1 0 0
1
1 0 01
0 1 0 2 5 6
0 0 0 3 6
0 0 0 4 6
1 0
3 1 2 0 0 0 3
A
1 A= 4
3 5 7 8
1 6 9
B
5 B= 0
−1 0 0 0 1 2 3 1 4 5
1 4
3 1 5 6 7 8 9
5 4 A= 3 2
B=
6
5 · 0
−1 0 0 0 1 2 3 1 4 5
5 4 3 2 1
8 = 38
0 7 2 29 62 2 44
1
5 4 32 · 6 1
5 4 3 2 1
=
1111 40 61
A
B
2
A×B = B×A A×B
1 2 3 1 2 A= 2 4 6 B= 3 5 3 6 9 7 9 1 2 3 5 7 7
A·B =
B·A= A·B = B·A
2
·
14
3 5 1 2 13
26 8
1
·
2 1
=
14
2
4
2 4
=
4
B×A
1 A= 3
0 1 0 0
0 2 4
1 0 0 3 1 2 0 0 4 1 −1 B = −2 3 0
1 −2 0 1 0
∼
2 2 1 4
1 0 0 5 1
0 0 1 2 0 1
L → L − 3L 2
2
1
L3 → 14 L3
1 ∼ 0 0
0 0 1 0 0 1
L → L − 2L 2
2
3
2
1 0 8 1616 L → L + L L → L + 2L2L ∼ 0 1 6 1111 0 0 −2 −9 L →L −L 1 0 0 −20 L → L − 8L 16 11 ∼ 0 1 0 −16 L → L − 6L
−1 2 5 3 2 1 1 4 2
0 8 1 6 0 0 1 99//2
2
2
1
1
2
3
3
2
1
1 2
L3 → − L3
0 0
1
9/2
1
1
3
2
2
3
x + x2x++ 1
1
1
3x2 + 2x3 + 3x4 − 7x5 6x2 +x3 +5x5 −2x4 +5x 3x2 − x3 + 2x5
1 3 2 3 −7 14 1 3 2 3 −7 14 1 3 2 6 1 −2 5 −2 ∼ 0 0 −3 −8 19 −30 ∼ 0 0 1 3 −1 0 2 −1 0 0 −3 −3 9 −15 0 0 1 3 2 3 −7 14 1 3 0 3 −5 10 1 19//3 10 ∼ 0 0 1 0 −1 2 ∼ 0 ∼ 0 0 1 88//3 −19 0 0 0 1 0 0 0 1 −2 3 0 −2 3 x + 3x3x + x = 1 xx −− x2x ==23 ⇒ x = 3 + 2x2x 1 3 4
x5 = α
2
5
5
5
4
5
x2 = β
x4 = 2 + α x3 = 2 + α x1 = 1 − 3β − α
(1 − 3β − α,β, 2 + α, 3 + 2α, 2 α, α)
= 14 = −2 = −1
2 3 −7 14 19 −30 −3 −8 0 5 −10 15 3 0 0 1 1 0 1 0 −1 2 0 0 1 −2 3
1 A= 2
−2 3 −1 3 −1 2 3 1 2 3
1 −2 3 −1 1 −2 0 3 −4 5 ∼ 0 1 01 07 −17/3 6 7/3 0 17 5 /3 ∼ 0 ∼ 0 1 −4/3 1 0 A= 1
2 1 3 4 1
∼
1 2 1 0 1 2 1 3 0 4 1 3 1 0 −3 0 1 2 0 0 −3 0
∼
1 0 0
0 1 0 0 0
0
13 13
0 −2 0 1 1 0 0 6
0 0 1 0 0 0 1 −17 17//7
7/3 −17 17//3
0 0
1 0 2 1 0 1 3 −1
38/9 11//7 −11
3 −1 −4/3 5/3 6 −7
1 2 1 0 1 0 0 0 0 1 2 1 0 1 0 0 ∼ 0 1 −1 1 −1 0 1 0 0 −7 −1 −1 −4 0 0 1 1 0 0 −2 2 −1 −1 0 1 −2 0 0 −2 0 1 2 1 0 1 0 0 1 0 1 0 0 ∼ 0 0 −3 0 −1 −1 1 0 0 −1 −1 1 0 6 −4 7 0 1 0 0 13 13 6 −4 7 0 1 1 0 0 0 −1414//18 −1/9 4/9 2 −1 −1 0 0 1 0 0 13/18 −7/3 −1/18 2/3 0 −2/3 1/3 ∼ 0 0 1 0 1/3 1/3 −1/3 1/3 1/3 −1/3 0 −25 25/1/3 0 8/03 0 1313//3−71/9 −1/9 04/90 01/31 −2525//18 8/18 13/18 0 1 0 0 13/18 −7/3 −1/18 −1/6 0 0 1 0 1/3 1/3 −1/3 0
0 1 1 −1
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 1 −25 25//18
4/9
13 13//18
1/6
1/3 −1/6 0 1/6
1 2 1
1 1
2 0 3 0 1 −1 2
3
∆ij = (−1)i+j |Aij | det( det(A) = a ∆ + · · · + a ∆ 2 = 1( −1) · 3 + 2(−1) · 1 = 1(1) · 3 + 2(−1) · 1 = 1 3 2 0 3 0 = (0)∆ + (0)∆ + (2)∆ −1 2 1 2 ∆ = (−1) 1 3 = 1·1 = 1 i1
1 det 11 det 1 1
1+1
i1
in
1+2
13
23
33
33
3+3
det( det(B ) = 2 · 1 = 2
in
1 0 0
1 2
2 3 3 0 3 0 −1
1
1 det( det(A) = 0 1 det( det(B ) = 2 3
0 1
=1
2 3 3 0 0 −1
−1
A
= 3∆
13
=
1 0 0 1
+ (0)∆23 + (−1)∆33
∆ = (−1) = (1)(−9) = −9 2 ∆ = (−1) = (1)(−1) = −1 3 1 2 3 1 0 0 1 2 3 3 1 0 0 0 0 1 0 ∼ 0 −1 −6 −2 1 0 ∼ 0 1 6 0 −6 −10 −3 0 1 0 −6 −10 −1 0 0 1 1 0 −9 −3 2 0 1 0 0 3/26 −1/13 9/26 0 1 6 2 −1 0 ∼ 0 1 0 −1/3 5/13 −3/13 0 0 26 9 −6 1 0 0 1 9/26 −6/26 1/26 3/26 −1/13 9/26 B = −1/3 5/13 −3/13 1+3
13
3+3
33
1 2
2 3 3 0
∼
2 3 1 2
3 0
1 0 0 2 −1 0 0 1 −3
−1
9/26 −6/26
1/26
2
2
R
R
V = R2 = {(x, y); x, y ∈ R} (x1, y1 ) + (x ( x2 , y2 ) = (x ( x1 + x2 , y1 + y2 ) a · (x, y) = (ax,ay ( ax,ay)) u = (x1 , y1 ) v = (x2 , y2 ) (A1 ) (u + v) + w = u + (v (v + w) (A2 ) u + v = v + u (A3 ) ∃0 = (0, (0, 0) ∈ R2 ∀u ∈ R2 u + 0 = u (A4 ) ∀u = (x1 , x2) ∈ R2 ∃(−u) = (−x1 , −x2 ) ∈ R2 u + (−u) = 0 (M 5 ) (ab) ab)v = a(bv) bv ) (M 6 ) (a + b)v = au + bv (M 7 ) a(u + v ) = au + av (M 8 ) 1u = u
∀u,v,w ∈ V
∀a, b ∈ R
w = (x ( x3 , y3 )
W = {(x,y,z,t) x,y,z,t) ∈ R4|x + t = 0 e z − t = 0}
•
u, v ∈ W
u = (x1 , y1 , z1 , t1 ) u+v
•
u ∈ W k ∈ R
u = (x1 , y1, z1 , t1 ) = = = =
u, v ∈ S
4 2
R
u = (x1 , y1) u+v
•
u ∈ W k ∈ R
k (x1 , y1 , z1 , t1 ) k (−t1 , y1 , t1 , t1 ) (−kt1 , ky1 , kt1 , kt1 ) (ku) ku) ∈ W
R
S = {(x, y) ∈ R2 |y = −x}
•
v = (x2 , y2) = =
(x1 , y1 ) + (x ( x2 , y2 ) (x1 , −x1 ) + (x ( x2 , −x2)
= =
(x1 + x2 , −x1 − x2 ) (u + v ) ∈ W
u = (x1 , y1) ku
= k(x1 , y1 ) = k(x1 , −x1 ) = (kx1 , −kx1 ) (ku) ku) ∈ W
= W
4
R
2
S = {(x, y)|x + 3y 3y = 2 }
•
u, v ∈ S
v = (x2 , y2 , z2 , t2 )
= (x1 , y1 , z1 , t1 ) + (x ( x2 , y2 , z2 , t2) = (−t1 , y1 , t1 , t1 ) + ( −t2 , y2 , t2 , t2 ) = (−t1 − t2, y1 + y2 , t1 + t2 , t1 + t2 ) = (u + v) ∈ W
ku
W
4
R
R
u = (x1 , y1) u+v
v = (x2 , y2)
= = = =
(x1 , y1 ) + (x ( x2 , y2 ) (2 − 3y1 , y1 ) + (2 − 3y2 , y2 ) (4 − 3y1 − 3y2 , y1 + y2 ) (u + v) ∈ / W 2
R
S = {(x,y,z) x,y,z)|x = z 2 }
•
u, v ∈ W
3
R
u = (x1 , y1, z1 ) u+v
v = (x2, y2 , z2)
= (x1 , y1 , z1 ) + (x (x2, y2 , z2) 2 = (z1 , y1 , z1 ) + (z ( z22 , y2, z2 ) = (z12 + z22, y1 + y2 , z1 + z2 ) = 2z1z2 , y1 + y2 , z1 + z2 ) = (u + v ) ∈ / S (z12 + z22 + 2z
S
3
R
2
αv u+v
R
α u = (x, y)
v = (x , y )
u + v = (x + y , x + y ) u + v = (xx , yy ) (3x + 3x 3x , 5x + 5x 5x ) u + v = (3x
u + v = (x + y , x + y )
w = (x1 , y1 )
(A1 ) (u + v )+ w = (x + y , x + y )+ (x1 , y1 ) = (x + y + y1 , x + y + x1 ) = (x, y)+ (y + y1 , x + x1 ) = u + (v (v + w) (A2 ) u + v = (x ( x + y , x + y ) = (y + x, y + x ) = (y ( y , x ) + (y, ( y, x) = v+u 2 2 (A3 ) ∃0 = (0, (0, 0) ∈ R ∀u ∈ R u + 0 = u (A4 ) u = (x, y) (−u) = (−x, −y ) u + (−u) = (x ( x − y, y − x) = x= 0 y (M 5 ) (ab) ab)v = a(bv) bv ) (M 6 ) (a + b)v = (ax + bx, bx, ay + by) by) = (ax, ax, by) by) + (by,bx ( by,bx)) = av + bv (M 7 ) a(u + v) = au + av (M 8 ) 1u = u u + v = (xx , yy ) (A1 ) (A2 ) (A3 ) (A4 ) (M 5 ) (M 6 ) (M 7 ) (M 8 )
v = (x ( x1 , y1 )
(u + v ) + w = (xx , yy ) + (x1, y1 ) = (xx x1 , yy y1 ) = (x, y) + (x + x1, y + y1 ) = u + (v + w) u + v = (xx ( xx , yy ) = (x ( x x, y y ) = v + u (0, 0) ∈ R2 ∀u ∈ R2 u + 0 = 0 ∃0 = (0, u = (x, y) (−u) = (−x, −y ) u + (−u) = ( −x2, −y2 ) = x, y = 0 0 (ab) ab)v = a(bv) bv ) (a + b)v = au + bv a(u + v) = a(xx , yy ) = (axx ,ayy ) = au + av 1u = u
u + v = (3x (3x + 3x 3x , 5x + 5x 5x )
w = (3 ( 3x1, 5x1 )
(A1 ) (u + v) + w = (3x (3x + 3x , 5x + 5x ) + (3x (3x1 , 5y1 ) = (3x (3x + 3x + 3x1 , 5x + 5x + 5x1) = (3x, (3x, 5x) + (3x (3x + 3x 3x1 , 5x + 5x 5x1 ) = u + (v (v + w) (A2 ) u + v = (3 ( 3x + 3x 3x , 5x + 5x 5x ) = (3x (3x + 3x, 3x, 5x + 5x 5x) = (3x (3x , 5x ) + (3x, (3x, 5x) = v + u 2 2 (A3 ) ∃0 = (0, (0, 0) ∈ R ∀u ∈ R u + 0 = u (A4 ) ∀u = (3x, (3x, 5x) ∈ R2 ∃(−u) = (−3x, −5x) ∈ R2 u + (−u) = 0 (M 5 ) (ab) ab)v = a(bv) bv ) (M 6 ) (a + b)v = (a + b)(3x, )(3x, 5x) = a(3x, (3x, 5x) + b(3x, (3x, 5x) = av + bv (M 7 ) a(u + v ) = a(3x (3x + 3x 3x , 5x + 5x 5x ) = (3ax (3ax + 3ax 3ax , 5ax + 5ax 5ax ) = (3ax, (3ax, 5ax) ax) + (3ax (3ax , 5ax ) = au + av (M 8 ) 1u = u W = {(x,y,z,t) x,y,z,t) ∈ R4 | x + t = 0
•
u, v ∈ W
u = (x1 , y1, z1 , t1 ) u+v
z − t = 0}
v = (x2 , y2 , z2 , t2 )
= (x1 , y1 , z1 , t1 ) + (x ( x2 , y2 , z2 , t2 ) = (−t1 , y1 , t1 , t1 ) + ( −t2 , y2, t2 , t2 ) = (−t1 − t2 , y1 + y2, t1 + t2, t1 + t2) = (u + v) ∈ W
4
R
•
u ∈ W k ∈ R
u = (x1 , y1 , z1 , t1) ku
W
4
R
= k(x1 , y1 , z1 , t1 ) = k(−t1 , y1 , t1 , t1 ) = (−kt1 , ky1 , kt1 , kt1 ) = (ku) ku) ∈ W
p( p(x) = x3 − 5x2 + 1 q (x) = 2x4 + 5x 5x − 6
r (x) = x2 − 5x + 2
a(x3 − 5x2 + 1) + b(2x (2x4 + 5x 5x − 6) + c(x2 − 5x + 2) = ax3 − 5ax2 + a + 2bx 2bx4 + 5bx 5bx − 6b) + cx2 − 5cx + 2c 2c = (2b) + x3 (a) + x2 (−5a + c) + x(5b (5b − 5c) + (a ( a − 6b + 2c 2c) = x4 (2b
x4 (2b (2b) + x3 (a) + x2 (−5a + c) + x(5b (5b − 5c) + (a ( a − 6b + 2c 2c) = 0 · x4 + 0 · x3 + 0 · x2 + 0 · x + 0
2b = 0 ⇒ b = 0 −a 5=a0+ c = 0 ⇒ c = 0
p(x), q(x), r(x)} { p(
a=b=c=0
0 2
1 −5 0 1 0 0 5 −6 0 0 1 −5 2
0 2
1 −5 0 1 0 0 5 −6 0 0 1 −5 2
∼
← ← ←
0x4 + x3 − 5x2 + 0x 0x + 1 4 3 2 2x + 0x 0x + 0x 0x + 5x 5x − 6 4 3 2 0x + 0x 0x + 1x 1 x − 5x + 2
2 0 0 5 −6 1 0 1 −5 0 1 ∼ 0 0 0 1 −5 2 0 1 0 0 5/2 −3 ∼ 0 1 0 −25 11 0 0
1
−5
2
0 0 5/2 −3 1 −5 0 1 0 1 −5 2
3
A = {(−1, 3, 2), 2), (2, (2, −2, 1)}
R
x det( det(A) = −1 2
3x + 4y 4y + 2z 2z − 6z + 4x 4x + y = 0
y z 3 2 −2 1
= 3x + 4y4y + 2z2z − 6z + 4x4x + y
7x + 5y 5 y − 4z = 0
⇒ 3
R
S = {(x,y,z) x,y,z) ∈ R3 ;
p( p(x) = x3 − 5x2 + 1 q (x) = 2x4 + 5x 5x − 6
p( p(x) = x3 − 5x2 + 1 q(x) = 2x 2 x4 + 5x 5x − 6
7x + 5y 5y − 4z = 0}
r (x) = x2 − 5x + 2
r(x) = x2 − 5x + 2
p( p(x) = aq( aq(x) + br( br(x) 0 · x4 + x3 − 5x2 + 1
= a(2x (2x4 + 5x 5x − 6) + b(x2 − 5x + 2) = 2ax4 + 5ax 5ax − 6a + bx2 − 5bx + 2b 2b 4 2 = 2ax + bx + x(5a (5a − 5b) + (2b (2b − 6a) 4 3 2 = 2ax + 0 · x + bx + (5a (5a − 5b)x + (2b (2b − 6a)
x4 → x3 → p( p(x) p(x), q(x), r(x)} { p(
2a = 0
a=0 1= 0 q (x)
r(x)
3
A = {(−1, 3, 2), 2), (2, (2, −2, 1)}
R
x det( det(A) = −1 2
3x + 4y 4y + 2z 2z − 6z + 4x 4x + y = 0
y z 3 2 −2 1
= 3x + 4y4y + 2z2z − 6z + 4x4x + y
7x + 5y 5 y − 4z = 0
⇒ 3
R
S = {(x,y,z) x,y,z) ∈ R3 ;
7x + 5y 5y − 4z = 0}
V = {(0, (0, 0, 0), 0), (1, (1, 2, 3), 3), v1 } v1
0·v = v1
V
B = {1, x − 1, x2 − 3x + 1} B
v = 2x 2 x2 − 5x + 6
P 2
0 = α(1)+ β (x − 1) + γ (x2 − 3x + 1)
0 0 1
P 2
0 1 −3 B
0 0 0 1 1 −3 1 1 −1 ∼ 0 1
1 −1 1
1 −3 1 −1 0 0 1
← ← ←
0 · x2 + 0 · x + 1 0 · x2 + 1 · x − 1 1 · x2 − 3 · x + 1
1 ∼ 0
−3 1 1 0 0 0 1
1 ∼ 0
B P 2 {1, x , x }) 2
v = a(1) + b(x − 1) + c(x2 − 3x + 1) 2x2 − 5x + 6
= a + bx − b + cx2 − 3cx + c = cx2 + (b (b − 3c)x + (a (a + c)
c = 2 ba =+ c3c==6 −5⇒ ⇒a =b 6=−−25=+43(2) = −5 + 6 = 1 v = 4(1) + 1(x 1(x − 1) + 2(x 2(x2 − 3x + 1)
0 1 0 0
0 0 1
S 1 = {(a,b,c,d) a,b,c,d)| a + b + c + d = 0}
Dim( Dim(S 1 )
S 1 = {(a,b,c,d) a,b,c,d)| a + b + c + d = 0} d = −a − b − c (a,b,c, −a − b − c) = a(1, (1, 0, 0, −1) + b(0, (0, 1, 0, −1) + c(0, (0, 0, 1, −1) S = [(1, [(1, 0, 0, −1), 1), (0, (0, 1, 0, −1), 1), (0, (0, 0, 1, −1)] (1, 0, 0, −1), 1), (0, (0, 1, 0, −1), 1), (0, (0, 0, 1, −1)} {(1, (1, 0, 0, −1), 1), (0, (0, 1, 0, −1), 1), (0, (0, 0, 1, −1)} S 1 {(1, Dim( Dim(S 1 ) = 3
M 2×2 M 2×2
1 0 0 1 0 0 0 0
0
0
,
1 0
M 2x2
0 0 0
,
M 2×2
1
M 2×2
M 2×2
M 2×2
1 0 0 1 0 0 0 0 1 1 0 0
,
0 0
,
1
0
,
0 1
,
1 1
M 2×2 M 2×2
