(2, 1, 1, 1) , (1, (1, 1, 1, 1) , (3, (3, 1, 1, 2) , (0, (0, 1, 2, 1) , (2, (2, −1, 1, −1)} {(2,
R4
R4 A A
A=
2 1 1 1 3 1 0 1 2 −1
1 1 1 1 1 2 2 1 1 −1
1 1 1 0 −1 −1 0 −2 −2 0 1 2 0 −3 −1
1 −1 −1 1 −1
1 0 0 0 0
1 1 0 0 0
1 1 0 1 2
1 1 1 0 2
1 0 0 0 0
1 1 0 0 0
1 1 1 0 0
1 1 0 1 2
E 1 ↔ E 2 ∼
−E 2 ∼
E 3 ↔ E 4 ∼
1 2 3 0 2
1 1 1 1 −1
1 1 1 1 1 2 2 1 1 −1
1 1 1 1 0 1 1 1 0 −2 −2 −1 0 1 2 1 0 −3 −1 −1
E 5 − 2E 4 ∼
1 0 0 0 0
1 1 0 0 0
1 1 1 0 2
1 1 0 1 2
1 0 0 0 0
1 1 0 0 0
1 1 1 0 0
U
1 1 0 1 0
E 2 − 2E 1 E 3 − 3E 1 ∼ E 4 − 2E 1
E 3 + 2E 2E 2 E 4 − E 2 ∼ E 5 + 3E 3E 2
E 5 − 2E 3 ∼
= U
R4 R
4
U
(2, 1, 1, 1) , (1, (1, 1, 1, 1) , (3, (3, 1, 1, 2) , (0, (0, 1, 2, 1) , (2, (2, −1, 1, −1) = (2, K
n
K
n
A
= (1, (1, 1, 1, 1) , (0, (0, 1, 1, 1) , (0, (0, 0, 1, 0) , (0, (0, 0, 0, 1) U
R4 R
4
R4 R4 (2, 1, 1, 1) , (1, (1, 1, 1, 1) , (3, (3, 1, 1, 2) , (0, (0, 1, 2, 1) , (2, (2, −1, 1, −1)} {(2,
R4 R4
R4
A
R
U
4
U
R
4
R4
A
A
A
U
A
A Fila3 ↔ Fila4 A (2, 1, 1, 1) , (1, (1, 1, 1, 1) , (3, (3, 1, 1, 2) , (0, (0, 1, 2, 1)} {(2,
Fila1 ↔ Fila2
R4 R4
(1, 1, 1, 1, 1) , (2, (2, 1, 2, 1, 2) , (1, (1, 2, 1, 2, 1) , (0, (0, 0, 0, 0, 1)} {(1, K 5 K = Z7
Z57
Z57
Z57 A A
A=
1 0 0 0
1 2 1 0
1 1 2 0 1 1 6 0
1 2 1 0 1 0 0 0
1 1 2 0 1 1 6 0
1 2 1 1 1 0 0 1
E 2 + 5E 5E 1 ∼ E 3 − E 1
E 3 + E 2 ∼ 1 0 0 0
1 1 0 0
1 0 0 0
1 1 0 0
1 0 0 0
1 6 1 0
1 0 0 0
1 6 1 0
1 0 0 1
1 0 0 0
1 1 0 0
1 0 0 0
1 1 0 0
1 0 0 1
1 0 1 0
= U
E 2 ↔ E 3 ∼
E 3 ↔ E 4 ∼
U A {(1, 1, 1, 1, 1) , (2, 1, 2, 1, 2) , (1, 2, 1, 2, 1) , (0, 0, 0, 0, 1)}
Z57 Z
5 7
Z57 {(1, 1, 1, 1, 1) , (2, 1, 2, 1, 2) , (1, 2, 1, 2, 1) , (0, 0, 0, 0, 1)}
Z57 {(1, 1, 1, 1, 1) , (0, 1, 0, 1, 0) , (0, 0, 0, 0, 1)}
(1, 1, 1, 1, 1) , (2, 1, 2, 1, 2) , (1, 2, 1, 2, 1) , (0, 0, 0, 0, 1) = (1, 1, 1, 1, 1) , (0, 1, 0, 1, 0) , (0, 0, 0, 0, 1)
{(1, 1, 1, 1, 1) , (0, 1, 0, 1, 0) , (0, 0, 0, 0, 1)} S v S = {(1, 2, 1, 1) , (1, 1, 0, 2)} v = (1, 1, 3, 1)
Z45 A
1 2 1 1 1 1 0 2
Z45 Z45
E 2 + 4E 1 ∼
1 2 1 1 0 4 4 1
B = {(1, 2, 1, 1) , (1, 1, 0, 2) , (0, 0, 1, 0) , (0, 0, 0, 1)} v B
(1, 1, 3, 1) = x1 · (1, 2, 1, 1) + x2 · (1, 1, 0, 2) + x3 · (0, 0, 1, 0) + x4 · (0, 0, 0, 1)
x1 2x1 x1 x1
+ +
x2 x2
+ 2x2
+ x3 +
+ x4
= = = =
1 1 3 1
Z45
B
1 1 0 0
1
2 1 0 0
1
1 0 1 0
3
1 2 0 1
1
E 3 − E 2 ∼ E 4 + E 2
E 1 + E 2 ∼
E 2 + 3E 1 E 3 + 4E 1 ∼ E 4 + 4E 1
1 1 0 0
1
0 4 0 0
4
0 4 1 0
2
0 1 0 1
0
1 1 0 0
1
0 4 0 0
4
0 0 1 0
3
0 0 0 1
4
1 0 0 0
0
0 4 0 0
4
0 0 1 0
3
0 0 0 1
4
x1 x2 x3 x4
=0 =1 =3 =4
4E 2 ∼
1 0 0 0
0
0 1 0 0
1
0 0 1 0
3
0 0 0 1
4
Mm×n (K ) K
A ∈ M m×n (K )
A=
¯b ∈ M n×1 (K )
a11 a21
a12 a22
··· ···
a1n a2n
am1
am2
··· ···
amn
v = (1, 1, 3, 1)
v = (0, 1, 3, 4)B
B
; ¯b =
b1 b2 bn
A · ¯b ¯ A · b ∈ M m×1 (K )
A · ¯b =
a11 a21
am1
=
a12 a22
··· ···
a1n a2n
am2
··· ···
amn
b1 b2
·
bn
a11 b1 + a12 b12 + . . . + a1n bn a21 b1 + a22 b2 + . . . + a2n bn
am1 b1 + am2 b2 + . . . + amn bn
=
a11 a21
b1 +
am1
a12 a22 am2
b2 + . . . +
a1n a2n amn
=a ¯1 b1 + ¯a2 b2 + . . . + ¯an bn
a ¯j =
a1j a2j amj
A
=
bn
; 1≤j≤n
m×n
A n
A · ¯b
¯b
n×1 ¯b
a11 x1 + a12 x2 + . . . + a1n xn = b1 a21 x1 + a22 x2 + . . . + a2n xn = b2 am1 x1 + am2 x2 + . . . + amn xn = bm
A=
a11 a21
a12 a22
··· ···
a1n a2n
am1
am2
··· ···
amn
; X =
x1 x2 xn
; ¯b =
b1 b2 bm
A · X = ¯b A · X A
A
a ¯1 , a ¯2 , . . . , a ¯n
a ¯1 x1 + ¯a2 x2 + . . . + ¯an xn = ¯b u = (b1 , b2 , . . . , bm ) ∈ K m v1 , v2 , . . . , vh ∈ K m vj
v1
v2
··· ··· ···
(1, 2, 1, 1) , (1, 1, 1, 2)
vh
A
x1 x2 xh
=
b1 b2 bm
u ∈ / v1 , v2 , . . . , vh
R4 v = (1, 0, 1, 0) R4
(x,y,z,t)
(x,y,z,t) ∈ (1, 2, 1, 1) , (1, 1, 1, 2) ⇐⇒ x1 , x2 ∈ R (x,y,z,t) = x1 · (1, 2, 1, 1) + x2 · (1, 1, 1, 2) ⇐⇒ x1 2x1 x1 x1
+ x2 + x2 + x2 + 2x2
= = = =
x y z t
⇐⇒
1
1
x
2
1
y
1
1
z
1
2
t
1 1
x
2 1
y
1 1
z
1 2
t
1
1
E 2 − 2E 1 E 3 − E 1 ∼ E 4 − E 1
x
0 −1
y − 2x
0
1
t−x
0
0
z−x
1
1
x
0 −1
y − 2x
0
0
z−x
0
1
t−x
E 3 + E 2 ∼
1
1
0 −1
y − 2x
0
0
−3x + y + t
0
0
z−x
−3x + y + t = 0 (x,y,z,t) ∈ R / x−z =0 (1, 2, 1, 1) , (1, 1, 1, 2) =
−3x + y + t = 0 (x,y,z,t) ∈ R / x−z = 0 4
v = (1, 0, 1, 0) ∈ R4 =0 −3 + 0 + 0 1−1 =0
/ (1, 2, 1, 1) , (1, 1, 1, 2) ⇒v∈
x−y+z −t = 0 (x,y,z,t) / 2x + z + t = 0
1 2
−1 1 −1 0
1
1
0 0
v ∈ (1, 2, 1, 1) , (1, 1, 1, 2)
x − y 2x
z − x = 0 (⇔ (1, 2, 1, 1) , (1, 1, 1, 2)
x − z = 0) 4
E 3 ↔ E 4 ∼
x
−3x + y + t = 0
+ z + z
R4
− t = 0 + t = 0
E 2 − 2E 1 ∼
1 0
−1
1 −1
2 −1
3
0 0
x −
y 2y
+ z − z
− t = 0 + 3t = 0
x −
↔
= −z + t = z − 3t
y 2y
x y z t
y
x = − 12 z − 12 t y = 12 z − 32 t z ∈ R t ∈ R
1 1 1 3 (x,y,z,t) ∈ R / (x,y,z,t) = − z − t, z − t, z, t ; z, t ∈ R 2 2 2 2 4
1 1 1 3 − z − t, z − t, z, t / z, t ∈ R 2 2 2 2
=
1 1 1 3 z − , , 1, 0 + t − , − , 0, 1 / z, t ∈ R 2 2 2 2
=
=
(x,y,z,t) /
1 1 1 3 − , , 1, 0 , − , − , 0, 1 2 2 2 2
x−y+z −t =0 2x + z + t = 0
− 12 , 12 , 1, 0 , − 12 , − 32 , 0, 1
=
− 12 , 12 , 1, 0 , − 12 , − 32 , 0, 1
(1, 2, 1, 1), (1, 1, 1, 2), (3, 2, 3, 2)
A A A=
1 2 1 1 1 1 1 2 3 2 3 2
A
E 2 + 4E 1 ∼ E 3 + 2E 1
1 2 1 1 0 4 0 1 0 0 0 0
1 2 1 1 0 4 0 1 0 1 0 4
E 3 + E 2 ∼
= U U
4 5
Z
(1, 2, 1, 1), (1, 1, 1, 2), (3, 2, 3, 2) = (1, 2, 1, 1), (0, 4, 0, 1)
Z45
{(1, 2, 1, 1), (0, 4, 0, 1)}
x−y+z−t+u= 0 (x,y,z,t,u)/ 2x + z + t − u = 0 x−y+z−t+u= 0 2x + z + t − u = 0 (x,y,z,t,u)/ x+t+u =0
W 1 =
W 2 =
Z55
x−y+z−t+u = 0 (x,y,z,t,u)/ 2x + z + t − u = 0
x−y+z−t+u = 0 2x + z + t − u = 0 (x,y,z,t,u)/ x+t+u= 0 W 1
W 2 ⊆ W 1
W 2
W 1 ∩ W 2 = W 2 W 1 ∪ W 2 = W 1
W 1 + W 2 = W 1 ∪ W 2
W 1 + W 2 = W 1
W 1 ∩ W 2 = W 2 W 1 + W 2 = W 1
(1, 2, 1, 1) , (1, 1, 1, 2) , (3, 1, 2, 1) , (2, 0, 1, −1)
R4
(2, 3, −1, 2)
W 1 = (1, 2, 1, 1) , (1, 1, 1, 2) , (3, 1, 2, 1) , (2, 0, 1, −1) W 2 = (2, 3, −1, 2)
A A
1 1 3 2
2 1 1 0
1 1 1 2 2 1 1 −1
E 2 − E 1 E 3 − 3E 1 ∼ E 4 − 2E 1
1 2 1 1 0 −1 0 1 0 −5 −1 −2 0 −4 −1 −3
−E 2 ∼
1 0 0 0
2 1 5 − −4
1 1 0 −1 1 − −2 −1 −3
E 3 + 5E 2 ∼ E 4 + 4E 2 1 0 0 0
1 0 0 0
2 1 1 1 0 −1 0 −1 −7 0 −1 −7
2 1 1 1 0 −1 0 −1 −7 0 0 0
E 4 − E 3 ∼
= U
W 1 = (1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7)
{(1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7)} (2, 3, −1, 2) {(2, 3, −1, 2)}
W 1 W 2 .
W 2 =
W 1 + W 2 W 1 + W 2 = (1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) , (2, 3, −1, 2)
A=
1 0 0 2
2 1 1 1 0 −1 0 −1 −7 3 −1 2 1 0 0 0
2 1 1 1 0 −1 0 −1 −7 0 −3 −1
E 4 − 2E 1 ∼
E 4 − 3E 3 ∼
U
1 2 1 1 0 1 0 −1 0 0 −1 −7 0 −1 −3 0 1 0 0 0
2 1 1 1 0 −1 0 −1 −7 0 0 20
E 4 + E 2 ∼
= U
A
B = {(1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) , (2, 3, −1, 2)} W 1 + W 2 (W 1 + W 2 ) = 4 W 1 + W 2
B
4
R
W 1 + W 2
⇒ W 1 + W 2 = R4
(W 1 + W 2 ) =
W 1 +
W 2 −
(W 1 ∩ W 2 )
(W 1 ∩ W 2 ) =
W 1 +
W 2 −
(W 1 + W 2 )
=3+1−4= 0 W 1 ⊕ W 2 = R4
W 1 ∩ W 2 = {(0, 0, 0, 0)} W 1 W 2 W 1 + W 2 W 1 ∩ W 2 (W 1 + W 2 ) =
W 1 +
W 2 −
(W 1 ∩ W 2 )
W 1 + W 2
•
W 1 W 2
W 1 + W 2 W 1 ∩ W 2
•
W 1 ∩W 2 W 1 W 1 = (1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7)
{(1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7)} (x,y,z,t) R4
W 1
(x,y,z,t) ∈ (1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) ⇐⇒ x1 , x2 , x3 ∈ R (x,y,z,t) = x1 · (1, 2, 1, 1) + x2 · (0, 1, 0, −1) + x3 · (0, 0, −1, −7) ⇐⇒ x1 2x1 x1 x1
⇐⇒
+
x2
− x2
− x3 − 7x3
= x = y = z = t
1
0
0
x
2
1
0
y
1
0 −1
z
1 −1 −7
t
1
0
0
x
2
1
0
y
1
0 −1
z
1 −1 −7
t
E 2 − 2E 1 E 3 − E 1 ∼ E 4 − E 1
1 0
0
x
0 1
0
y − 2x
0 0 −1
z−x
0 0 −7
−3x + y + t
1
0
0
x
0
1
0
y − 2x
0
0 −1
z−x
0 −1 −7
t−x
E 4 − 7E 3 ∼
E 4 + E 2 ∼
1 0
0
x
0 1
0
y − 2x
0 0 −1
z−x
0 0
0
4x + y − 7z + t
4x + y − 7z + t = 0 W 1 = (1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) = (x,y,z,t) ∈ R4 /4x + y − 7z + t = 0
W 2 = (2, 3, −1, 2)
R4
(x,y,z,t)
(x,y,z,t) ∈ (2, 3, −1, 2) ⇐⇒ x1 ∈ R (x,y,z,t) = x1 · (2, 3, −1, 2) ⇐⇒ 2x1 3x1 −x1 2x1
2
x
3
y
−1
z
2
t
E 1 ↔ E 3 ∼
= = = =
x y z t
−1
z
3
y
2
x
2
t
⇐⇒
2
x
3
y
−1
z
2
t
E 2 + 3E 1 E 3 + 2E 1 ∼ E 4 + 2E 1
−1
z
0
y + 3z
0
x + 2z
0
t + 2z
y + 3z = 0 x + 2z = 0 2z + t = 0 W 2 = (2, 3, −1, 2) =
y + 3z = 0 4 (x,y,z,t) ∈ R / x + 2z = 0 2z + t = 0
W 1 = (1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) = (x,y,z,t) ∈ R4 /4x + y − 7z + t = 0
W 2 = (2, 3, −1, 2) =
W 1 ∩ W 2 =
y + 3z = 0 4 (x,y,z,t) ∈ R / x + 2z = 0 2z + t = 0
4 1
−7 1
0
0 1
3
0
0
1 0
2
0
0
0 0
2
1
0
1
0
2 0
0
0
1
3 0
0
0
1 −15 1
0
0
0
0
2 1
4x + y − 7z + t = 0 y + 3z = 0 (x,y,z,t) ∈ R4 / x + 2z = 0 2z + t = 0
4x + y − 7z + t = 0 y + 3z = 0 x + 2z = 0 2z + t = 0
E 1 ↔ E 3 ∼
E 3 − E 2 ∼
1 0
2 0
0
3 0
0
4 1 −7 1
0
0 0
0
0 1
2 1
1
0
2
0
0
0
1
3
0
0
0
0 −18 1
0
0
0
0
2
1
E 3 − 4E 1 ∼
E 3 ↔ E 4 ∼
1 0
2 0
0
0 1
3 0
0
0 0
2 1
0
0 0
−18 1
0
x y
E 4 + 9E 3 ∼
+2z +3z 2z
= = +t = 10t =
1 0 2
0
0
0 1 3
0
0
0 0 2
1
0
0 0 0
10
0
0 0 0 0
(x,y,z,t) = (0, 0, 0, 0)
W 1 ∩ W 2 = {(0, 0, 0, 0)}
x−y+z −t = 0 (x,y,z,t) / 2x + z + t = 0
W 1 =
R4
(2, 3, −1, 2)
x−y+z−t = 0 (x,y,z,t) / 2x + z + t = 0
W 2 = (2, 3, −1, 2) W 1 (W 1 ) = = W 2 (W 2 ) = 1 (2, 3, −1, 2) W 1 W 1 2 − 3 + (−1) − 2 = −4 = 0 ⇒ (2, 3, −1, 2) ∈ / W 1 W 1 W 2
2 W 2 W 1
x−y+z −t+u = 0 W 1 = (x,y,z,t,u)/ 2x + z + t − u = 0 x−y +z −t+u =0 2x + z + t − u = 0 W 2 = (x,y,z,t,u)/ x+t+u= 0
1 −1 1
1
0
2
0 1
1 −1
0
1
0 0
1
0
1
Z55
W 1 W 2 W 2
W 2 ⊆ W 1
−1
E 2 + 3E 1 ∼ E 3 − E 1
1 −1 1 −1 1
0
0
2 4
3 2
0
0
1 4
2 0
0
E 2 ↔ E 3 ∼
E 3 + 3E 2 ∼
1
−1 1 −1 1
0
0
1
4
2 0
0
0
2
4
3 2
0
1 −1 1 −1 1
0
0
1 4
2 0
0
0
0 1
4 2
0
W 1 = (1, 2, 1, 1), (1, 1, 1, 2), (3, 1, 2, 1), (2, 0, 1, −1) W 2 = (1, 1, 1, 1), (2, 3, −1, 2) R4 W 1
B = {(1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7)} E = {(1, 1, 1, 1), (2, 3, −1, 2)}
W 2
W 1 = W 2 (W 1 + W 2 ) =
W 1 +
W 2 −
(W 1 ∩ W 2 )
W 1 + W 2 = R4
{(1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) , (2, 3, −1, 2)}
R4 4=3+2−
(W 1 ∩ W 2 ) ⇒
(W 1 ∩ W 2 ) = 1 ⇒ W 2 W 1
W 2 ⊆ W 1 ⇒ W 1 ∩ W 2 = W 2
(W 1 ∩ W 2 ) =
W 2 = 2 W 1 W 2 W 1 ⊆ W 2 W 1
W 2 W 1
W 2
W 2
•
W 1 W 2
•
W 2 W 1 ⊆ W 2 W 1
W 2 W 2
W 1 W 2
W 2 ⊆ W 1 W 2
W 1
W 1 = W 2 ⇔
W 1 ⊆ W 2 y W 2 ⊆ W 1
x−y+z =0 2x − z = 0
R3 x+y+z =0
1 −1 2
x −
1
0
0 −1
0
y 2y
+ z − 3z
E 2 − 2E 1 ∼
= 0 = 0
←→
1 −1 0
x −
z
1 3 z, z, z /z ∈ R 2 2
0
2 −3
0
y 2y
= −z = 3z
x
y
y
x = y = z ∈ R
1
=
z
1
z z 2 2 3
1 3 , , 1 /z ∈ R 2 2
=
x + y + z = 0 ↔ x = −y − z
1 3 , ,1 2 2
= (1, 3, 2)
x
y
z
x = −y − z y ∈ R z ∈ R
.
{(−y − z , y , z) /y,z ∈ R} = {y · (−1, 1, 0) + z · (−1, 0, 1) /y, z ∈ R} = (−1, 1, 0) , (−1, 0, 1) .
B = {(−1, 1, 0) , (−1, 0, 1) , (1, 3, 2)}
Z45
W =
{(x,y,z,t) /x − y + z − t = 0}
x − y + z − t = 0 ↔ x = y − z + t (= y + 4z+) x
y z
t
x = y + 4z + t y ∈ R z ∈ R t ∈ R
.
W = (y + 4z + t,y,z,t) /y,z,t ∈ Z5
= {y · (1, 1, 0, 0) + z · (4, 0, 1, 0) + t · (1, 0, 0, 1) /y,z,t ∈ Z5 } = (1, 1, 0, 0) , (4, 0, 1, 0) , (1, 0, 0, 1) . (1, 1, 0, 0) ∈ W
(0, 1, 0, 0) ∈ / W (0, 0, 1, 0) ∈ / W (0, 0, 0, 1) ∈ / W
B = {(1, 1, 0, 0) , (0, 1, 0, 0) , (0, 0, 1, 0) , (0, 0, 0, 1)} 5 5
Z
Z47
x−y+z −t =0 2x − z = 0
1
−1
1
−1
0
2
0
−1
0
0
x−y+z = 0 2x − z + t = 0
E 2 + 5E 1 ∼
1 −1 1 −1
0
0
0
2 4
2
x −
y 2y
+ z + 4z x
− t = 0 + 2t = 0
↔
x −
y 2y
y
z
= −z + t = −4z − 2t
t
y 2y = −4z − 2t = 3z + 5t → y = 5z + 6t ↓
Z7 y
x
x − 5z − 6t = −z + t → x = 4z
x = 4z y ∈ 5z + 6t z ∈ Z7 t ∈ Z7
.
{(4z, 5z + 6t,z,t) /z,t ∈ Z7 } = {z · (4, 5, 1, 0) + t · (0, 6, 0, 1) /z,t ∈ Z7 } = (4, 5, 1, 0) , (0, 6, 0, 1) .
1 −1 2
x −
1 0
0
0 −1 1
0
y 2y x
+ +
z 4z
y
E 2 + 5E 1 ∼
= 0 + t = 0
↔
1 −1 1 0
0
0
0
2 4 1
x − z
y 2y
= −z = −4z − t
t
Z7
x − 5z − 3t = −z → x = 4z + 3t
y
2y = −4z − t = 3z + 6t → y = 5z + 3t ↓ y
x
x = 4z + 3t y ∈ 5z + 3t z ∈ Z7 t ∈ Z7
.
{(4z + 3t, 5z + 3t,z,t) /z,t ∈ Z7 } = {z · (4, 5, 1, 0) + t · (3, 3, 0, 1) /z,t ∈ Z7 } = (4, 5, 1, 0) , (3, 3, 0, 1) .
(4, 5, 1, 0) , (0, 6, 0, 1) (4, 5, 1, 0) , (3, 3, 0, 1)
Z47 B = {(4, 5, 1, 0) , (0, 6, 0, 1) , (3, 3, 0, 1) , (0, 0, 0, 1)} ,
B
4 0 3 0
5 6 3 0
1 0 0 0
0 1 1 1
E 3 + E 1 ∼
4 0 0 0
5 6 1 0
1 0 1 0
0 1 1 1
E 3 + E 2 ∼
B
4 0 0 0
5 6 0 0
1 0 1 0
R3
x−y+z =0 2x − z = 0 (2, 1, 2)
(1, 1, 1)
x−y+z = 0 2x − z = 0
1 −1 2
1
0
0 −1
0
E 2 − 2E 1 ∼
1 −1 0
1
0
2 −3
0
0 1 2 1
x −
y 2y
+ z − 3z
x
y
= =
0 0
↔
x −
y 2y
= −z = 3z
,
z
y x = y = z ∈ R
z 2 3 z 2
,
1 3 z, z, z /z, ∈ R 2 2 =
1
=
1 3 , ,1 2 2
z·
1 3 , , 1 /z ∈ R 2 2
= (1, 3, 2) .
R3 (1, 1, 1) (2, 1, 2)
B
B = {(1, 0, 0) , (1, 3, 2) , (a,b,c)} (1, 1, 1)
(2, 1, 2) (1, 1, 1) = (2, 1, 2)B ⇐⇒ (1, 1, 1) = 2 · (1, 0, 0) + 1 · (1, 3, 2) + 2 · (a,b,c) 1 = 3 + 2a 1 = 3 + 2b 1 = 2 + 2c
B=
a = −1 ↔ b = −1 c = − 12
1 → (a,b,c) = −1, −1, − 2
1 (1, 0, 0) , (1, 3, 2) , −1, −1, − 2
.
.
K n
1 − x + x2 , x + x2 , 2 − x + x2
C = 1, x, x2 Q2 [x]
Q2 [x] . Q2 [x]
a + bx + cx2
a + bx + cx2 = a · 1 + b · x + c · x2 ⇒ a + bx + cx2 = (a, b, c)C
Q3
Q2 [x]
Q3 ↔ ↔ (a, b, c)
Q2 [x] 2
a + bx + cx
Q2 [x] 1 − x + x2 x + x2 2 − x + x2
Q3 ↔ ↔ (1, −1, 1) ↔ (0, 1, 1) ↔ (2, −1, 1)
{(1, −1, 1) , (0, 1, 1) , (2, −1, 1)}
Q3 A U A=
1 −1 1 0 1 1 2 −1 1
U
E 3 − 2E 1 ∼
1 −1 1 0 1 1 0 0 −2
1 −1 1 0 1 1 0 1 −1
E 3 − E 2 ∼
= U A
(1, −1, 1) , (0, 1, 1) , (2, −1, 1) ∈ Q3
Q3 = 3
1 − x + x2 , x + x2 , 2 − x + x2
Q2 [x]
•
1 1 2 1
,
1 0 2 1
C =
M 2×2 (Z5 )
a b c d
=a·
1 0 0 0
,
0 0 2 1
1 0 , 0 0 a b c d
+b·
,
0 1 0 0
0 1 0 0
,
0 1 0 0
M 2×2 (Z5 )
0 0 1 0
,
0 0 0 1
M 2×2 (Z5 )
+c·
0 0 1 0
+d·
0 0 0 1
⇒
a b c d
= (a,b,c,d)C
Z45
M 2×2 (Z5 )
M 2×2 (Z5 ) ↔ Z45 a b ↔ (a,b,c,d) c d
M 2×2 (Z5 ) 1 1 2 1 1 0 2 1 0 0 2 1 0 1 0 0
Z45
↔
↔ (1, 1, 2, 1) ↔ (1, 0, 2, 1) ↔ (0, 0, 2, 1) ↔ (0, 1, 0, 0)
{(1, 1, 2, 1) , (1, 0, 2, 1) , (0, 0, 2, 1) , (0, 1, 0, 0)}
Z45 Z45 A U
A=
1 1 0 0
1 0 0 1
2 2 2 0
1 1 1 0
E 2 + 4E 1 ∼ 1 0 0 0
1 4 0 0
2 0 2 0
U A
1 0 0 0
1 0 1 0
1 4 0 1
2 0 2 0
1 0 1 0
E 4 + 4E 2 ∼
= U
Z45
A 4 5
Z
U 4 5
Z
Z
4 5
B = {(1, 1, 2, 1) , (0, 4, 0, 0) , (0, 0, 2, 1) , (0, 0, 0, 1)} (0, 0, 0, 1)
A A
Z45 B˜ = {(1, 1, 2, 1) , (1, 0, 2, 1) , (0, 0, 2, 1)(0, 0, 0, 1)} M 2×2 (Z5 )
• W 1 =
• W 2 =
B=
B˜ =
1 1 2 1
1 2
1 1
0 4 0 0
,
1 0 2 1
,
,
,
0 0 2 1
0 0 2 1
1 1 0 1 1 0
,
1 1 1 1 1 0
P (x) = a + bx + cx2 /
,
1 2 1
0 1
0 0 0 1
,
2 1 0
P (2) = 0 P (1) = 0
0 0
,
.
M 3×2 (Z5 )
(Z5 )2 [X ]
(Z5 )2 [X ]
Z5
M 2×3 (Z5 ) 1 1 0 1 1 0 1 1 1 1 1 0 1 2 2 1 1 0
↔
Z65
↔ (1, 1, 0, 1, 1, 0)
↔ (1, 1, 1, 1, 1, 0)
↔ (1, 2, 2, 1, 1, 0)
(1, 1, 0, 1, 1, 0) , (1, 1, 1, 1, 1, 0) , (1, 2, 2, 1, 1, 0) ,
Z65
Z65 a · (1, 1, 0, 1, 1, 0) + b · (1, 1, 1, 1, 1, 0) + c · (1, 2, 2, 1, 1, 0) ;
a,b,c ∈ Z5
a,b,c ∈ Z5
(a + b + c, a + b + 2c, b + 2c, a + b + c, a + b + c, 0) ;
W 1 =
a+b+c b + 2c a+b+c
a + b + 2c a+b+c 0
/a, b, c ∈ Z5
ecuaciones paramtricas
W 1 3×2
1 0 1
a·
1 1 0
1 1 1 1 1 0
+b·
a+b+c b + 2c a+b+c
1 2 2 1 1 0
+c·
a + b + 2c a+b+c 0
a+b+c b + 2c a+b+c
a + b + 2c a+b+c 0
=
a+b+c b + 2c a+b+c
a + b + 2c a+b+c 0
1 1 0 1 1 0
= a·
+b·
1 1 1 1 1 0
+c·
1 2 2 1 1 0
Z65 W = (1, 1, 0, 1, 1, 0) , (1, 1, 1, 1, 1, 0) , (1, 2, 2, 1, 1, 0) W 1
Z65
(x,y,z,t,r,s)
(x,y,z,t,r,s) ∈ (1, 1, 0, 1, 1, 0) , (1, 1, 1, 1, 1, 0) , (1, 2, 2, 1, 1, 0) ⇐⇒ x1 , x2 , x3 ∈ Z5 (x,y,z,t,r,s) = x1 · (1, 1, 0, 1, 1, 0) + x2 · (1, 1, 1, 1, 1, 0) + x3 · (1, 2, 2, 1, 1, 0) ⇐⇒
1 1 0 1 1 0
1 1 1 1 1 0
1 2 2 1 1 0
·
x1 x2 x3
=
x y z t r s
x1 x1 x1 x1
+ +
x2 x2 x2 + 2x + x2
+ x3 + 2x3 + 2x3 + x3 + x3 0
= x = y = z = t = r = s
⇐⇒
1 1 1
x
1 1 2
y
0 1 2
z
1 1 1
t
1 1 1
r
0 0 0
s
1
1 1
x
1
1 2
y
0
1 2
z
1
1 1
t
1
1 1
r
0
0 0
s
E 2 − 2E E 4 − E 1 ∼ E 5 − E 1
1 1 1
x
0 0 1
y−x
0 1 2
z
0 0 0
t−x
0 0 0
r−x
0 0 0
s
1
1 1
x
0
1 2
z
0
0 1
y−x
0
0 0
t−x
0
0 0
r−x
0
0 0
s
E 2 ↔ E 3 ∼
t r
W =
W 1 =
• W 2 =
x y z t r s
− x = 0 − x = 0 s = 0
4x + t = 0 ↔ 4x + r = 0 s = 0 W
4x + t (x,y,z,t,r,s) ∈ Z / 4x + r s
= 0 = 0 = 0
6 5
W 1
4x + t = 0 M ( Z )/ ∈ 3×2 5 4x + r = 0 s = 0
P (2) = 0 P (x) = a + bx + cx2 / P (1) = 0
ecuaciones implcitas
(Z5 )2 [X ]
P (2) = 0 ⇐⇒ a + 2b + 4c = 0 P (x) = b + 2cx → P (1) = 0 ⇐⇒ b + 2c = 0
W 2 =
a + 2b + 4c = 0 P (x) = a + bx + cx / b + 2c = 0 2
ecuaciones implcitas W 2
Z5 a + 2b + 4c = 0 b + 2c = 0
a
b
↔
a + 2b = b =
W 2 W 2 = P (x) = 3cx + cx2 /c ∈ Z5
= c · 3x + x2 /c ∈ Z5 = 3x + x2
→ a + 6c = c → a + c = c →a=0
c a = 0 b = 3c c ∈ Z5
c 3c
ecuaciones paramtricas sistema generador
