1 n(n1)2 a
n2
0 an nan1
0 0 an
.
(3) (1) tJ2 =
0 0 00 0 01 0 0
(J + atJ2)3 =0 1 00 0 1a 0 0
3
=
0 0 1a 0 00 a 0
0 1 00 0 1a 0 0
=a 0 00 a 00 0 a
= aE3 . (J + atJ2)3n = (aE3)n = anE3.
(4) (1) J2 =
0 0 10 0 00 0 0
(J2 + atJ)3 =0 0 1a 0 00 a 0
3
=
0 a 00 0 aa2 0 0
0 0 1a 0 00 a 0
=a
2 0 0
0 a2 0
0 0 a2
= a2E3 . (J2 + atJ)3n = (a2E3)n = a2nE3.5. (1) A 1 TA : R
5 R5 , j = 1, 2, 3, 4, 5 , TA(ej) = Aej = (A j ) TA(e1) = e2, TA(e2) = e4, TA(e3) = e5, TA(e4) = e3, TA(e5) = e1 . , ej
e1TA7 e2
TA7 e4TA7 e3
TA7 e5TA7 e1 e2
TA7 e4TA7 e3
TA7 e5TA7 e1
TA7 e2 e3TA7 e5
TA7 e1TA7 e2
TA7 e4TA7 e3
e4TA7 e3
TA7 e5TA7 e1
TA7 e2TA7 e4 e5
TA7 e1TA7 e2
TA7 e4TA7 e3
TA7 e5
,(A5 j
)= A5ej = TA5(ej) = TATATATATA(ej) = ej j = 1, 2, 3, 4, 5
. A5 = E5 .
(2) R5 e1, e2, e
3, e
4, e
5 ,
tA =(e3 e
1 + e
5 e
4 e
2
) A tA =
(e1 2e2 e3 e4
). A tAej = (A tA j ) =
ej j = 22ej j = 2 ,((A tA)
5 j )= (A tA)
5ej =
ej j = 225ej j = 2, (A tA)5 =
(e1 32e2 e3 e4
).
6. A = (aij) 3 aji = aij i, j = 1, 2, 3 , aii = aii
a11 = a22 = a33 = 0 . a32 = a, a13 = b, a21 = c A =
0 c bc 0 ab a 0
. , 3 B B =
0 r qr 0 pq p 0
. AB =bq cr bp cpaq ap cr cq
ar br ap bq
,BA =
bq cr aq arbp ap cr brcp cq ap bq
AB = BA bp aq =ar cp = cq br = 0 . a =
abc
, b =pqr
, 2 2 (2), , AB = BA b a . B
A , b a , AB = BA B A
.
7. (1) a1 = 1, b1 = 0 . A2 = 2A 2En a2 = 2, b2 = 2. k
15
, Ak = akA + bkEn , A Ak+1 = akA2 + bkA = ak(2A 2En) + bkA =(2ak + bk)A 2akEn ak+1 = 2ak + bk, bk+1 = 2ak Ak+1 = ak+1A + bk+1E3 ,k + 1 .
(2) ak+2 = 2ak+1 + bk+1 = 2ak+1 2ak. , ak+2 ak+1 = (ak+1 ak) , {ak+1 ak}k=1 a2 a1 = , . ak+1 ak = k , k+1 ,
ak+1k+1
akk
=1
,
{ akk
}k=1
a1
=1
,
1
. ,
akk
=k
ak = kk1, bk = 2ak1 = (1 k)k. Ak = kk1A+ (1 k)kEn.
8. (1) AP =
(a a+ b
c c+ d
), xA+ yP + zE2 =
(ax+ y + z bx+ y
cx dx+ y + z
) AP = xA+ yP + zE2
a = ax+ y + z, a+ b = bx+ y, c = cx, c+ d = dx+ y + z . c 0 , 3
, = x , 1 z = y, 4 c = y + z = 0 . c = 0.
c = 0 , AP =
(a a+ b
0 d
), xA + yP + zE2 =
(ax+ y + z bx+ y
0 dx+ y + z
) x = , y = a,
z = a AP = xA+ yP + zE2 .
(2)
(p q
0 r
)= uA + vP + wE2 , au + v + w = p, bu + v = q, du + v + w = r u =
p ra d
,
v =q(a d) b(p r)
a d, w =
ar dp q(a d) + b(p r)a d
.
9. X =
(p q
r s
). A =
(a b
c d
), AX XA =
(br cq aq + b(s p) dq
ar + c(p s) + dr br + cq
)
. q = r = 0 , p = s , AX = XA b = c = 0 . ,A =
dp asp s
E2 +a dp s
X , =dp asp s
, =a dp s
. q = 0 , AX = XA
c =br
q, d = a b(p s)
q, A =
1
q
(aq bq
br aq bp+ bs
)=aq bp
qE2 +
b
qX . ,
q = 0 = aq bpq
, =b
q. r = 0 , AX = XA b = cq
r, d = a c(p s)
r
, A =1
r
(ar cq
cr ar cp+ cs
)=ar cp
rE2 +
c
rX . , r = 0 = ar cp
r,
=c
r.
10. (1) Nn =(0 e1 en1) Nnej =
0 j = 1ej1 2 j n . N lnej =0 1 j lejl l + 1 j n
, 1 l n 1 N ln 1 l, l+ 1 j n, j ej , l n N ln .(2) NmX i, i < m X i+1, m. NmX = O
, X 1.
(3) XNn j , j > 1 X j 1, 1. XNn = O , X n.
(4) NmX XNn , NmX = XNn 2 i m xi1 = 0,1 j n 1 xmj = 0 1 i m 1, 1 j n 1 xi+1 j+1 = xij
16
.
NmX XNn =
x21 x22 x11 x2j x1j1 x2n x1n1...
......
...
xi+1 1 xi+1 2 xi1 xi+1 j xij1 xi+1n xin1...
......
...
xm1 xm2 xm1 1 xmj xm1 j1 xmn xm1n10 xm1 xmj1 xmn1
m n , j i < n m xij = 0 . i = m 1 j n 1 xmj = 0 , k < i m j i < nm xij = 0 , j k < nm , k + 1 m (j + 1) (k + 1) = j k < nm xkj = xk+1 j+1 = 0 . ai = xmi n ,1 j m x1nm+j = x2nm+j+1 = = xi nm+i+j1 = = xmj+1n = aj1 , N lm (1), .
x1nm+1 x1nm+2 x1nm+j x1nx2nm+1 x2nm+2 x2nm+j x2n
......
......
xi nm+1 xi nm+2 xi nm+j xin...
......
...
xmnm+1 xmnm+2 xmnm+j xmn
=
a0 a1 aj1 am1a0 aj2 am2
. . ....
...
0 a0 ami. . .
...
a0
=
m1l=0
alNlm
, X =(O
m1l=0
alNlm
)=m1l=0
al
(O N lm
).
m n , i > j xij = 0 . j = 1 2 i m xi1 = 0 , 1 j < k j < i xij = 0 , k < i , j k 1 < i 1 xik = xi1 k1 = 0 . bj = x1 j+1 , 1 j m bj1 = x1j = x2j+1 = = xi j+i1 = = xnj+1n , N ln (1), .
x11 x12 x1j x1nx21 x22 x2j x2n...
......
...
xi1 xi2 xij xin...
......
...
xn1 xn2 xnj xnn
=
b0 b1 bj1 bn1b0 bj2 bn2
. . ....
...
0 b0 bni. . .
...
b0
=
n1l=0
blNln
, X =
n1l=0 blN lnO
= n1l=0
bl
(N ln
O
).
(5) , (aEm+Nm)X = X(bEn+Nn) (a b)xi1+xi+1 1 = 0 (i = 1, 2, . . . ,m 1),(a b)xm1 = 0, (a b)xij + xi+1 j xij1 = 0 (i = 1, 2, . . . ,m 1, j = 2, 3, . . . , n), (a b)xmj = xmj1
17
(j = 2, 3, . . . , n) .
(aEm +Nm)X =
ax11 + x21 ax12 + x22 ax1j + x2j ax1n + x2n...
......
...
axi1 + xi+1 1 axi2 + xi+1 2 axij + xi+1 j axin + xi+1n...
......
...
axm1 1 + xm1 axm1 2 + xm2 axm1 j + xmj axm1 n + xmnaxm1 axm2 axmj axmn
X(bEn +Nn) =
bx11 bx12 + x11 bx1j + x1j1 bx1n + x1n1...
......
...
bxi1 bxi1 + xi1 bxij + xij1 bxin + xin1...
......
...
bxm1 1 bxm1 2 + xm1 1 bxm1 j + xm1 j1 bxm1n + xm1n1bxm1 bxm2 + xm1 bxmj + xmj1 bxmn + xmn1
a = b xm1 = 0, xi1 1 =
xi1a b
(i = 2, 3, . . . ,m), xmj =xmj1a b
(j = 2, 3, . . . , n) xi1 = xmj = 0
(i = 1, 2, . . . ,m, j = 1, 2, . . . , n) . xij =xi+1 j + xij1
a b(i = 1, 2, . . . ,m 1, j = 2, 3, . . . , n)
j i xij = 0 . j i = 1 m (i, j) = (m, 1) , , j i = k 1 (2 m k n 2) xij = 0 , j i = k , j (i + 1) = (j 1) i = k 1 xi+1 j = xij1 = 0 xij =
xi+1 j + xij1a b
= 0 .
(aEm +Nm)X = X(bEn +Nn) X .
11. A, X (i, j) aij , xij , 1 = 0, s = n1 + n2 + + ns1 (s = 2, 3, . . . , k) . ,ap+i q+j , xp+i q+j (i = 1, 2, . . . , np, j = 1, 2, . . . , nq) (i, j) np nq Apq, Xpq . AX, XA (i, j) yij , zij , yp+i q+j , zp+i q+j (i = 1, 2, . . . , np, j = 1, 2, . . . , nq) (i, j)
np nq Ypq, Zpq ,
yp+i q+j =
ns=1
ap+i sxs q+j =
kt=1
nts=1
ap+i t+sxt+s q+j =
kt=1
(AptXtq (i, j))
zp+i q+j =
ns=1
xp+i sas q+j =
kt=1
nts=1
xp+i t+sat+s q+j =
kt=1
(XptAtq (i, j))
Ypq =kt=1
AptXtq, Zpq =kt=1
XptAtq . p = q Apq , App = apEnp, Ypq = AppXpq = apXpq, Zpq = XpqAqq = aqXpq . AX = XA ,
p, q = 1, 2, . . . , k Ypq = Zpq apXpq = aqXpq (ap aq)Xpq = O . p = q ap = aq Xpq . Xi = Xii Xi ni , X
X =
X1 0
X2. . .
0 Xk
.
12. (1) L =
(0 11 0
) L2 = E2 , I =
(L O
O L
), J =
(O E2E2 O
), K =
(O LL O
)
I2 =
(L O
O L
)(L O
O L
)=
(L2 O
O (L)2
)=
(E2 OO E2
)= E4, J2 =
(O E2E2 O
)(O E2E2 O
)=
18
(E2 OO E2
)= E4, K2 =
(O LL O
)(O LL O
)=
((L)2 OO (L)2
)=
(E2 OO E2
)= E4,
IJ =
(L O
O L
)(O E2E2 O
)=
(O LL O
)= K, JK =
(O E2E2 O
)(O LL O
)=
(L O
O L
)= I,
KI =
(O LL O
)(L O
O L
)=
(O (L)2
L2 O
)=
(O E2E2 O
)= J , JI = J(JK) = J2K = E4K = K,
KJ = K(KI) = K2I = E4I = I, IK = I(IJ) = I2J = E4J = J .(2) AA = (aE4+bI+cJ+dK)(aE4bIcJdK) = a(aE4+bI+cJ+dK)E4b(aE4+bI+cJ+dK)Ic(aE4+
bI+cJ+dK)Jd(aE4+bI+cJ+dK)K = a2E24+abIE4+acJE4+adKE4abE4Ib2I2bcJIbdKIacE4JbcIc2J2cdKJadE4KbdIKcdJKd2K2 = a2E4+abI+acJ+adKabI+b2E4bcJIbdKIacJbcIJ+c2E4cdKJadKbdIKcdJK+d2E4 = (a2+b2+c2+d2)E4bc(IJ+JI)cd(JK+KJ)bd(KI+IK) =AE4. b, c, d b, c, d,AA = (aE4bIcJdK)(aE4+bI+cJ+dK) =(aE4 + (b)I + (c)J + (d)K)(aE4 (b)I (c)J (d)K) = (a2 + (b)2 + (c)2 + (d)2)E4 = A2E4.(3) A = aE4+bI+cJ+dK,B = pE4+qI+rJ+sK H AB = (aE4+bI+cJ+dK)(pE4+qI+rJ+sK) =
p(aE4 + bI + cJ + dK)E4 + q(aE4 + bI + cJ + dK)I + r(aE4 + bI + cJ + dK)J + s(aE4 + bI + cJ + dK)K =
apE24 + bpIE4 + cpJE4 + dpKE4 + aqE4I + bqI2 + cqJI + dqKI + arE4J + brIJ + crJ
2 + drKJ + asE4K +
bsIK + csJK + dsK2 = apE4 + bpI + cpJ + dpK + aqI bqE4 cqK + dqJ + arJ + brK crE4 drI + asK bsJ + csI dsE4 = (ap bq cr ds)E4+(bp+ aq dr+ cs)I +(cp+ dq+ ar bs)J +(dp cq+ br+ as)K H
. A = O |A| = 0 , (2) A(
1
A2A
)=
(1
A2A
)A = E4 , A ,
A1 =1
A2A =
a
A2E4 +
bA2
I +cA2
J +dA2
K H .
(4) X = xE4 + yI + zJ + wK (x, y, z, w R) X2 = (x2 y2 z2 w2)E4 + 2xyI + 2xzJ + 2xwK X2 + E4 = O x2 y2 z2 w2 = 1 xy = xz = xw = 0 . x = 0 y = z = w = 0 , x2 = 1 , x . x = 0 , y, z, w y2 + z2 + w2 = 1
. , y = cos,z2 + w2 = sin (0 ) z2 + w2 = sin2 z = sin cos,
w = sin sin (0 < 2) . X2 + E4 = O X H {cosI + sin cosJ + sin sinK| 0 , 0 < 2} .
19
I 4 1
1. .
(1)
(1 a
0 1
)(2)
1 a b0 1 c0 0 1
(3)1 a b c
0 1 d e
0 0 1 f
0 0 0 1
2. A =
(aij) n.
(1) B =(bij) n AB , i = 1, 2, . . . , n , AB (i, i)
aiibii .
(2) A , i = 1, 2, . . . , n aii = 0 , A A , i = 1, 2, . . . , n A (i, i)1
aii.
3. n A , Am = O m , En +A .
4. 1 f : R3 R2
101
( 21
),
011
(12
),
002
( 46
), f
.
5. a . f : R4 R4 e1, e1+e2, e1e3, e1e3e4
1
2
12
,
0
123
,0
1
1
2
,
12
3
2 a
1. , f .
6. a, b, , , ax+ by + z = 0 R3 H . R3 1 f x
H f(x) = x , x H f(x) = x, f
.
7. a, b, , , R3 , u, v , H . R3
1 f x H f(x) = x , x H f(x) = x
, f .
(1) u =
1a2a 1
, v = a1 a22a2 2a+ 1
(2) u =1a1
, v =01b
8. (1) 2 A R2 1 TA , y = 2x y = x , (1, 1) (1, 1) . A 4, A .
(2) 2 B R2 1 TB , y = x y =1
2x , y = 2x
y = 2x . B 5 , B .
9. v = ae1 + be2 + ce3 R3 , v R3 , v
R3 P .
(1) R3 p .
(2) R3 p P .
(3) .
20
(4) P .
10. R3 , x 12
= y + 1 =z 21
. , (1), (2)
R3 1 f : R3 R3 .(1) x f(x) = 3x .
(2) x f(x) = 2x .
11. R3 2 u =1
3
122
, e3 =001
.(1) u e3 , 1. v .
(2) w v e3 , z v u , w, z 1
, .
(3) 1 f : R3 R3 f(v) = v, f(w) = z, f(e3) = u . , f .
12. u =
101
, v = 011
, u, v H . , (1), (2), (3) R3 1 f : R3 R3 .(1) x H f(x) = 2x .
(2) x H f(x) H .
(3) H H (H = H), f H H .
13. f : R4 R4 2e1 + e4 e1 , e1 e2 e2 , e2 + ae3 e3 , (2a 1)e3 + 2e4 e4 1. f A , A .
14. , R , S , T (cos
sin
).
(1) T S, ST , 1 1.
(2) , R ST = T S .
15. () c, d . R2 u =
(x
y
), v =
(z
w
), u v u v
u v =
(xz + cyw
xw + yz + dyw
), .
(1) v , u 7 u v R2 1, u , v 7 u v R2 1.
(2) i R2 , i i = e1 , i .(3) 0 n R2 , n n = 0 , n .(4) 0 p, q R2 , p q = 0, p p = p, q q = q , 0 e1 p p p = p .(5) 0 p, q R2 , p q = 0, p p = p, q q = q , p, q .
(6) (5), R2 x x = sp+ tq , a b = 0 a, b .
21
4
1. (1) A =
(1 a
0 1
), B =
(1 b
0 1
), AB =
(1 a+ b
0 1
) AB = E2 , a+ b = 0
. , b = a A1 =
(1 a0 1
).
(2) A =
1 a b0 1 c0 0 1
, B =1 d e0 1 f0 0 1
AB =1 a+ d b+ af + e0 1 c+ f0 0 1
AB = E3 , a+d = b+af+e = c+f = 0. , d = a, f = c, e = afb = acb
A1 =
(1 a0 1
).
(3) A =
1 a b c
0 1 d e
0 0 1 f
0 0 0 1
, B =1 g h i
0 1 j k
0 0 1 l
0 0 0 1
AB =1 a+ g b+ aj + h c+ bl + ak + i
0 1 d+ j e+ dl + k
0 0 1 f + l
0 0 0 1
AB = E4 , a + g = b + aj + h = c + bl + ak + i = d + j = e + dl + k = f + l = 0
. , g = a, j = d, l = f , h = aj b = ad b, k = dl e = df e,
i = bl ak c = ae+ bf adf c A1 =
1 a ad b ae+ bf adf c0 1 d df e0 0 1 f0 0 0 1
.
2. (1) AB =(cij). i > k aik = 0 , cij =
nk=1
aikbkj =nk=i
aikbkj .
k > j bkj = 0 , i > j cij = 0 , cii = aiibii . AB (i, i)
aiibii .
(2) A , A1 =(aij), (i) AA1 (i, i) aiiaii . , AA
1 = En
aiiaii = 1 i = 1, 2, . . . , n , aii = 0 , aii =1
aii.
i = 1, 2, . . . , n aii = 0 , i, j = 1, 2, . . . , n bij , bij . i > j
bij = bij = 0 , i = 1, 2, . . . , n bii = bii =
1
aii. j i r 1 (1 r n 1)
i, j bij bij , j i = r , bij bij bij = 1
aii
jk=i+1
aikbkj ,
bij = 1
ajj
jk=i+1
bikakj . , B =(bij), B =
(bij), AB =
(cij), BA =
(cij). i > k
aik = 0, k > j bkj = 0 , i > j cij =nk=1
aikbkj =nk=i
aikbkj = 0, i > k bik = 0,
k > j akj = 0 , i > j cij =nk=1
bikakj =nk=i
bikakj = 0, i = j bii = bii =
1
aii
cii =nk=1
aikbki =nk=i
aikbki = aiibii = 1, cii =
nk=1
bikaki =nk=i
bikaki = biiaii = 1, i < j bij , b
ij
cij =nk=1
aikbkj =jk=i
aikbkj = 0 cij =
nk=1
bikakj =jk=i
bikakj = 0 . AB = BA = En
, B = BEn = B(AB) = (BA)B = En = B . AB = BA = En B A
, A .
22
3. B = En A+A2 + + (1)kAk + + (1)m1Am1 = En +m1k=1
(1)kAk ,
(En +A)B = B +AB = En +
m1k=1
(1)kAk +A
(En +
m1k=1
(1)kAk)
= En +
m1k=1
(1)kAk +A+m1k=1
(1)kAk+1
B(En +A) = B +BA = En +
m1k=1
(1)kAk +
(En +
m1k=1
(1)kAk)A = En +
m1k=1
(1)kAk +A+m1k=1
(1)kAk+1
,
m1k=1
(1)kAk +A+m1k=1
(1)kAk+1 = A+m1k=1
(1)kAk +mk=2
(1)k1Ak
= AA+m1k=2
(1)kAk +m1k=2
(1)k1Ak + (1)m1Am
=
m1k=2
((1)k + (1)k1)Ak =m1k=2
((1)k1 + (1)k1)Ak = O
, (En+A)B = B(En+A) = En . , En+A, B = En+m1k=1
(1)kAk
.
4. f(e1 + e3) =
(2
1
), f(e2 + e3) =
(1
2
), f(2e3) =
(4
6
)
f(e1) + f(e3) =
(2
1
) (1), f(e2) + f(e3) =
(1
2
) (2), 2f(e3) =
(4
6
) (3)
(3) f(e3) =
(2
3
), (1), (2) f(e1) =
(0
2
), f(e2) =
(15
). , f
(0 1 22 5 3
).
5. , f(e1) =
1
2
12
, f(e1 + e2) =
0
123
, f(e1 e3) =0
1
1
2
, f(e1 e3 e4) =
12
3
2 a
f(e1) =
1
2
12
(1), f(e1) + f(e2) =
0
123
(2), f(e1) f(e3) =0
1
1
2
(3),
f(e1) f(e3) f(e4) =
12
3
2 a
(4) . (2) (1), (1) (3), (3) (4) , f(e2) =1315
,
f(e3) =
1
1
20
, f(e4) =
1
12a
f
1 1 1 12 3 1 11 1 2 22 5 0 a
.
23
5. ae1 + be2 + e3 H , f .
af(e1) + bf(e2) + f(e3) = (ae1 + be2 + e3) ()
e1 ae3, e2 be3 H , H . f f(e1) af(e3) = (e1 ae3), f(e2) bf(e3) = (e2 be3) , f(e1) = af(e3) + (e1 ae3),f(e2) = bf(e3) + (e2 be3) , ().
(a2 + b2 + 1)f(e3) = a( )e1 + b( )e2 + (a2+ (b2 + 1))e3
f(e3) =
a()a2+b2+1b()a2+b2+1
(a2+b2)+a2+b2+1
f(e1) =
(b2+1)+a2a2+b2+1ab()a2+b2+1a()a2+b2+1
, f(e2) =
ab()a2+b2+1
(a2+1)+b2a2+b2+1b()a2+b2+1
, f .
(b2+1)+a2a2+b2+1
ab()a2+b2+1
a()a2+b2+1
ab()a2+b2+1
(a2+1)+b2a2+b2+1
b()a2+b2+1
a()a2+b2+1
b()a2+b2+1
(a2+b2)+a2+b2+1
6. (1) (1 a2)u+ av = e1 (a2 3a+ 1)e3 v au = e2 (a 1)e3 H , f .
f(e1) (a2 3a+ 1)f(e3) = (e1 (a2 3a+ 1)e3) (i)
f(e2) (a 1)f(e3) = (e2 (a 1)e3) (ii)
, w =
pqr
H (w, (1a2)u+av) = (w,vau) = 0 p = r(a23a+1),q = r(a 1) , r = 1
a2 3a+ 1a 11
= (a2 3a+ 1)e1 + (a 1)e2 + e3H . , f .
(a2 3a+ 1)f(e1) + (a 1)f(e2) + f(e3) = ((a2 3a+ 1)e1 + (a 1)e2 + e3) (iii)
(i), (ii) f(e1) = (a2 3a+1)f(e3) +(e1 (a2 3a+1)e3), f(e2) = (a 1)f(e3) +(e2 (a 1)e3) , (iii)
((a23a+1)2+(a1)2+1)f(e3) = ()(a23a+1)e1+()(a1)e2+((a46a3+12a28a+2)+)e3
, f(e3) =
()(a23a+1)
(a23a+1)2+(a1)2+1()(a1)
(a23a+1)2+(a1)2+1((a23a+1)2+(a1)2)+(a23a+1)2+(a1)2+1
. f(e1) =((a1)2+1)+(a23a+1)2
(a23a+1)2+(a1)2+1()(a1)(a23a+1)(a23a+1)2+(a1)2+1
()(a23a+1)(a23a+1)2+(a1)2+1
,
f(e2) =
()(a1)(a23a+1)(a23a+1)2+(a1)2+1
((a23a+1)2+1)+(a1)2(a23a+1)2+(a1)2+1
()(a1)(a23a+1)2+(a1)2+1
. f .((a1)2+1)+(a23a+1)2
(a23a+1)2+(a1)2+1()(a1)(a23a+1)(a23a+1)2+(a1)2+1
()(a23a+1)(a23a+1)2+(a1)2+1
()(a1)(a23a+1)(a23a+1)2+(a1)2+1
((a23a+1)2+1)+(a1)2(a23a+1)2+(a1)2+1
()(a1)(a23a+1)2+(a1)2+1
()(a23a+1)(a23a+1)2+(a1)2+1
()(a1)(a23a+1)2+(a1)2+1
((a23a+1)2+(a1)2)+(a23a+1)2+(a1)2+1
24
(2) u = e1 + ae2 + e3, v = e2 + be3 , f(u) = u, f(v) = v f
.
f(e1) + af(e2) + f(e3) = (e1 + ae2 + e3) (i), f(e2) + bf(e3) = (e2 + be3) (ii)
, w =
pqr
u, v (w,u) = (w,v) = 0 p+ aq + r = q + br = 0 . , p = r(ab 1), q = br , r = 1
ab 1b1
= (ab 1)e1 be2 + e3H . , f
(ab 1)f(e1) bf(e2) + f(e3) = ((ab 1)e1 be2 + e3) (iii)
. (i) (ii) a f(e1) (ab 1)f(e3) = e1 + a()e2 +( ab)e3 f(e1) = (ab 1)f(e3)+e1 + a()e2 +( ab)e3 , (ii) f(e2) = bf(e3)+(e2 + be3), (iii)
((ab1)2+ b2+1)f(e3) = (ab1)()e1+(b()a(ab1)())e2+(b2+ (ab1)(ab))e3
, f(e3) =
(ab1)()(ab1)2+b2+1
b()a(ab1)()(ab1)2+b2+1
b2+(ab1)(ab)(ab1)2+b2+1
. f(e1) =
(b2+1)+(ab1)2(ab1)2+b2+1
a(b2+1)b(ab1)(a+b)(ab1)2+b2+1
(b2+1)+(ab1)b(a+b)(ab1)2+b2+1
,
f(e2) =
b(ab1)()(ab1)2+b2+1
b2+ab(ab1)(ab2)(ab1)2+b2+1
b(+(ab1)(ab2))(ab1)2+b2+1
. f
(b2+1)+(ab1)2(ab1)2+b2+1
b(ab1)()(ab1)2+b2+1
(ab1)()(ab1)2+b2+1
a(b2+1)b(ab1)(a+b)(ab1)2+b2+1
b2+ab(ab1)(ab2)(ab1)2+b2+1
b()a(ab1)()(ab1)2+b2+1
(b2+1)+(ab1)b(a+b)(ab1)2+b2+1
b(+(ab1)(ab2))(ab1)2+b2+1
b2+(ab1)(ab)(ab1)2+b2+1
. = , .
(b2+1)+(ab1)2(ab1)2+b2+1
b(ab1)()(ab1)2+b2+1
(ab1)()(ab1)2+b2+1
b(ab1)()(ab1)2+b2+1
(a2b22ab+2)+b2(ab1)2+b2+1
b()(ab1)2+b2+1
(ab1)()(ab1)2+b2+1
b()(ab1)2+b2+1
b2++(ab1)2(ab1)2+b2+1
7. (1) A =
(a b
c d
). TA y = 2x
(1
2
) y = x
(1
1
)
, TA
((1
2
))= k
(1
1
) k . A
(1
2
)= k
(1
1
) a+2b = k,
c + 2d = k . TA
((1
1
))=
(1
1
), A
(1
1
)=
(1
1
) a + b = 1, c + d = 1 .
a+ 2b = ka+ b = 1 ,c+ 2d = kc+ d = 1 , a b, c d , a = k + 2,
b = k 1, c = k + 2, d = k 1 . , A 4 ad bc = 4
25
, , 2k = 4 . k = 2 a = 4, b = 3, c = 0, d = 1 ,
A =
(4 30 1
).
(2) B 2
(a b
b c
). TB y = x
(1
1
) y =
1
2x
(2
1
), TB
((1
1
))= k
(2
1
) k , TB y = 2x
(1
2
) y = 2x
(1
2
), TB
((1
2
))= l
(1
2
)
l . B
(1
1
)= k
(2
1
) a + b = 2k, b + c = k , B
(1
2
)= l
(1
2
)
a+2b = l, b+2c = 2l .
a+ b = 2ka+ 2b = l ,b+ c = kb+ 2c = 2l , a b, b c
, a = 4k l, b = 2k+ l , b = 2k+ 2l, c = k 2l . b = 2k+ l = 2k+ 2l l = 4k , a = 8k, b = 6k, c = 7k . , B 5 ac b2 = 5 , , 20k2 = 5 . k = 1
2
a = 4, b = 3, c = 72 a = 4, b = 3, c = 7
2, B =
(4 33 72
) B =
(4 33 72
).
8. (1) tv , tv p v (tv p,v) = 0. , p =
xyz
,t =
(p,v)
(v,v)=ax+ by + cz
a2 + b2 + c2. tv =
ax+ by + cz
a2 + b2 + c2
abc
= 1a2 + b2 + c2
a2x+ aby + acz
abx+ b2y + bcz
acx+ bcy + c2z
,
1
a2 + b2 + c2
a2 ab ac
ab b2 bc
ac bc c2
.(2) P q , q p v q p = sv . q v
(q,v) = 0. q = p + sv , (p + sv,v) = 0. , s = (p,v)(v,v)
= ax+ by + cza2 + b2 + c2
.
q = p + sv =
xyz
ax+ by + cza2 + b2 + c2
abc
= 1a2 + b2 + c2
(b2 + c2)x aby acz
abx+ (a2 + c2)y bczacx bcy + (a2 + b2)z
, 1
a2 + b2 + c2
b2 + c2 ab acab a2 + c2 bcac bc a2 + b2
.(3) p r ,
1
2(p + r) p
1
2(p + r) =
ax+ by + cz
a2 + b2 + c2v . , r =
2(ax+ by + cz)
a2 + b2 + c2vp = 1
a2 + b2 + c2
(a2 b2 c2)x+ 2aby + 2acz
2abx+ (a2 + b2 c2)y + 2bcz2acx+ 2bcy + (a2 b2 + c2)z
,
1
a2 + b2 + c2
a2 b2 c2 2ab 2ac
2ab a2 + b2 c2 2bc2ac 2bc a2 b2 + c2
.
26
(4) P p u,1
2(p + u) p P
1
2(p + u) =
p ax+ by + cza2 + b2 + c2
v . u = p 2(ax+ by + cz)a2 + b2 + c2
v =1
a2 + b2 + c2
(a2 + b2 + c2)x 2aby 2acz
2abx+ (a2 b2 + c2)y 2bcz2acx 2bcy + (a2 + b2 c2)z
,
1
a2 + b2 + c2
a2 + b2 + c2 2ab 2ac2ab a2 b2 + c2 2bc2ac 2bc a2 + b2 c2
.
9.
211
= 2e1 + e2 e3 ,011
= e2 + e3 102
= e1 +2e3 . f(2e1 + e2 e3) = 3(2e1 + e2 e3), f(e2 + e3) = 2(e2 + e3), f(e1 +2e3) = 2(e1 +2e3), f
2f(e1) + f(e2) f(e3) = 6e1 + 2e2 3e3, f(e2) + f(e3) = 2e2 2e3, f(e1) + 2f(e3) = 2e1 4e3
. 2 3 f(e2) = f(e3) 2e2 2e3, f(e1) = 2f(e3) 2e1 4e3 ,
1 f(e3) =
53
2376
, f(e2) =
13
4376
, f(e1) =
4343
53
. f
43
13
53
43
43
23
5376
76
.
10. (1) v e3 v =
ab0
. (u,v) = 13(a+ 2b) v u
a+ 2b = 0 , v a2 + b2 = 1 . b = a2
a2+ b2 = 1 , a > 0 a =25. b = 1
5 v =
15
210
.(2) w e3 w =
cd0
. (v,w) = 15(2c d) w v
2c d = 0 , w c2 + d2 = 1 . d = 2c
c2 + d2 = 1 , c < 0 c = 15. b = 2
5w =
15
120
. z =
pqr
(u, z) = 13(p+2q+2r), (v, z) =
15(2p q) w u v
p+2q+2r = 0 2p q = 0 , z p2+ q2+ r2 = 1. q = 2p, r = p
2 q = 5p
2 p2 + q2 + r2 = 1 , p < 0 p = 2
35.
q = 435, r =
5
35 z =
1
35
245
.
27
(3) v =15(2e1 e2), w =
15(e1 2e2) , f
15(2f(e1) f(e2)) = f(v) = v,
15(f(e1) 2f(e2)) = f(w) = z . 2f(e1) f(e2) =
5v, f(e1) + 2f(e2) =
5z ,
(1), (2) f(e1) =15(2v z) = 1
15
1425
, f(e2) = 15(v 2z) = 1
15
21110
. ,f(e3) = u =
1
3
122
f 115
14 2 52 11 105 10 10
.
11. w =
pqr
u, v (w,u) = (w,v) = 0 p + r = q r = 0 ., p = r, q = r , r = 1 w =
111
= e1 + e2 + e3 H . f(w) u v H w , f(w) = kw k
. H H x = aw + su + tv (a 0), aw
H f f(aw) = akw , H f H
akw = aw + bu+ cv b, c . a(k 1)w = bu+ cv , w , (w,u) = (w,v) = 0 a(k 1)w2 = b(w,u) + c(w,v) = 0 , a = 0 w2 = 3 , k = 1 . f(w) = w f(e1) + f(e2) + f(e3) = e1 + e2 + e3 () ., u = e1 + e3, v = e2 e3 f(e1) + f(e3) = 2(e1 + e3), f(e2) f(e3) = 2(e2 e3) f(e1) = 2(e1 + e3) f(e3), f(e2) = 2(e2 e3) + f(e3) . () f(e3) f(e3) =
1
3e1
1
3e2 +
5
3e3 , f(e1) =
5
3e1 +
1
3e2 +
1
3e3, f(e2) =
1
3e1 +
5
3e2
1
3e3 .
f 1
3
5 1 11 5 11 1 5
.12. A(2e1 + e4) = e1, A(e1 e2) = e2, A(e2 + ae3) = e3, A((2a 1)e3 + 2e4) = e4 A(2e1 + e4 e1 e2 e2 + ae3 (2a 1)e3 + 2e4
)=(e1 e2 e3 e4
)= E4. ,
A1 = (2e1 + e4 e1 e2 e2 + ae3 (2a 1)e3 + 2e4) =
2 1 0 0
0 1 1 00 0 a 3
1 0 2a 1 2
.
13. (1) R() =
(cos sinsin cos
), Q() =
(cos 2 sin 2
sin 2 cos 2
) S, T R(), Q()
1. T S, ST , ,
Q()R() =
(cos 2 sin 2
sin 2 cos 2
)(cos sinsin cos
)=
(cos(2 ) sin(2 )sin(2 ) cos(2 )
)
R()Q() =
(cos sinsin cos
)(cos 2 sin 2
sin 2 cos 2
)=
(cos(2 + ) sin(2 + )
sin(2 + ) cos(2 + )
)
1. T S, ST
(cos( 2
)sin( 2
)), (cos( + 2 )sin( + 2
)) 28
.
(2) T S, ST, ,
(cos(2 ) sin(2 )sin(2 ) cos(2 )
),
(cos(2 + ) sin(2 + )
sin(2 + ) cos(2 + )
)
1 ST = T S cos(2 + ) = cos(2 ) sin(2 + ) =sin(2 ) . , 2( ) + + 2 .
14. (1) u v =
(xz + cyw
xw + yz + dyw
)=
(z cw
w z + dw
)(x
y
)=
(x cy
y x+ dy
)(z
w
) u 7 u v, v 7 u v
(z cw
w z + dw
),
(x cy
y x+ dy
) R2 1.
(2) i =
(u
v
) R2 . i i =
(u2 + cv2
2uv + dv2
) i i = e1
u2 + cv2 = 1 (i)2uv + dv2 = 0 (ii) . (ii) v = 0 2u + dv = 0 , v = 0 (i) u2 = 1 , u
. u = dv2, (i)
(d2
4+ c
)v2 = 1 d
2
4+ c < 0
d2 + 4c < 0 . d2 + 4c < 0 , u = dv2, (d2 + 4c)v2 = 4 =
1
d2 + 4c
(u, v) = (d,2) () i =
(d
2
) i i = e1 , i i = e1
i
(d
2
). d2 + 4c < 0 i i = e1 .
(3) n =
(u
v
) R2 . n n =
(u2 + cv2
2uv + dv2
) n n = 0
u2 + cv2 = 0 (i)2uv + dv2 = 0 (ii) . (ii) v = 0 2u + dv = 0 , v = 0 (i) u = 0 , n 0
. u = dv2, (i)
(d2
4+ c
)v2 = 0 , v = 0 d
2
4+ c = 0
d2 + 4c = 0 . , v = 2k u = dk n n = k
(d2
)
. d2 + 4c = 0 n n = 0 , n n = 0
n k
(d2
)(k 0).
(4) 0 p, q R2 , pq = 0, pp = p, q q = q , p = e1 . , p = e1 q = e1 q = p q = 0 , q 0 . 0 e1 p pp = p , q = e1p p = e1 q = 0 , (1) p q = p (e1p) = p e1p p = pp = 0, q q = (e1p) (e1p) = (e1p) e1 (e1p) p =e1 e1 p e1 e1 p+ p p = e1 p p+ p = e1 p = q .(5) (4) 0 e1 p p p = p .
p =
(a
b
) p p =
(a2 + cb2
2ab+ db2
) p p = p a, b
a2 + cb2 = a (i)2ab+ db2 = b (ii) . b = 0 (i) a 0 1 p 0 e1 .
p = 0, e1 b = 0 , (ii) a =1 db
2. (i) (d2 + 4c)b2 = 1
d2 + 4c > 0 . d2 + 4c > 0 =1
d2 + 4c b2 = 2,
a =1 db
2 b = , a = 1 d
2, p p = p p p =
(1d
2
)
(1+d
2
)
, = 0 p 0 e1 . , d2 + 4c > 0 .
29
d2 + 4c > 0 , 0 p, q R2 p q = 0, p p = p, q q = q , (4), p 0 e1 , q 0 e1 , , p, q
(1d
2
)
(1+d
2
). , p = q p = p p = p q = 0 p = 0
. p = q p =
(1d
2
), q =
(1+d
2
) p =
(1+d
2
), q =
(1d
2
).
, q = e1 p p q = p (e1 p) = p e1 p p = p p = 0 .(6) q = kp k kp = k(p p) = p (kq) = p q = 0 , k = 0 p = 0 . q = 0 , p, q 0 . p = kq
k . R2 x x = sp+ tq . 0
a, b R2 a b = 0 , a = sp+ tq, b = up+ vq (1) q p = p q = 0
a b = (sp+ tq) (up+ vq) = u(sp+ tq) p+ v(sp+ tq) q = sup p+ tuq p+ svp q + tvq q = sup+ tvq
p q = 0 su = tv = 0 . s = 0 , a = 0 t = 0 v = 0 . u = 0 , b = 0 v = 0 t = 0 . a b = 0 a, b (a, b) = (xp, yq) (yq, xp) (x, y 0).
30
I 5 1
1. 1. , a .
(1)
x+ 2y z = 32x+ 4y 3z = 5 (2)2x+ y + 3z = 03x 2y + 4z = 1 (3)
x+ 2y = 1
2x 4y = 1
3x+ 5y = 2
(4)
x 3y 2z = 1
2x 2y 3z = 1
x+ 5y = 1
(5)
2x+ 3y z = 2
3x+ 4y z = 1
5x+ 6y z = 3
(6)
2x 5y + 3z = 7
x 3y + 2z = 1
4x 5y + z = 18
(7)
x+ 2y z = 1
3x+ 2y + z = 1
x 2y + 3z = 0
(8)
2x y z = 12
x 3y + 2z = 1
4x 5y + z = 18
(9)
x+ 2y z = 1
3x+ 4y z = 1
5x+ 6y z = 1
(10)
3x y 4z = 5
x+ y z = 2
x+ 5y = 3
(11)
2x+ 5y + 3z = 4
3x+ 8y + 5z = 5
3x+ 5y + 2z = 11
(12)
3x 5y 5z = 4
2x 2y 3z = 3
5x+ y 6z = 7
(13)
x+ 2y + 3z = 4
2x+ y + 3z = 0
2x+ 3y + z = 1
(14)
x+ y z = 6
2x+ 4y + 6z = 2
3x 2y + z = 5
(15)
(a 2)x y 2z = 0
x+ (a 2)y + 2z = 0
2x+ 2y + (a+ 1)z = 0
(16)
x y = 2
3x y + z = 2
2x y + 2z = 1
y z = 1
(17)
x+ y + 2z + 3w = 6x+ 2y z + 4w = 8 (18)3x+ 2y + z + w = 45x+ y + 2z + 3w = 6
(19)
2x 4y 5w = 9
4x+ 3y + 11z 2w = 4
5x+ 2y + 12z 4w = 7
(20)
x+ 2y 3z 2w = 3
x+ 3y + z w = 2
2x+ 5y 2z 3w = 5
(21)
2x+ 3y 4z w = 1
x+ y z + 2w = 2
y 2z 5w = a
(22)
x 2y + 4z w = 6
3x+ 12y 6z + w = 8
5x y + 11z 3w = 3
(23)
x+ 2y + z + 2w = 1
2x y 3z w = 3
x+ 8y + 9z + 8w = 9
(24)
x+ 2y + 4z 6w = 1
3x+ y + 7z + 7w = 8
x+ 9y + 21z + 19w = 14
2x+ 7y + 11z + 5w = 1
(25)
x 3y + 3z + 2w = 7
4x+ 3y 2z + w = 8
5x+ 2y + z + 3w = 3
5x+ 6y + z + 3w = 7
(26)
3x+ 2y 3z + w = 2
4x+ 3y 4z + 2w = 1
2x+ y + 2z w = 4
6x+ 3y + 2z 2w = 5
(27)
x+ 2y w = 1
x+ y + 2z = 3
2x+ z + w = 8
x 2y z + w = 2
(28)
3y + 3z 2w = 4
x+ y + 2z + 3w = 2
x+ 2y + 3z + 2w = 1
x+ 3y + 4z + 2w = 1
(29)
2x+ 4y + z w = 1
x+ 2y z + w = 2
2x+ y + z + 2w = 2
x+ 3y + 2z 3w = 0
(30)
x+ z + 2w = 6
2x+ y + 4z + w = 3
4x 3y 4z + w = 3
x+ y + 2z + w = 4
31
(31)
2x+ 3y + 2z + w = 1
4x+ 2y z + w = 2
2x y z 2w = 1
2x+ y + 2z + 3w = 1
(32)
x+ y + 2z + w = 1
x+ y + 3z + 2w = 2
2x 2y + 2z w = 1
x+ y + w = 1
(33)
y z + w = 4
x+ 2y + z + w = 1
2x+ y + 5z + 6w = 3
x+ y + 2z + 2w = 1
(34)
2x+ y + 2z + 3w = 1
x y + 2z + 2w = 2
3x+ y + 2w = 1
4x+ 2y z + w = 0
(35)
x+ 2y z + 3w = 3
x y + 3z 6w = 3
2x+ 3y z + 3w = 3
x+ 2y + w = 2
(36)
x+ 2y + 4z 6w = 7
3x+ y + 7z + 7w = 9
x+ 9y + 11z + 19w = 11
2x+ 7y + 11z + 5w = a
(37)
2x y z + 2w = a
x+ 2y z w = b
x y + 2z w = c
(38)
x+ 2y z + 3w + 4v = 5
z 2w + 4v = 2
2x+ 4y z + 3w + 2v = 5
(39)
x+ 2y + 3z + w = 1
2x+ 5y + 8z + 3w = 3
x+ 3y + 5z + 2w = a+ 3
4x+ 3y + 2z + aw = a
(40)
3x 2y + z + v = 2
x y + z 2w + v = 1
2x+ y 3z + w + 3v = 2
(41)
2y + 4z + 2w = 2
x+ y + 3z + 2w = 2
x+ 2y + 3z + w = b
2x y + aw = 1
(42)
x+ 2y 2z + w + 3v = 2
2x+ y + 2z + v = 3
2x 3y + 2z w + 2v = 1
(43)
x 2y + z + aw = a+522x+ y 3az = 1
x y + 2az + 2w = 0
2x+ 2y + (a 1)z (a+ 2)w = b
(44)
x+ 2y z + 3w 2v = 1
2x+ 4y + z + 3w 3v = 2
x 2y + 2z 4w v = 1
3x+ 6y + 6w 5v = 3
(45)
11x+ 12y + 13z + 14u+ 15v = a
6x+ 7y + 8z + 9u+ 10v = 0
x+ 2y + 3z + 4u+ 5v = 0
x+ 4y + 9z + 16u+ 25v = b
2. 1, a, b, p, q, r, s , .
(1)
2x+ z = p
x+ 2y + z + w = q
y w = r
x+ y z + 4w = s
(2)
x 3y 3z + w = p
x 2y 2z + 2w = q
2x 5y 5z + 3w = r
x y z + 3w = s
(3)
x y + z + w = p
2x 3y + z w = q
x y 2z 2w = r
2x 5y + aw = s
(4)
x 2y + 3z w = p
2x 3y + az = q
3x 5y + (a+ 3)z w = r
4x 7y + (a+ 6)z 2w = s
(5)
x+ y 2w = p
2x y + 3z 2w = q
3x+ 3z 4w = r
4x+ y + 3z 6w = s
(6)
x y z = p
2x+ y 2z + w = q
y 4z + w = r
x 3z + w = s
(7)
2x+ z = p
x+ 4y z + w = q
y w = r
x+ y + z + 2w = s
(8)
x 2y + z w = p
x+ y z + 2w = q
2x+ y w = r
3y 2z + 3w = s
(9)
x+ y = p
y + w = q
x+ z = r
z + w = s
32
(10)
x+ 2y z = p
y + 3z 2w = q
x+ 3y + 2z 2w = r
x+ y 4z + 2w = s
(11)
x+ 2y + 2z + w = p
x+ 3y + z + 2w = q
3x+ 7y + 5z + 4w = r
2x+ 5y + 3z + 3w = s
(12)
x 2y + 2z w = p
x y z + 2w = q
2x 3y + z + w = r
x+ 3y 5z + 4w = s
(13)
x+ 2y + 2z + w = p
x y + z + 2w = q
2x+ y + az + w = r
3x+ 2y + 4z + aw = s
(14)
x+ y + az = p
x z = q
bx y z = r
(15)
ax+ y + z = p
x+ ay + z = q
x+ y + az = r
(16)
ax+ y + z + w = p
x+ ay + z + w = q
x+ y + az + w = r
x+ y + z + aw = s
(17)
x y + z + 2w = p
y z w = q
2x y + az + 3w = r
ax 2y + 2z + 3w = s
(18)
x+ ay = p
y + az = q
z + aw = r
x+ 3y + 3z + w = s
(19)
x+ 2y 3z w = p
2x 3y + z = q
3x y 2z w = r
(5 4a)x (a+ 4)y + (5a 1)z + (2a 1)w = s
(20)
x+ y az + (a 2)w = p
x y + z + (2 a)w = q
2x+ (a 3)y + 2z + 2w = r
(a+ 1)x+ 2y 2z 2w = s
(21)
x y + z w = p
x 2y 2z + 2w = q
2x 3y z + w = r
(a+ b)x (a+ 2b)y + (a 2b)z (a 2b)w = s
3.
x+ (k + 1)y + 2kz = k + 1
(k 1)x+ (3k 1)y + kz = k 1
(2k 1)x+ (7k 1)y + 3kz = 3k 1
2 k , .
4. , 1
( 3)x y 2z = 0
x+ ( 3)y + 2z = 0
2x+ 2y + z = 0
x = y = z = 0
, , .
33
5
1. (1)
(1 2 1 32 4 3 5
)(1,1) 1
(1 2 1 30 0 1 1
)(2,3) 3
(1 2 0 4
0 0 1 1
),
x+ 2y = 4z = 1 . y = t , x = 2t+ 4
y = t
z = 1
(t ).
(2)
(2 1 3 0
3 2 4 1
) 1 12
(1 12
32 0
3 2 4 1
)(1,1) 1
(1 12
32 0
0 72 12 1
) 2 27
(1 12
32 0
0 1 17 27
)(2,2) 2
(1 0 107
17
0 1 17 27
),
x+ 107 z = 17y + 17z = 27 . z = 7t ,
x = 10t+ 17y = t 27z = 7t
(t ).
(3)
1 2 12 4 13 5 2
(1,1) 1
1 2 10 6 10 1 1
(3,2) 2
1 0 10 0 50 1 1
, x = 1
0 = 5
y = 1
, , .
(4)
1 3 2 12 2 3 11 5 0 1
(1,1) 1
1 3 2 10 4 1 10 8 2 2
2 14
1 3 2 10 1 14 140 8 2 2
(2,2) 2
1 0 54
14
0 1 1414
0 0 0 0
,x 54z = 14y + 14z = 14 . z = 4t,
x = 5t 14y = t+ 14z = 4t
(t ).
(5)
2 3 1 23 4 1 15 6 1 3
2 1
2 3 1 21 1 0 15 6 1 3
1 2
1 1 0 12 3 1 25 6 1 3
(1,1) 1
1 1 0 10 1 1 40 1 1 8
(2,2) 2
1 0 1 50 1 1 40 0 0 4
, x+ z = 15
y z = 4
0 = 4
, , .
(6)
2 5 3 71 3 2 14 5 1 18
1 2
1 3 2 12 5 3 74 5 1 18
(1,1) 1
1 3 2 10 1 1 50 7 7 14
(2,2) 2
1 0 1 160 1 1 50 0 0 21
, x z = 16
y z = 5
0 = 21
, ,
.
34
(7)
1 2 1 13 2 1 11 2 3 0
(1,1) 1
1 2 1 10 4 4 40 4 4 1
2 14
1 2 1 10 1 1 10 4 4 1
(2,2) 2
1 0 1 10 1 1 10 0 0 3
, x+ z = 16
y z = 1
0 = 3
, ,
.
(8)
2 1 1 121 3 2 14 5 1 18
1 2
1 3 2 12 1 1 124 5 1 18
(1,1) 1
1 3 2 10 5 5 100 7 7 14
2 15 1 3 2 10 1 1 2
0 7 7 14
(2,2) 2
1 0 1 70 1 1 20 0 0 0
,x z = 7y z = 2 . z = t
,
x = t+ 7
y = t+ 2
z = t
(t ).
(9)
1 2 1 13 4 1 15 6 1 1
(1,1) 1
1 2 1 10 2 2 20 4 4 4
2 12
1 2 1 10 1 1 10 4 4 4
(2,2) 2
1 0 1 10 1 1 10 0 0 0
, x+ z = 1y z = 1 . z = t ,
x = t 1
y = t+ 1
z = t
(t ).
(10)
3 1 4 51 1 1 21 5 0 3
1 3
1 5 0 31 1 1 23 1 4 5
(1,1) 1
1 5 0 30 4 1 10 16 4 4
2 14 1 5 0 30 1 14 14
0 16 4 4
(2,2) 2
1 0 54
74
0 1 14 14
0 0 0 0
, x 54z = 74y + 14z = 14
. z = 4t ,
x = 5t 74y = t 14z = 4t
(t ).
(11)
2 5 3 43 8 5 53 5 2 11
2 1
2 5 3 41 3 2 13 5 2 11
1 2
1 3 2 12 5 3 43 5 2 11
(1,1) 1 1 3 2 10 1 1 2
0 4 4 8
2 1
1 3 2 10 1 1 20 4 4 8
(2,2) 2
1 0 1 70 1 1 20 0 0 0
,
x z = 7y + z = 2 . z = t , x = t+ 7
y = t 2
z = t
(t ).
35
(12)
3 5 5 42 2 3 35 1 6 7
1 2
1 3 2 12 2 3 35 1 6 7
(1,1) 1
1 3 2 10 4 1 10 16 4 2
3 2 4
1 3 2 10 4 1 10 0 0 2
, x 3y 2z = 1
4y + z = 1
0 = 2
,
, .
(13)
1 2 3 42 1 3 02 3 1 1
(1,1) 1
1 2 3 40 3 3 80 7 7 9
2 2 3
1 2 3 40 3 3 80 1 1 7
2 3
1 2 3 40 1 1 70 3 3 8
(2,2) 2
1 0 1 180 1 1 70 0 0 29
,x+ z = 18
y + z = 7
0 = 29, , .
(14)
1 1 1 62 4 6 23 2 1 5
(1,1) 1
1 1 1 60 2 8 140 1 2 23
2 12
1 1 1 60 1 4 70 1 2 23
(2,2) 2 1 0 5 130 1 4 7
0 0 6 30
2 16
1 0 5 130 1 4 70 0 1 5
(3,3) 3
1 0 0 120 1 0 130 0 1 5
,
x = 12
y = 13
z = 5
.
(15)
a 2 1 21 a 2 22 2 a+ 1
(2,1) 1
0 (a 1)(a 3) 2(a 3)1 a 2 20 2(a 3) a 3
1 2 1 a 2 20 (a 1)(a 3) 2(a 3)
0 2(a 3) a 3
1 1
1 2 a 20 (a 1)(a 3) 2(a 3)0 2(a 3) a 3
() , a = 3 () 1 1 20 0 00 0 0
, a = 3 () 2 3
1 2 a 20 2(a 3) a 30 (a 1)(a 3) 2(a 3)
2 12(a3) 3 1a3 1 2 a 20 1 12
0 a 1 2
(2,2) 2
1 0a2
2
0 1 120 0 a+32
(). a = 3 () =1 0
12
0 1 120 0 0
, a = 3 () 3
2a+3
1 0 a+22
0 1 120 0 1
(3,3) 3
1 0 00 1 00 0 1
. , a = 3 x = y = z = 0 . a = 3 , x = s + 2t, y = s, z = t (s, t ) , a = 3 , x = 1
2t, y =
1
2t,
z = t (t ) .
36
(16)
1 1 0 23 1 1 22 1 2 10 1 1 1
(1,1) 1 1 1 0 20 2 1 4
0 1 2 3
0 1 1 1
2 4 1 1 0 20 1 1 10 1 2 3
0 2 1 4
(2,2) 2
1 0 1 10 1 1 10 0 3 2
0 0 3 2
(3,3) 3 1 0 0 130 1 0 530 0 3 2
0 0 0 0
,x = 13y = 53
3z = 2
.
x = 13y = 53
z = 23
.
(17)
(1 1 2 3 6
1 2 1 4 8
)(1,1) 1
(1 1 2 3 6
0 1 3 1 2
)(2,2) 2
(1 0 5 2 4
0 1 3 1 2
),
x+ 5z + 2w = 4y 3z + w = 2 . z = s, w = t ,
x = 5s 2t+ 4
y = 3s t+ 2
z = s
w = t
(s, t )
.
(18)
(3 2 1 1 4
5 1 2 3 6
) 2 1
(3 2 1 1 4
2 1 1 2 2
) 1 2
(1 3 0 1 22 1 1 2 2
)(1,1) 1 (
1 3 0 1 20 7 1 4 2
),
x+ 3y w = 27y + z + 4w = 2 . y = s, w = t ,
x = 3s+ t+ 2
y = s
z = 7s 4t 2
w = t
(s, t ).
(19)
2 4 0 5 94 3 11 2 45 2 12 4 7
3 2
2 4 0 5 94 3 11 2 41 1 1 2 3
1 3
1 1 1 2 34 3 11 2 42 4 0 5 9
(1,1) 1
1 1 1 2 30 7 7 6 80 2 2 1 3
2 3 3
1 1 1 2 30 1 1 3 10 2 2 1 3
(2,2) 2 1 0 2 1 40 1 1 3 1
0 0 0 5 5
3 15
1 0 2 1 40 1 1 3 10 0 0 1 1
(3,4) 2
1 0 2 0 30 1 1 0 20 0 0 1 1
, x+ 2z = 3
y + z = 2
w = 1
. z = t ,
x = 2t+ 3
y = t 2
z = t
w = 1
(t ).
37
(20)
1 2 3 2 31 3 1 1 22 5 2 3 5
(1,1) 1
1 2 3 2 30 1 4 1 10 1 4 1 1
(2,2) 2
1 0 11 4 50 1 4 1 10 0 0 0 0
,
x 11z 4w = 5y + 4z + w = 1 . z = s, w = t ,
x = 11s+ 4t+ 5
y = 4s t 1
z = s
w = t
(s, t ).
(21)
2 3 4 1 11 1 1 2 20 1 2 5 a
1 2
1 1 1 2 22 3 4 1 10 1 2 5 a
(1,1) 1
1 1 1 2 20 1 2 5 30 1 2 5 a
(2,2) 2
1 0 1 7 50 1 2 5 30 0 0 0 a+ 3
, a = 3 , a = 3
x+ z + 7w = 5y 2z 5w = 3 . z = s, w = t ,
x = s 7t+ 5
y = 2s+ 5t 3
z = s
w = t
(s, t
).
(22)
1 2 4 1 63 12 6 1 85 1 11 3 3
(1,1) 1
1 2 4 1 60 18 18 4 260 9 9 2 33
2 3 2
1 2 4 1 60 0 0 0 400 9 9 2 33
, x 2y + 4z w = 6
0 = 40
9y 9z + 2w = 33
,
, .
(23)
1 2 1 2 12 1 3 1 31 8 9 8 9
(1,1) 1
1 2 1 2 10 5 5 5 50 10 10 10 10
2 15
1 2 1 2 10 1 1 1 10 10 10 10 10
(2,2) 2
1 0 1 0 10 1 1 1 10 0 0 0 0
, x z = 1y + z + w = 1 . z = s, w = t
,
x = s 1
y = s t+ 1
z = s
w = t
(s, t ).
(24)
1 2 4 6 13 1 7 7 8
1 9 21 19 14
2 7 11 5 1
(1,1) 1 1 2 4 6 10 5 5 25 50 7 17 25 13
0 3 3 17 3
15 1 2 4 6 10 1 1 5 10 7 17 25 13
0 3 3 17 3
38
(2,2) 2
1 0 2 16 30 1 1 5 10 0 10 10 200 0 0 2 0
3 110 1 0 2 16 30 1 1 5 10 0 1 1 20 0 0 2 0
(3,3) 3 1 0 0 18 10 1 0 6 30 0 1 1 20 0 0 2 0
(4,4) 4 1 0 0 0 10 1 0 0 30 0 1 0 2
0 0 0 2 0
,
x = 1
y = 3
z = 2
w = 0
.
(25)
1 3 3 2 74 3 2 1 85 2 1 3 3
5 6 1 3 7
(1,1) 1 1 3 3 2 70 15 14 7 360 17 14 7 380 21 14 7 42
2 3 1 1 3 3 2 70 2 0 0 20 17 14 7 380 21 14 7 42
2 12 1 3 3 2 70 1 0 0 1
0 17 14 7 380 21 14 7 42
(2,2) 2 1 0 3 2 40 1 0 0 1
0 0 14 7 210 0 14 7 21
2
114
1 0 3 2 40 1 0 0 1
0 0 1 12 32
0 0 14 7 21
(3,3) 3 1 0 0 12
12
0 1 0 0 1
0 0 1 12 32
0 0 0 0 0
, x+ 12w =
12
y = 1
z + 12w = 32
. w = 2t ,
x = t+ 12y = 1
z = t 32w = 2t
(t ).
(26)
3 2 3 1 24 3 4 2 12 1 2 1 46 3 2 2 5
1 3 1 1 1 5 2 64 3 4 2 12 1 2 1 46 3 2 2 5
(1,1) 1 1 1 5 2 60 1 16 6 230 1 12 5 160 3 32 14 41
2 1 1 1 5 2 60 1 16 6 230 1 12 5 160 3 32 14 41
(2,2) 2 1 0 11 4 170 1 16 6 230 0 4 1 70 0 16 4 28
3
114
1 0 11 4 170 1 16 6 230 0 1 14
74
0 0 16 4 28
(3,3) 3 1 0 0 54
94
0 1 0 2 5
0 0 1 1474
0 0 0 0 0
, x 54w =
94
y + 2w = 5
z 14w =74
. w = 4t ,
x = 5t 94y = 8t+ 5
z = t+ 74
w = 4t
(t ).
39
(27)
1 2 0 1 11 1 2 0 32 0 1 1 8
1 2 1 1 2
(1,1) 1 1 2 0 1 10 3 2 1 20 4 1 3 100 4 1 2 3
2 4 1 2 0 1 10 1 1 1 50 4 1 3 100 4 1 2 3
2 1 1 2 0 1 10 1 1 1 50 4 1 3 100 4 1 2 3
(2,2) 2 1 0 2 1 9
0 1 1 1 50 0 3 1 100 0 5 2 17
4 3
2
1 0 2 1 9
0 1 1 1 50 0 3 1 100 0 1 0 3
3 4 1 0 2 1 9
0 1 1 1 50 0 1 0 3
0 0 3 1 10
(3,3) 3 1 0 0 1 3
0 1 0 1 20 0 1 0 3
0 0 0 1 1
4 1 1 0 0 1 3
0 1 0 1 20 0 1 0 3
0 0 0 1 1
(4,4) 4 1 0 0 0 2
0 1 0 0 10 0 1 0 3
0 0 0 1 1
,
x = 2
y = 1
z = 3
w = 1
.
(28)
0 3 3 2 41 1 2 3 2
1 2 3 2 1
1 3 4 2 1
1 2 1 1 2 3 2
0 3 3 2 41 2 3 2 1
1 3 4 2 1
(1,1) 1 1 1 2 3 2
0 3 3 2 40 1 1 1 10 2 2 1 3
2 3
1 1 2 3 2
0 1 1 1 10 3 3 2 40 2 2 1 3
(2,2) 2 1 0 1 4 3
0 1 1 1 10 0 0 1 10 0 0 1 1
(3,4) 4 1 0 1 0 7
0 1 1 0 20 0 0 1 10 0 0 0 0
,x+ z = 7
y + z = 2
w = 1
. z = t,
x = t+ 7
y = t 2
z = t
w = 1(t ).
(29)
2 4 1 1 11 2 1 1 22 1 1 2 21 3 2 3 0
1 2 1 2 1 1 22 4 1 1 12 1 1 2 21 3 2 3 0
(1,1) 1 1 2 1 1 20 0 3 3 30 3 3 0 60 1 3 4 2
3 13
1 2 1 1 20 0 3 3 30 1 1 0 20 1 3 4 2
2 3 1 2 1 1 20 1 1 0 20 0 3 3 30 1 3 4 2
(2,2) 2
40
1 0 1 1 20 1 1 0 20 0 3 3 30 0 4 4 4
3 13 1 0 1 1 20 1 1 0 20 0 1 1 10 0 4 4 4
(3,3) 3 1 0 0 2 10 1 0 1 10 0 1 1 10 0 0 0 0
,
x+ 2w = 1
y w = 1
z w = 1
. w = t ,
x = 2t 1
y = t+ 1
z = t 1
w = t
(t ).
(30)
1 0 1 2 6
2 1 4 1 34 3 4 1 31 1 2 1 4
(1,1) 1 1 0 1 2 6
0 1 6 5 15
0 3 8 7 270 1 3 3 10
(2,2) 2 1 0 1 2 6
0 1 6 5 15
0 0 10 8 18
0 0 3 2 5
4 3 3
1 0 1 2 6
0 1 6 5 15
0 0 1 2 3
0 0 3 2 5
(3,3) 3 1 0 0 0 3
0 1 0 7 30 0 1 2 3
0 0 0 4 4
4 14 1 0 0 0 3
0 1 0 7 30 0 1 2 3
0 0 0 1 1
(4,4) 4
1 0 0 0 3
0 1 0 0 4
0 0 1 0 1
0 0 0 1 1
,
x = 3
y = 4
z = 1
w = 1
.
(31)
2 3 2 1 1
4 2 1 1 22 1 1 2 12 1 2 3 1
(1,1) 1 2 3 2 1 1
0 4 5 1 00 2 1 1 00 2 0 2 0
2 4 2 3 2 1 1
0 2 0 2 00 2 1 1 00 4 5 1 0
2 12
2 3 2 1 1
0 1 0 1 00 2 1 1 00 4 5 1 0
(2,2) 2 2 0 2 4 1
0 1 0 1 00 0 1 1 0
0 0 5 5 0
(3,3) 3 2 0 0 2 1
0 1 0 1 00 0 1 1 0
0 0 0 0 0
,
2x+ 2w = 1
y w = 0
z + w = 0
. w = t,
x = t+ 12y = t
z = t
w = t
(t).
(32)
1 1 2 1 11 1 3 2 2
2 2 2 1 11 1 0 1 1
(1,1) 1 1 1 2 1 10 0 1 1 3
0 4 2 3 10 2 2 2 0
2 4 1 1 2 1 10 2 2 2 0
0 4 2 3 10 0 1 1 3
2 12
1 1 2 1 10 1 1 1 0
0 4 2 3 10 0 1 1 3
(2,2) 2 1 0 1 0 10 1 1 1 0
0 0 2 1 1
0 0 1 1 3
3 4 1 0 1 0 10 1 1 1 0
0 0 1 1 3
0 0 2 1 1
41
(3,3) 3
1 0 0 1 40 1 0 0 30 0 1 1 3
0 0 0 1 5
(4,4) 4 1 0 0 0 1
0 1 0 0 30 0 1 0 20 0 0 1 5
,
x = 1
y = 3
z = 2
w = 5
.
(33)
0 1 1 1 41 2 1 1 12 1 5 6 3
1 1 2 2 1
1 2 1 2 1 1 10 1 1 1 42 1 5 6 3
1 1 2 2 1
(1,1) 1 1 2 1 1 10 1 1 1 40 3 3 4 50 1 1 1 2
(2,2) 2
1 0 3 1 70 1 1 1 40 0 0 7 70 0 0 2 2
3 17 1 0 3 1 70 1 1 1 40 0 0 1 10 0 0 2 2
(3,3) 3 1 0 3 0 6
0 1 1 0 30 0 0 1 10 0 0 0 0
,
x+ 3z = 6
y z = 3
w = 1
. z = t ,
x = 3t+ 6
y = t 3
z = t
w = 1
(t ).
(34)
2 1 2 3 1
1 1 2 2 23 1 0 2 1
4 2 1 1 0
1 2 1 1 2 2 22 1 2 3 1
3 1 0 2 1
4 2 1 1 0
(1,1) 1 1 1 2 2 20 1 6 7 50 2 6 8 70 2 7 9 8
1
1 1 2 2 20 1 6 7 50 2 6 8 70 2 7 9 8
(2,2) 2 1 0 4 5 3
0 1 6 7 50 0 6 6 3
0 0 5 5 2
4 1 3 1 0 4 5 3
0 1 6 7 50 0 1 1 1
0 0 5 5 2
(3,3) 3
1 0 0 1 10 1 0 1 10 0 1 1 1
0 0 0 0 3
,
x+ w = 1
y w = 1
z + w = 1
0 = 3
,
, .
(35)
1 2 1 3 31 1 3 6 32 3 1 3 31 2 0 1 2
(1,1) 1 1 2 1 3 30 1 2 3 00 1 1 3 30 0 1 2 1
(2,2) 2 1 0 5 9 30 1 2 3 00 0 3 6 30 0 1 2 1
3 13 1 0 5 9 30 1 2 3 00 0 1 2 10 0 1 2 1
(3,3) 3 1 0 0 1 20 1 0 1 2
0 0 1 2 10 0 0 0 0
,
x w = 2
y + w = 2
z 2w = 1
. w = t ,
x = t 2
y = t+ 2
z = 2t 1
w = t
(t ).
42
(36)
1 2 4 6 73 1 7 7 9
1 9 11 19 11
2 7 11 5 a
(1,1) 1 1 2 4 6 70 5 5 25 300 7 7 25 18
0 3 3 17 a+ 14
2 15 1 2 4 6 70 1 1 5 60 7 7 25 18
0 3 3 17 a+ 14
(2,2) 2
1 0 2 4 5
0 1 1 5 60 0 0 60 60
0 0 0 32 a+ 32
3 160 1 0 2 4 5
0 1 1 5 60 0 0 1 1
0 0 0 32 a+ 32
(3,4) 4 1 0 2 0 1
0 1 1 0 10 0 0 1 1
0 0 0 0 a
,
x+ 2w = 1
y + z = 1
w = 1
0 = a
, a = 0 , a = 0
z = s ,
x = 2s+ 1
y = s 1
z = s
w = 1
(s ).
(37)
2 1 1 2 a1 2 1 1 b1 1 2 1 c
1 3
1 2 1 1 b2 1 1 2 a1 1 2 1 c
(1,1) 1 1 2 1 1 b0 3 3 0 a+ 2b
0 3 3 0 c b
(2,2) 2
1 0 1 1 2a3
b3
0 3 3 0 a+ 2b0 0 0 0 a+ b+ c
, x+ z w = 2a+b33y 3z = a+ 2b
0 = a+ b+ c
. a+b+c = 0 z = s, w = t,
x = s t+ 2a+b3y = s+ a+2b3
z = s
w = t
(s, t ) . a+ b+ c = 0 , .
(38)
1 2 1 3 4 50 0 1 2 4 22 4 1 3 2 5
(1,1) 1
1 2 1 3 4 50 0 1 2 4 20 0 1 3 6 5
(2,3) 3 1 2 0 1 8 30 0 1 2 4 2
0 0 0 1 10 3
(3,4) 4
1 2 0 0 2 00 0 1 0 24 40 0 0 1 10 3
, x+ 2y 2v = 0
z + 24v = 4
w + 10v = 3
. y = s, v = t ,
x = 2s+ 2t
y = s
z = 24t+ 4
w = 10t+ 3
v = t
(s, t ).
43
(39)
1 2 3 1 1
2 5 8 3 3
1 3 5 2 a+ 3
4 3 2 a a
(1,1) 1 1 2 3 1 1
0 1 2 1 1
0 1 2 1 a+ 2
0 5 10 a 4 a 4
(2,2) 2 1 0 1 1 10 1 2 1 1
0 0 0 0 a+ 1
0 0 0 a+ 1 a+ 1
3 4 1 0 1 1 10 1 2 1 1
0 0 0 a+ 1 a+ 1
0 0 0 0 a+ 1
() , a = 1 () 1 0 1 1 10 1 2 1 1
0 0 0 0 0
0 0 0 0 0
, x z w = 1y + 2z + w = 1 , z = u, w = v
,
x = u+ v 1
y = 2u v + 1
z = u
w = v
(u, v). a = 1 () 3 4 1a+1
1 0 1 1 10 1 2 1 1
0 0 0 1 1
0 0 0 0 1
,
x z w = 1
y + 2z + w = 1
w = 1
0 = 1
, ,
.
(40)
3 2 1 0 1 21 1 1 2 1 12 1 3 1 3 2
1 2
1 1 1 2 1 13 2 1 0 1 22 1 3 1 3 2
(1,1) 1 1 1 1 2 1 10 1 2 6 2 1
0 3 5 5 1 0
(2,2) 2
1 0 1 4 1 00 1 2 6 2 10 0 1 13 7 3
(3,3) 3
1 0 0 9 6 30 1 0 20 12 50 0 1 13 7 3
, x 9w + 6v = 3
y 20w + 12v = 5
z 13w + 7v = 3
. w = s, v = t
,
x = 9s 6t+ 3
y = 20s 12t+ 5
z = 13s 7t+ 3
w = s
v = t
(s, t ).
(41)
0 2 4 2 2
1 1 3 2 21 2 3 1 b
2 1 0 a 1
1 3
1 2 3 1 b
1 1 3 2 20 2 4 2 2
2 1 0 a 1
(1,1) 1 1 2 3 1 b
0 3 6 3 b+ 2
0 2 4 2 2
0 3 6 a+ 2 2b+ 1
44
2 3
1 2 3 1 b
0 2 4 2 2
0 3 6 3 b+ 2
0 3 6 a+ 2 2b+ 1
3 12 1 2 3 1 b
0 1 2 1 1
0 3 6 3 b+ 2
0 3 6 a+ 2 2b+ 1
(2,2) 2 1 0 1 1 b 20 1 2 1 1
0 0 0 0 b 10 0 0 a 1 2(b 1)
,
x z w = b 2
y + 2z + w = 1
0 = b 1
(a 1)w = 2(b 1)
. b = 1,
a = 1 w = 0 , z = t ,
x = t+ b 2
y = 2t+ 1
z = t
w = 0
(t ), b = 1, a = 1
, z = s, w = t ,
x = s+ t+ b 2
y = 2s t+ 1
z = s
w = t
(s, t ) . b = 1
, .
(42)
1 2 2 1 3 22 1 2 0 1 32 3 2 1 2 1
(1,1) 1
1 2 2 1 3 20 3 6 2 5 10 1 2 1 8 5
2 3 1 2 2 1 3 20 1 2 1 8 5
0 3 6 2 5 1
(2,2) 2
1 0 2 1 13 80 1 2 1 8 50 0 0 1 19 14
(3,4) 4
1 0 2 0 6 60 1 2 0 11 90 0 0 1 19 14
, x+ 2z + 6v = 6
y 2z 11v = 9
w + 19v = 14
. z = s, v = t
,
x = 2s 6t+ 6
y = 2s+ 11t 9
z = s
w = 19t+ 14
v = t
(s, t ).
(43)
1 2 1 a a+522 1 3a 0 11 1 2a 2 02 2 a 1 a 2 b
(1,1) 1 1 2 1 a a+520 5 3a 2 2a a 60 3 2a+ 1 a+ 2 a+520 2 a+ 1 a 2 a+ b+ 5
3 2 2 1 2 1 a a+520 1 a 4 a+ 2b+ 40 3 2a+ 1 a+ 2 a+520 2 a+ 1 a 2 a+ b+ 5
(2,2) 2 1 0 1 2a a 8 5a+8b+2120 1 a 4 a+ 2b+ 40 0 1 a a 10 7a+12b+2920 0 1 a a 10 3a+ 5b+ 13
3 1 4
45
1 0 1 2a a 8 5a+8b+2120 1 a 4 a+ 2b+ 40 0 1 a a 10 7a+12b+2920 0 0 0 a2b32
a = 2b 3 . a = 2b 3
1 0 4b+ 7 2b 11 3 b0 1 2b+ 3 4 10 0 2b+ 4 2b 13 4 b0 0 0 0 0
() . b = 2 , () 3 12b+4 1 0 4b+ 7 2b 11 3 b0 1 2b+ 3 4 10 0 1 2b+132b+4
4b2b+4
0 0 0 0 0
(3,3) 3 1 0 0 4b
2+36b+472b+4
2b27b162b+4
0 1 0 4b2+24b+232b+4
2b23b82b+4
0 0 1 2b+132b+44b2b+4
0 0 0 0 0
,
x+ 4b
2+36b+472b+4 w =
2b27b162b+4
y + 4b2+24b+232b+4 w =
2b23b82b+4
z 2b+132b+4 w =4b2b+4
. w = t ,
x = 2b27b162b+4
4b2+36b+472b+4 t
y = 2b23b82b+4
4b2+24b+232b+4 t
z = 4b2b+4 +2b+132b+4 t
w = t
(t )
. b = 2 , () =
1 0 1 7 50 1 1 4 10 0 0 9 60 0 0 0 0
3 19 1 0 1 7 50 1 1 4 10 0 0 1 230 0 0 0 0
(3,4) 4 1 0 1 0 130 1 1 0 530 0 0 1 230 0 0 0 0
,x z = 13y z = 53w = 23
. z = t,
x = t+ 13
y = t 53z = t
w = 23(t ).
(44)
1 2 1 3 2 12 4 1 3 3 21 2 2 4 1 13 6 0 6 5 3
(1,1) 1 1 2 1 3 2 10 0 3 3 1 00 0 1 1 3 20 0 3 3 1 0
2 3 1 2 1 3 2 10 0 1 1 3 20 0 3 3 1 00 0 3 3 1 0
(2,3) 3 1 2 0 2 5 30 0 1 1 3 20 0 0 0 10 60 0 0 0 10 6
3 110 1 2 0 2 5 30 0 1 1 3 20 0 0 0 1 350 0 0 0 10 6
(3,5) 5 1 2 0 2 0 0
0 0 1 1 0 150 0 0 0 1 350 0 0 0 0 0
, x+ 2y + 2w = 0
z w = 15v = 35
. y = s, w = t ,
x = 2s 2t
y = s
z = t+ 15
w = t
v = 35
(s, t ).
46
(45)
11 12 13 14 15 a
6 7 8 9 10 0
1 2 3 4 5 0
1 4 9 16 25 b
1 3
1 2 3 4 5 0
6 7 8 9 10 0
11 12 13 14 15 a
1 4 9 16 25 b
(1,1) 1 1 2 3 4 5 0
0 5 10 15 20 00 10 20 40 40 a0 2 6 12 20 b
2 15 1 2 3 4 5 0
0 1 2 3 4 0
0 10 20 30 40 a0 2 6 12 20 b
(2,2) 2 1 0 1 2 3 00 1 2 3 4 0
0 0 0 0 0 a
0 0 2 6 12 b
3 4 1 0 1 2 3 00 1 2 3 4 0
0 0 2 6 12 b
0 0 0 0 0 a
3 12 1 0 1 2 3 00 1 2 3 4 0
0 0 1 3 6 b20 0 0 0 0 a
(3,3) 3
1 0 0 1 3 b20 1 0 3 8 b0 0 1 3 6 b20 0 0 0 0 a
a = 0 . a = 0 ,
x+ u+ 3v = b2
y 3u 8v = b
z + 3u+ 6v = b2
. u = s, v = t ,
x = s 3t+ b2y = 3s+ 8t b
z = 3s 6t+ b2u = s
v = t
(s, t
).
2. (1) 1.2 0 1 0 p
1 2 1 1 q
0 1 0 1 r1 1 1 4 s
1 4 1 1 1 4 s1 2 1 1 q
0 1 0 1 r2 0 1 0 p
(1,1) 1 1 1 1 4 s0 1 2 3 q s0 1 0 1 r0 2 3 8 p 2s
(2,2) 2
1 0 3 7 2s q0 1 2 3 q s0 0 2 4 q + r s0 0 7 14 p+ 2q 4s
2 12 1 0 3 7 2s q0 1 2 3 q s0 0 1 2 q+rs20 0 7 14 p+ 2q 4s
(3,3) 3 1 0 0 1 q+3r+s20 1 0 1 r0 0 1 2 q+rs20 0 0 0 2p3q7rs2
, 2p 3q 7r s = 0 .
,
x+ w = q+3r+s2
y + w = r
z 2w = q+rs2
. w = t ,
x = t+ q+3r+s2y = t r
z = 2t+ q+rs2
w = t
(t )
.
47
(2)
1 3 3 1 p1 2 2 2 q2 5 5 3 r1 1 1 3 s
(1,1) 1 1 3 3 1 p0 1 1 1 q p0 1 1 1 r 2p0 2 2 2 s p
(2,2) 2 1 0 0 4 3q 2p0 1 1 1 q p0 0 0 0 r q p0 0 0 0 s 2q + p
, r = p+ q s = p+ 2q . ,
x+ 4w = 3q 2py + z + w = q p . z = t, w = u ,
x = 4u 2p+ 3q
y = t u p+ q
z = t
w = u
(t, u )
.
(3)
1 1 1 1 p2 3 1 1 q1 1 2 2 r2 5 0 a s
(1,1) 1 1 1 1 1 p0 1 1 3 q 2p0 2 1 1 p+ r0 3 2 a 2 s 2p
2 1 1 1 1 1 p0 1 1 3 2p q0 2 1 1 p+ r0 3 2 a 2 s 2p
(2,2) 2 1 0 2 4 3p q0 1 1 3 2p q0 0 1 5 5p 2q + r0 0 1 a+ 7 4p 3q + s
(3,3) 3 1 0 0 6 7p+ 3q 2r0 1 0 2 3p+ q r0 0 1 5 5p 2q + r0 0 0 a+ 2 p q r + s
, a = 2 , 4 1a+2 (4, 4)
4,
1 0 0 0 7ap+3aq2ar20p10r+6sa+20 1 0 0 3ap+aqar8p4r+2sa+20 0 1 0 5ap2aq+ar+15p+q+7r5sa+20 0 0 1 pqr+sa+2
,
x = 7ap+3aq2ar20p10r+6sa+2
y = 3ap+aqar8p4r+2sa+2
z = 5ap2aq+ar+15p+q+7r5sa+2
w = pqr+sa+2
. a = 2 , s = p+ q + r
, s = p+ q+ r
x 6w = 7p+ 3q 2r
y 2w = 3p+ q r
z + 5w = 5p 2q + r
, w = t ,
x = 6t 7p+ 3q 2r
y = 2t 3p+ q r
z = 5t+ 5p 2q + r
w = t
(t ).
(4)
1 2 3 1 p2 3 a 1 0 q3 5 a+ 2 1 r4 7 a+ 5 2 s
(1,1) 1 1 2 3 1 p0 1 a 7 2 q 2p0 1 a 7 2 r 3p0 1 a 7 2 s 4p
(2,2) 2
48
1 0 2a 11 3 3p+ 2q0 1 a 7 2 q 2p0 0 0 0 r p q0 0 0 0 s 2p q
, 1 r = p + q, s = 2p + q
, 1
x+ (2a 11)z + 3w = 3p+ 2qy + (a 7)z + 2w = 2p+ q
. , z = t, w = u ,
x = (2a 11)t 3u 3p+ 2q
y = (a 7)t 2u 2p+ q
z = t
w = u
(t, u ) .
(5)
1 1 0 2 p2 1 3 2 q3 0 3 4 r4 1 3 6 s
(1,1) 1 1 1 0 2 p0 3 3 2 q 2p0 3 3 2 r 3p0 3 3 2 s 4p
(2,3) 3 1 1 0 2 p0 3 3 2 q 2p0 0 0 0 r q p0 0 0 0 s q 2p
, 1 r = p + q, s = 2p + q
, 1
x+ y 2w = p3y + 3z + 2w = 2p+ q .
, y = t, w = u ,
x = t+ 2u+ p
y = t
z = t 2u3 2pq
3
w = u
(t, u ) .
(6)
1 1 1 0 p2 1 2 1 q0 1 4 1 r1 0 3 1 s
(1,1) 1 1 1 1 0 p0 1 4 1 2p+ q0 1 4 1 r0 1 4 1 p+ s
(2,2) 2 1 0 3 1 p q0 1 4 1 2p+ q0 0 0 0 2p q + r0 0 0 0 p q + s
, , r = 2p+ q, s = p+ q
,
x+ 3z w = p qy 3z + w = 2p+ q . z = t, w = u
,
x = 3t+ u p q
y = 3t+ u 2p q
z = t
w = u
(t, u ).
49
(7)
2 0 1 0 p
1 4 1 1 q0 1 0 1 r1 1 1 2 s
(4,1) 1 0 2 1 4 p 2s0 3 2 1 q s0 1 0 1 r1 1 1 2 s
(3,2) 2 0 0 1 2 p 2r 2s0 0 2 4 q + 3r s0 1 0 1 r1 0 1 1 r + s
(1,3) 3 0 0 1 2 p 2r 2s0 0 0 0 q 2p+ 7r + 3s0 1 0 1 r1 0 0 1 p r s
, 1 q = 2p 7r 3s , 1
x w = p r s
y w = r
z 2w = p 2r 2s
. , w = t ,
x = t+ p r s
y = t r
z = 2t p+ 2r + 2s
w = t
(t )
.
(8)
1 2 1 1 p1 1 1 2 q2 1 0 1 r0 3 2 3 s
(1,1) 1 1 2 1 1 p0 3 2 3 q p0 3 2 3 r + 2p0 3 2 3 s
(2,2) 2 1 0 13 1
p+2q3
0 3 2 3 q p0 0 0 0 r + p+ q
0 0 0 0 s+ p q
, r = p q, s = p+ q
,
x 13z + w =p+2q
3
3y 2z + 3w = q p. z = t, w = u ,
x = t3 u+p+2q
3
y = 2t3 upq3
z = t
w = u
(t, u ).
(9)
1 1 0 0 p
0 1 0 1 q
1 0 1 0 r
0 0 1 1 s
(1,1) 1 1 1 0 0 p
0 1 0 1 q
0 1 1 0 r p0 0 1 1 s
(2,2) 2 1 0 0 1 p q0 1 0 1 q
0 0 1 1 r p+ q0 0 1 1 s
(3,3) 3
1 0 0 1 p q0 1 0 1 q
0 0 1 1 r p+ q0 0 0 0 s+ p q r
, s = p+ q + r
,
x w = p q
y + w = q
z + w = r p+ q
. w = t ,
50
x = t+ p q
y = t+ q
z = t p+ q + r
w = t
(t ).
(10)
1 2 1 0 p0 1 3 2 q1 3 2 2 r1 1 4 2 s
(1,1) 1 1 2 1 0 p0 1 3 2 q0 1 3 2 r p0 1 3 2 s p
(2,2) 2 1 0 7 4 p 2q0 1 3 2 q0 0 0 0 r p q0 0 0 0 s p+ q
, r = p+ q, s = p q
,
x 7z + 4w = p 2qy + 3z 2w = q . z = t, w = u ,
x = 7t 4u+ p 2q
y = 3t+ 2u+ q
z = t
w = u
(t, u ).
(11)
1 2 2 1 p
1 3 1 2 q
3 7 5 4 r
2 5 3 3 s
(1,1) 1 1 2 2 1 p
0 1 1 1 q p0 1 1 1 r 3p0 1 1 1 s 2p
(2,2) 2 1 0 4 1 3p 2q0 1 1 1 q p0 0 0 0 r 2p q0 0 0 0 s p q
, r = 2p+ q, s = p+ q ,
x+ 4z w = 3p 2qy z + w = q p . z = t, w = u ,
x = 4t+ u+ 3p 2q
y = t u p+ q
z = t
w = u
(t,
u ).
(12)
1 2 2 1 p1 1 1 2 q2 3 1 1 r1 3 5 4 s
(1,1) 1 1 2 2 1 p0 1 3 3 q p0 1 3 3 r 2p0 1 3 3 s+ p
(2,2) 2 1 0 4 5 2q p0 1 3 3 q p0 0 0 0 r p q0 0 0 0 s+ 2p q
, r = p+ q, s = 2p+ q
,
x 4z + 5w = p+ 2qy 3z + 3w = q p . z = t, w = u ,
51
x = 4t 5u p+ 2q
y = 3t 3u p+ q
z = t
w = u
(t, u ).
(13)
1 2 2 1 p
1 1 1 2 q2 1 a 1 r
3 2 4 4 s
(1,1) 1 1 2 2 1 p
0 3 1 1 q p0 3 a 4 1 r 2p0 4 2 1 s 3p
(2,4) 4 1 5 3 0 2p q0 3 1 1 q p0 6 a 5 0 r 3p+ q0 1 1 0 s 2p q
(4,2) 2 1 0 2 0 8p 6q + 5s0 0 2 1 5p+ 4q 3s0 0 a+ 1 0 9p+ 7q + r 6s0 1 1 0 s 2p q
() , a = 2
, () 3 1
a+1
1 0 2 0 8p 6q + 5s0 0 2 1 5p+ 4q 3s0 0 1 0 9p+7q+r6sa+10 1 1 0 s 2p q
(3,3) 3 1 0 0 0 (8a10)p(6a8)q+2r+(5a7)sa+10 0 0 1 (5a13)p+(4a10)q2r(3a9)sa+10 0 1 0 9p+7q+r6sa+10 1 0 0 (2a+7)p(a6)q+r+(a5)sa+1
,
x = (8a10)p(6a8)q+2r+(5a7)sa+1
y = (2a7)p+(a6)qr(a5)sa+1
z = 9p+7q+r6sa+1
w = (5a13)p+(4a10)q2r(3a9)sa+1
. a = 1 , () =
1 0 2 0 8p 6q + 5s0 0 2 1 5p+ 4q 3s0 0 0 0 9p+ 7q + r 6s0 1 1 0 s 2p q
,
r = 9p 7q + 6s ,
x 2z = 8p 6q + 5s
2z + w = 5p+ 4q 3s
y z = s 2p q
. z = t ,
x = 2t 8p 6q + 5s
y = t+ 2p+ q s
z = t
w = 2t+ 5p+ 4q 3s
(t ).
(14)
1 1 a p1 0 1 qb 1 1 r
(2,1) 1
0 1 a+ 1 p q1 0 1 q0 1 b 1 r bq
(1,2) 2 0 1 a+ 1 p q1 0 1 q
0 0 a+ b p (b+ 1)q + r
() , a = b , () 3 1
a+b
0 1 a+ 1 p q1 0 1 q0 0 1 p(b+1)q+ra+b
(3,3) 3
0 1 0(b1)p+(ab+1)q(a+1)r
a+b
1 0 0 p+(a1)q+ra+b0 0 1 p(b+1)q+ra+b
, x = p+(a1)q+ra+b
y = (b1)p+(ab+1)q(a+1)ra+b
z = p(b+1)q+ra+b
52
. a = b , () =
0 1 a+ 1 p q1 0 1 q0 0 0 p+ (a 1)q + r
, r = p (a 1)q ,
y + (a+ 1)z = p qx z = q . z = t ,
x = t+ q
y = (a+ 1)t+ p q
z = t
(t ) .
(15)
a 1 1 p1 a 1 q1 1 a r
(2,1) 1
0 1 a2 1 a p aq
1 a 1 q
0 1 a a 1 r q
() , a = 1 ,() 1,3
11a
0 1 + a 1aqpa1
1 a 1 q
0 1 1 qra1
(3,2) 2
0 0 a+ 2pq+(a+1)r
a11 0 a+ 1 q+ara10 1 1 qra1
() .
, a = 1,2 , () 1 1
a+2
0 0 1pq+(a+1)r(a1)(a+2)
1 0 a+ 1 q+ara10 1 1 qra1
(1,3) 3
0 0 1 pq+(a+1)r(a1)(a+2)1 0 0 (a+1)pqr(a1)(a+2)0 1 0 p+(a+1)qr(a1)(a+2)
,
x = (a+1)pqr(a1)(a+2)
y = p+(a+1)qr(a1)(a+2)
z = pq+(a+1)r(a1)(a+2)
. a = 2 , () =
0 0 0p+q+r
3
1 0 1 q+2r30 1 1 q+r3
, p = q r , x z =
q+2r3
y z = q+r3. z = t ,
x = t+ q+2r3
y = t qr3z = t
(t ). a = 1 ,
() =
0 0 0 p q1 1 1 q0 0 0 r q
, p = q = r ,
x + y + z = p . y = t, z = u ,
x = t u+ p
y = t
z = u
(t, u
).
(16)
a 1 1 1 p
1 a 1 1 q
1 1 a 1 r
1 1 1 a s
(2,1) 1 0 1 a2 1 a 1 a p aq1 a 1 1 q
0 1 a a 1 0 r q0 1 a 0 a 1 s q
() , a = 1 ,
() 1,3,4 1
1a
0 1 + a 1 1 aqpa11 a 1 1 q
0 1 1 0 qra10 1 0 1 qsa1
(3,2) 2 0 0 a+ 2 1 pq+(a+1)ra11 0 a+ 1 1 q+ara10 1 1 0 qra10 0 1 1 rsa1
53
(4,3) 3
0 0 0 a+ 3 pqr+(a+2)sa11 0 0 a+ 2 qr+(a+1)sa10 1 0 1 qsa10 0 1 1 rsa1
() . , a = 1,3 ,
() 1 1
a+3
0 0 0 1 pqr+(a+2)s(a1)(a+3)1 0 0 a+ 2 qr+(a+1)sa10 1 0 1 qsa10 0 1 1 rsa1
(1,4) 4 0 0 0 1 pqr+(a+2)s(a1)(a+3)1 0 0 0 (a+2)pqrs(a1)(a+3)0 1 0 0 p+(a+2)qrs(a1)(a+3)0 0 1 0 pq+(a+2)rs(a1)(a+3)
,
x = (a+2)pqrs(a1)(a+3)
y = p+(a+2)qrs(a1)(a+3)
z = pq+(a+2)rs(a1)(a+3)
w = pqr+(a+2)s(a1)(a+3)
. a = 3 , () =
0 0 0 0 p+q+r+s41 0 0 1 q+r+2s40 1 0 1 q+s40 0 1 1 r+s4
, p = q r s , x w = q+r+2s4y w = q+s4z w = r+s4
. w = t ,
x = t+ q+r+2s4
y = t qs4z = t rs4w = t
(t ). a = 1
, () =
0 0 0 0 p q1 1 1 1 q
0 0 0 0 r q0 0 0 0 s q
, p = q = r = s , x+ y + z + w = p . y = t, z = u, w = v ,
x = t u v + p
y = t
z = u
w = v
(t, u, v ).
(17)
1 1 1 2 p0 1 1 1 q2 1 a 3 ra 2 2 3 s
(1,1) 1 1 1 1 2 p0 1 1 1 q0 1 a 2 1 r 2p0 a 2 2 a 3 2a s ap
(2,2) 2 1 0 0 1 p+ q
0 1 1 1 q0 0 a 1 0 r 2p q0 0 0 1 a s ap (a 2)q
() , a = 1 , () 3 1a1 4 11a
1 0 0 1 p+ q
0 1 1 1 q0 0 1 0 r2pqa10 0 0 1 ap+(a2)qsa1
(3,3) 3 1 0 0 1 p+ q
0 1 0 1 r2p+(a2)qa10 0 1 0 r2pqa10 0 0 1 ap+(a2)qsa1
(4,4) 4 1 0 0 0 p+q+sa10 1 0 0 (a2)p+2(a2)q+rsa10 0 1 0 2pq+ra10 0 0 1 ap+(a2)qsa1
,
x = p+q+sa1
y = (a2)p+2(a2)q+rsa1
z = 2pq+ra1
w = ap+(a2)qsa1
. a = 1
54
, () =
1 0 0 1 p+ q
0 1 1 1 q0 0 0 0 r 2p q0 0 0 0 s p+ q
, r = 2p + q, s = p q
,
x+ w = p+ qy z w = q . z = t, w = u
,
x = u+ p+ q
y = t+ u+ q
z = t
w = u
(t, u ).
(18)
1 a 0 0 p
0 1 a 0 q
0 0 1 a r
1 3 3 1 s
(1,1) 1 1 a 0 0 p
0 1 a 0 q
0 0 1 a r
0 3 a 3 1 s p
(2,2) 2 1 0 a2 0 p aq0 1 a 0 q
0 0 1 a r
0 0 a2 3a+3 1 sp+(a 3)q
(3,3) 3 1 0 0 a3 p aq + a2r0 1 0 a2 q ar0 0 1 a r
0 0 0 (1a)3 sp+(a3)q(a23a+ 3)r
()
, a = 1 , () 4 1
(1a)3
1 0 0 a3 p aq + a2r0 1 0 a2 q ar0 0 1 a r
0 0 0 1 sp+(a3)q(a23a+3)r
(1a)3
(4,4) 4 1 0 0 0 (3a
23a+1)p+(3a2a)q+a2ra3s(1a)3
0 1 0 0 a2p(3a1)qar+a2s
(1a)3
0 0 1 0 ap(a23a)q+ras(1a)3
0 0 0 1 p+(a3)q(a23a+3)r+s
(1a)3
,
x = (3a23a+1)p+(3a2a)q+a2ra3s
(1a)3
y = a2p(3a1)qar+a2s
(1a)3
z = ap(a23a)q+ras(1a)3
w = p+(a3)q(a23a+3)r+s
(1a)3
. a = 1 , () =
1 0 0 1 p q + r0 1 0 1 q r0 0 1 1 r
0 0 0 0 s p 2q r
,
s = p + 2q + r ,
x+ w = p q + r
y w = q r
z + w = r
.
w = t ,
x = t p+ q r
y = t+ q r
z = t+ r
w = t
(t ).
(19)
1 2 3 1 p2 3 1 0 q3 1 2 1 r
5 4a a 4 5a 1 2a 1 s
(1,1) 1 1 2 3 1 p0 7 7 2 q 2p0 7 7 2 r 3p0 7(a 2) 7(2 a) 2(2 a) (4a 5)p+ s
55
(2,2) 2
1 2 3 1 p0 7 7 2 q 2p0 0 0 0 r p q0 0 0 0 (2a 1)p+ (a 2)q + s
, r = p+q,
s = (1 2a)p+ (2 a)q ,
x z 37w =3p+2q
7
7y + 7z + 2w = q 2p
. z = t, w = u ,
x = t+ 3u7 +3p+2q
7
y = t+ 2u7 +2pq
7
z = t
w = u
(t, u ).
(20)
1 1 a a 2 p1 1 1 2 a q2 a 3 2 2 ra+ 1 2 2 2 s
(1,1) 1 1 1 a a 2 p0 0 1 a 0 p+ q0 a 1 2(1 a) 2(a 1) r + 2p0 1 a (a 1)(a+ 2) a(1 a) s p(a+ 1)
()
, a = 1 , () 2,3,4 1
a1
1 1 a a 2 p0 0 1 0 p+qa10 1 2 2 r+2pa10 1 a+ 2 a sp(a+1)a1
(3,2) 2 1 0 2 a a 4 p(a3)ra10 0 1 0 p+qa10 1 2 2 r+2pa10 0 a 2 a sp(a1)+ra1
(2,3) 3 1 0 0 a 4 pr+q(2a)a10 0 1 0 p+qa10 1 0 2 r2qa10 0 0 2 a s+p+aq+ra1
() .
, a = 1, 2 , () 4 1
2a
1 0 0 a 4 pr+q(2a)a10 0 1 0 p+qa10 1 0 2 r2qa10 0 0 1 s+p+aq+r(a1)(2a)
(4,4) 4 1 0 0 0 2p4q2r+(a4)s(a1)(a2)0 0 1 0 p+qa10 1 0 0 2p+4q+ar+2s(a1)(a2)0 0 0 1 p+aq+r+s(a1)(2a)
,
x = 2p4q2r+(a4)s(a1)(a2)
y = 2p+4q+ar+2s(a1)(a2)
z = p+qa1w = p+aq+r+s(a1)(a2)
. a = 2
, () =
1 0 0 2 p r0 0 1 0 p+ q0 1 0 2 r 2q0 0 0 0 s+ p+ 2q + r
, s = p 2q r
,
x 2w = p r
z = p+ q
y + 2w = 2q + r
. w = t ,
x = 2t p r
y = 2t 2q + r
z = p q
w = t
(t ). a = 1 , () =
1 1 1 1 p0 0 0 0 p+ q
0 0 0 0 r + 2p
0 0 0 0 s 2p
,
56
q = p, r = 2p, s = 2p ,
x+ y z w = p . y = t, z = u, w = v ,
x = t+ u+ v + p
y = t
z = u
w = v
(t, u, v )
.
(21)
1 1 1 1 p1 2 2 2 q2 3 1 1 r
a+ b a 2b a 2b a+ 2b s
(1,1) 1 1 1 1 1 p0 1 3 3 q p0 1 3 3 r 2p0 b 3b 3b s (a+ b)p
(2,2) 2 1 0 4 4 2p q0 1 3 3 q p0 0 0 0 r p q0 0 0 0 s ap bq
, r = p+ q, s = ap+ bq
,
x+ 4z 4w = 2p qy 3z + 3w = q p . z = t, w = u ,
x = 4t+ 4u+ 2p q
y = 3t+ 3u+ p q
z = t
w = u
(t, u ).
3.
1 k + 1 2k k + 1k 1 3k 1 k k 12k 1 7k 1 3k 3k 1
(1,1) 1
1 k + 1 2k k + 10 3k k2 3k 2k2 k20 6k
