f f (x, y) = x+y -2
G = (-1, 1) ‚" R
a + b
a " b = .
1 + ab
H = {a, b}
a b
a a a
b a b
J = {a, b, c, d, e} •"
•" a b c d e
a d c e b a
b c e e a b
c e e d b c
d b a c e d
e a b c d e
•"
J
(G, ")
(R, " )
u + v
u " v = ,
2
" "
a 2 + b 5 : a, b " Q
(Z, ć%)
a ć% b = a + b + 2
(-1 ć% 5) ć% (6 ć% 2)
G = (R \ {-1})
x y = x + y + xy.
(G, )
P = (x, y, z) " R3 : x = 0, y = 0
(x, y, z) (k, l, m) = (kx, ly, mx + lz) .
(P, )
(X, " ) "
(A, +) (R, +)
A = R
A = N
"
A = a + b 2 : a, b " Z
R2 I Sx
x Sy y So
S = {I, Sx, Sy, So} ć% S
(S, ć%)
Z4 = {0, 1, 2, 3}
a •" b = a + b 4.
(Z4, •")
(S, ć%) (Z4, •")
(R, +) (R \ {0} , ·)
f : P R \ {0}
f (a, b, c) = ab
(P, ) (R \ {0} , ·)
P
A = (2, + ") ‚" R "
x " y = xy - 2x - 2y + 6
f : A R+
f (a) = a - 2,
(A, ") (R+, ·) R+ def {x " R : x > 0}
=
(Z, +, ·)
(Z, +, ·)
(Z, +, ·)
def
Z (3) = {p " Z : 3|p} (Z, +, ·)
(Z, +, ·) (Z (3) , +, ·)
(Z, +, ·)
R2, •",
(a, b) •" (x, y) = (a + x, b + y) ,
(a, b) (x, y) = (ax - by, ay + bx) ,
"
H = a + b 5 : a, b " Q
(R, +, ·)
R+
a " b = ab.
(R+, ·, ")
(R, +, ·)
R
a b = a + b + 1,
a ! b = a + b + ab.
(R, , ! )
(R \ {-1} , ! , ) (R \ {0} , ·)
(A, •", ) (B, " , )
p : A B p (a) = . . . (A, •", )
(B, " , )
(B, " , ) (A, •", )
x, y
x (2 + 3i) + y (4 - 5i) = 6 - 2i
y
x
+ = 1
3+2i 2-3i
z x + iy
x, y " R
(a - bi) z = a + bi
(a + bi) a - bi (1 - z) + (a - bi)2 (1 + z) = 0
1 + i - i -1 i -1 - i
"1 "+ " "
1 + i 3 1 - i 3 -1 + i 3 -1 - i 3
"
(1 + i) 1 - i 3
1+i
"
1+i 3
"
2 + i 12
1 - cos Ä… - i sin Ä… 1 + cos Ä… + i sin Ä…
1 + i tg Ä… 1 - i tg Ä…
1+i tg Ä…
1-i tg Ä…
z2 + z + 1 = 0
z2 + 2z + 5 = 0
z1 z2
z4 z1 z2 z3
z1 z2
zw z1 z2
z1 z2
z
|z - z0| = r z0 " C r > 0
|z - z0| < r z0 " C r > 0
|z - 2 + i| 3
Arg (z - z0) = Õ z0 " C Õ " [0, 2Ä„)
Arg (z - 2 - i) = Ä„
|z - 1 + i| = |z - i|
z-2i
= 1
z+1
|z - i| < |z + 1|
|z + i| |iz + 2|
Re z 3
Im (z + 1) > -1
|Re z| + |Im z| 1
Im z2 0
Re z2 + i = 1
(1 + i)8
9
"
3 + i
10
cos Ä…+i sin Ä…
"
1+i
3
144
1+i
"
1+i 3
(cos 3ć% + i sin 3ć%)100
"
(1+i)101 i- 3
( )102
"
i1523 3
( )103
1+i
2002 2002 2002
" " "
1
1 + i 3 + -1 + i 3 + 1 - i 3
22002
30 50 60
"
" "
1 1-i 3
· · (1 + i)40 · 3 - i 3 + i
2122 1-i
sin 2x cos 2x sin x cos x
sin 3x cos 3x sin x cos x
sin 4x cos 4x sin x cos x
(cos x + i sin x) + (cos x - i sin x)
cos x = ,
2
(cos x + i sin x) - (cos x - i sin x)
sin x = ,
2i
sin2 x cos2 x cos 2x
cos3 x cos 3x cos x
sin3 x sin 3x sin x
sin4 x cos4 x cos 4x cos 2x
"
z + |z| z + |z|
z = |z|, - |z|
|z + |z|| |z + |z||
z " C \ {r " R : r 0}
" "
4
"i 16
-11 + 60i
" "
4
-4 + 4 3i -8 + 8i 3
3
(3 + 4i)3
sin 15ć% cos 15ć%
"
6
-1 = {w1, w2, w3, w4, w5, w6}
w1 + w2 + w3 + w4 + w5 + w6,
w1 · w2 · w3 · w4 · w5 · w6.
z2 = i
z3 = -27
z8 = 28
z2 - 3z + 3 + i = 0
z2 - (2 + i) z - 1 + 7i = 0
z2 + 2z + 3 = 0
z4 - 30z2 + 289 = 0
z4 + (15 + 7i) z2 8 - 15i = 0
"
"+
2+i2 3
"
2z2 + 2 -1 + i 3 z + = 0
1-i 3
z |z| = 2z
z1, z2 " C
|z1 + z2|2 + |z1 - z2|2 = 2 |z1|2 + |z2|2 .
f (x)
x - x0 f (x0)
f (x) = 2x5 - 4x4 - x3 + x2 + 2x - 3 x0 = 2
f (x) = x4 + x2 + 2 x0 = -1
f (x) = 4x5 - 4x4 + 3x3 - 3x2 + 9x - 9 x0 = 1 + i
f (x) g (x)
f (x0)
f (x) = 3x4 + 2x3 - 7x - 1 g (x) = x + 4 x0 = -4
1
f (x) = x3 + x2 + x + 2 g (x) = 2x - 1 x0 =
2
f (x) = x4 + (3 - 8i) x3 - (21 + 18i) x2 - (33 - 20i) x + 7 + 18i
g (x) = x - x0 x0 = 2i - 1
a b
x4 - 3x3 + ax2 + bx + a x2 - 1
axn+1 + bxn + 1 (x - 1)2
x5 + ax3 + b
(x - 1) (x - 2) x + 1
(x + 1) (-2)
(x - 1) (x - 2) (x + 1)
x4 - 1
x4 + 81
x4 + x2 + 1
x4 - x2 + 1
x4 - 1 = 0
x6 - x3 + 1 = 0
x4 + 2x3 - 9x2 - 2x + 8 = 0
x5 - 4x3 - 2x2 + 3x + 2 = 0
x4 - 2x3 - 2x2 - 2x - 3 = 0
x4 - 2x3 + 5x2 - 8x + 4 = 0
x4 + 6x2 - 8ix - 3 = 0 C
Å„Å‚
ôÅ‚ x + y + z = 9
òÅ‚
xy + yz + zx = 27
ôÅ‚
ół
xyz = 27,
xy (x + y) = 30
x3 + y3 = 35.
w (x) = 3x4 -10x3 +a2x2 +a1x+a0
x1 : x2 : x3 : x4 = 1 : 2 : 3 : 4.
a0 a1 a2
p1 p2 p3 a3x3 +a2x2 +a1x+a0 = 0
a0, a1, a2, a3
p2 + p2 + p2
1 2 3
p3 + p3 + p3
1 2 3
1 1 1
+ +
p1 p2 p3
A = (x1, x2, x3) " R3 : 2x1 + x2 + x3 = 0
x •" y = (x1 + y1, x2 + y2, x3 + y3) ,
Ä… x = (Ä…x1, Ä…x2, Ä…x3) ,
x = (x1, x2, x3) " R3 y = (y1, y2, y3) " R3 Ä… " R R
C [0, 1] f : [0, 1] R
(f + g) (x) = f (x) + g (x) , x " [0, 1] ,
(Ä… · f) (x) = Ä…f (x) , x " [0, 1] ,
f, g " C [0, 1] Ä… " R R
V = R+
v •" w = v · w, v, w " V,
Ä… v = vÄ…, v " V, Ä… " R.
(R, V, •", )
V P
R
P = R3 V = (x, y, z) " R3 : x > 0
P = R3 V = (x, y, z) " R3 : x + y + z = 0
P = R3 V = (x, y, z) " R3 : x + 2y - z = 1
P = R3 V = (x, y, z) " R3 : x " Q
P = R3 V = (x, y, z) " R3 : yz 0
P = R3 V = (x, y, z) " R3 : x + y + z = x - y = 0
P = R4 V = (x, y, z, t) " R4 : x = z y = t
P = R4 V = (x, y, z, t) " R4 : x = z y = t
P = R5 V = (2x, x + y, 7, 13, x - y) " R5 : x, y " R
(R, V, +, ·)
V = (x, y, z) " R3 : x + 2y + 3z = 0 ‚" R3
R, R3, +, · e1 = (-2, 1, 0)
e2 = (-3, 0, 1) V V
V
e1 = (1, 1, 2, 1) , e2 = (1, -1, 0, 1) , e3 = (0, 0, -1, 1) , e4 = (1, 2, 2, 0)
R4 v = (1, 2, 2, 0)
e1 = (1, 2, 0) , e2 = (2, 1, -1) , e3 = (-4, -5, -1) , e4 = (3, -3, 1)
dim (e1, e2, e3, e4) (e1, e2, e3, e4)
L L
e1 = (1, 2, 0, 4) , e2 = (-1, 0, 5, 1) , e3 = (1, 6, 10, 14) ,
Ä…1e1 + Ä…2e2 + Ä…3e3 = ¸,
¸ = (0, 0, 0, 0) v = (0, 8, 20, 20) " (e1, e2, e3)
L
f1 = (1, 1, 1, 1, 1) , f2 = (0, 1, 1, 1, 1) , f3 = (0, 0, 1, 1, 1) , f4 = (0, 0, 0, 1, 1) , f5 = (0, 0, 0, 0, 1) ,
R5 x = (1, 0, 1, 0, 1)
W = (v1, v2, v3, v4)
L
v1 = (1, 2, 3, 4) , v2 = (4, 7, 10, 13) , v3 = (2, 3, 4, 5) , v4 = (3, 5, 7, 9)
v = (2, 1, 1, 2) W
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
0 0 2 3 1 4 -2 0
ïÅ‚ śł ïÅ‚ śł
ïÅ‚ śł ïÅ‚ śł
2 -2 -1 2 3 -2 4 -3
ïÅ‚ śł ïÅ‚ śł
· ,
ïÅ‚ śł ïÅ‚ śł
1 0 4 0 0 1 -1 0
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
-3 1 5 -1 2 2 5 1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 -4 7 0 -3 -2 4 2 0
ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł
2 ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
ðÅ‚ -5 8 · 1 4 -1 + 1 0 1 .
3 6 -9 2 5 0 0 -3 -2
0 0 2 0 3 0 1 0 2 0 3 0
2 12 4 2 7 3 3 12 4 2 7 3
1 0 4 0 9 0 0 0 4 0 9 0
+ ,
1 12 5 3 7 6 7 12 5 3 7 6
2 0 8 0 27 0 -1 0 8 0 27 0
-3 12 5 4 3 10 12 12 5 4 3 10
-1
24 11 13 17 19
35 59 71 52
51 13 32 40 46
12547 13447 42 70 77 54
+ + 61 11 14 50 56
28523 28423 43 68 72 52
62 20 7 13 52
29 49 65 50
80 24 45 57 70
ëÅ‚
öÅ‚2005
1 2 3 4 22 -6 -26 17
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
2 3 1 2 -17 5 20 -13
ìÅ‚ ÷Å‚
·
ìÅ‚ ÷Å‚
1 1 1 -1 -1 0 2 -1
íÅ‚
Å‚Å‚
1 0 -2 -6 4 1 5 3
2 -1 3 4 -5
1 20052004 20042005 4 -2 7 8 -7
0 20052005 20052005 -6 4 -9 -2 3
·
0 02005 1 3 -2 4 1 -2
20052005
-2 6 5 4 -3
îÅ‚ Å‚Å‚
2 2 2 2 5
ïÅ‚ śł
ïÅ‚ 2 2 2 5 2 śł
ïÅ‚ śł
ïÅ‚ śł
2 2 5 2 2 .
ïÅ‚ śł
ïÅ‚ śł
ðÅ‚ 2 5 2 2 2 ûÅ‚
5 2 2 2 2
îÅ‚ Å‚Å‚
3 2 -1 2 0 1
ïÅ‚ śł
ïÅ‚ 4 1 0 -3 0 2 śł
ïÅ‚ śł
ïÅ‚ śł
2 -1 -2 1 1 -3
ïÅ‚ śł
ïÅ‚ śł
ðÅ‚ 3 1 3 -9 -1 6 ûÅ‚
3 -1 -5 7 2 -7
a1 = (1, 2, 3, 4) , a2 = (2, 3, 4, 5) , a3 = (3, 4, 5, 6) , a4 = (4, 5, 6, 7) .
îÅ‚ Å‚Å‚-1
1 2 -1 -2
ïÅ‚ śł
ïÅ‚ śł
3 8 0 -4
ïÅ‚ śł
.
ïÅ‚ śł
2 2 -4 -3
ðÅ‚ ûÅ‚
3 8 -1 -6
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 2 -3 0 1 0 1 -2 0
ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł
3 2 ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
ðÅ‚ -4 · X + -5 -1 -3 = 5 1 4
2 -1 0 -4 -3 -3 6 4 5
T
1 0 0 -2 3 1 0 0 -1
· 2 1 + · X · = + 2
2 -6 -2 -1 1 0 0 0 1
3 -1 1 1 4 5 14 16
· X · + =
5 -2 1 1 6 7 9 10
T
1 2 0 0 3 5
+ · X =
0 1 3 3 5 9
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚T îÅ‚ Å‚Å‚
5 3 1 -8 0 -2 0 3 -1
ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł
X · 1 -3 -2 = 0 9 15 + -5 0 0 ûÅ‚
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚
5 2 1 1 0 1 0 0 -1
3 3 îÅ‚ Å‚Å‚
1 2 3 -10 -7 -1 250 -470 -230
ïÅ‚ śł
2 3 4 · 16 11 2 · 482 -238 2 ûÅ‚
ðÅ‚
3 1 2 -7 -5 -1 -478 242 722
ëÅ‚
îÅ‚ Å‚Å‚-1 îÅ‚ Å‚Å‚T öÅ‚ îÅ‚ Å‚Å‚
4 1 -2 1
1 1 -1 1 4 5 -1 2 1
ìÅ‚ ÷Å‚
1 3 1 4
ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł
ìÅ‚ ÷Å‚
+ X · 2 · 2 1 0 + 2 0 0 = · -2 1 0 ûÅ‚
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚
íÅ‚ Å‚Å‚
-2 6 0 5
1 -1 1 1 4 3 2 -1 -3
5 1 2 0
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 -1 2 1 1 0 0 -1
1 2 1 x 1
2 -1 4 2 -2 1 0 0
ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł
· · 2 1 -5 · y = 2 .
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
3 0 1 0 3 0 1 3
0 -1 -2 z 3
-2 1 -2 -1 -6 1 -2 5
Ä… " R
îÅ‚ Å‚Å‚
2 0 Ä…
ïÅ‚ śł
A = 0 1 1 ûÅ‚
ðÅ‚
1 0 0
A
x, y, z " R
Å„Å‚ Å„Å‚ Å„Å‚
ôÅ‚ x + y + 2z = 1 ôÅ‚ x + 2y + z = - ôÅ‚ x + y + z = 1
òÅ‚ òÅ‚ òÅ‚
x + y + 2z = 1 x + y - z = 2 x + y + z =
ôÅ‚ ôÅ‚ ôÅ‚
ół ół ół
x + y + 2z = 1 y + z = 1 x + y + z = 2
x1, x2, x3, x4 " R
Å„Å‚
ôÅ‚
x1 + x2 + x3 + x4 = 1
ôÅ‚
ôÅ‚
ôÅ‚
òÅ‚
x1 + x2 + x3 + x4 = 1
ôÅ‚
x1 + x2 + x3 + x4 = 1
ôÅ‚
ôÅ‚
ôÅ‚
ół
x1 + x2 + x3 + x4 = 1
Å„Å‚
Å„Å‚ Å„Å‚
ôÅ‚ x1 - x3 + x5 = 0
ôÅ‚
ôÅ‚ ôÅ‚ ôÅ‚
3x + 4y + z + 2w + 3t = 0 6x - 2y + 2z + 5w + 7t = 0
ôÅ‚ ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚ ôÅ‚ x2 - x4 + x6 = 0
òÅ‚ òÅ‚ òÅ‚
5x + 7y + z + 3w + 4t = 0 9x - 3y + 4z + 8w + 9t = 0
x1 - x2 + x5 - x6 = 0
ôÅ‚ ôÅ‚ ôÅ‚
4x + 5y + 2z + w + 5t = 0 6x - 2y + 6z + 7w + t = 0
ôÅ‚ ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚ ôÅ‚ x2 - x3 + x6 = 0
ół ół ôÅ‚
ôÅ‚
7x + 10y + z + 6w + 5t = 0 3x - y + 4z + 4w - t = 0
ół
x1 - x4 + x5 = 0
Å„Å‚
Å„Å‚
ôÅ‚ 8x1 + 6x2 + 5x3 + 2x4 = 21
ôÅ‚
ôÅ‚ ôÅ‚
10x1 + 23x2 + 17x3 + 44x4 = 25
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
ôÅ‚ 3x1 + 3x2 + 2x3 + x4 = 10 ôÅ‚
òÅ‚ òÅ‚
15x1 + 35x2 + 26x3 + 69x4 = 40
4x1 + 2x2 + 3x3 + x4 = 8
ôÅ‚ ôÅ‚
25x1 + 57x2 + 42x3 + 108x4 = 65
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
ôÅ‚ 7x1 + 4x2 + 5x3 + 2x4 = 18 ôÅ‚
ôÅ‚ ół
ôÅ‚
30x1 + 69x2 + 51x3 + 133x4 = 95
ół
3x1 + 2x2 + x3 + x4 = 15
Å„Å‚ Å„Å‚
ôÅ‚ ôÅ‚
2x1 + 2x2 - x3 + x4 = 4 24x1 + 14x2 + 30x3 + 40x4 + 41x5 = 28
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
òÅ‚ òÅ‚
4x1 + 3x2 - x3 + 2x4 = 6 36x1 + 21x2 + 45x3 + 61x4 + 62x5 = 43
ôÅ‚ ôÅ‚
8x1 + 5x2 - 3x3 + 4x4 = 12 48x1 + 28x2 + 60x3 + 82x4 + 83x5 = 58
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
ół ół
3x1 + 3x2 - 2x3 + 2x4 = 6 60x1 + 35x2 + 75x3 + 99x4 + 102x5 = 69
Å„Å‚
ôÅ‚ 2x1 - x2 + 3x3 - 7x4 = 5
òÅ‚
6x1 - 3x2 + x3 - 4x4 = 7
ôÅ‚
ół
4x1 - 2x2 + 14x3 - 31x4 = 18.
(x1, x2) , (y1, y2) = x1y1 + x2y2 (x1, x2) , (y1, y2) " R2
(x1, . . . , xn) , (y1, . . . , yn) = x1y1 + . . . + xnyn (x1, x2) , (y1, y2) " Rn
(x1, x2) , (y1, y2) = 2x1y1 + x1y2 + x2y1 + x2y2 (x1, x2) , (y1, y2) " R2
(x1, x2) , (y1, y2) = x1y1 - x2y2 (x1, x2) , (y1, y2) " R2
(x1, x2, x3, x4) , (y1, y2, y3, y4) = 3x1y1 + x2y2 + x3y3 + 2x4y4 + x2y4 + x4y2
(x1, x2, x3, x4) , (y1, y2, y3, y4) " R4
b
f, g = f (x) g (x) dx f, g " C [a, b]
a
x, y = xT Ay
A
(X, ·, · ) x = x, x
x " X x, y " X t " R
x 0
x = 0 Ô! x = 0
tx = |t| x
x + y x + y
(X, ·, · ) x, y " X
x + y 2 = x 2 + 2 x, y + y 2
x - y 2 = x 2 - 2 x, y + y 2
x 2 - y 2 = x - y, x + y = x + y, x - y
1
x, y = x + y 2 - x - y 2
4
x - z 2 + y - z 2 - x - y 2 = 2 x - z, y - z
f : R2 × R2
1 1
f (x, y) = x1y1 + x1y2 + x2y1 + x2y2,
2 2
x = (x1, x2) y = (y1, y2) R2
e1 = (1, 0) e2 = (0, 1)
a = (1, -1, 2) b = (2, 0, 3) c = (-1, 2, -1) " R3
a b a c b c a - 2b + c a + b
a b a c b c
R3
Õ ((x1, x2, x3) , (y1, y2, y3)) = x1y1 + x1y2 + x2y1 + 2x2y2 + x3y3
(x1, x2, x3) , (y1, y2, y3) " R3
p, q " R2005 p, q p - q
Ä„
p = 2 q = 3 (p, q) =
3
5Ä„
p = 1 q = 2 (p, q) =
6
u = 2a + 3b a, b " R777 a = 3
Ä„
b = 1 (a, b) =
3
u v
7
u = 6m + 4n v = 2m + 10n m = 5 n = 3 (m, n) = Ä„
4
u = 5a + b v = 2a + 3b a b
" "
a + b 2 · a2 + b2 a, b " R
"
2a + 4b + c + 2d 5 a2 + b2 + c2 + d2 a, b, c, d " R
" "
2a + 3b 10 · 4a2 + b2 a, b " R
a b a + b a - b
a b a+3b
7a - b a - 4b 7a - 2b
A (1, 2) B (3, 1) C (-1, -1)
- -
-
BC P BC AP ABC
P = (x, y, z, t) " R4 : 4x - z = 2y - 3z + 2t = 0
(e1, e2, e3)
L
e1 = (1, 2, 2, -1) , e2 = (1, 1, -5, 3) , e3 = (3, 2, 8, 7) .
B = {e1, e2, e3, e4} R4
e1 = (2, 1, 3, -1) , e2 = (7, 4, 3, -3) , e3 = (1, 1, -6, 0) , e4 = (5, 7, 7, 8) .
B = {e1, e2, e3} R3
e1 = (1, 2, 3) , e2 = (2, 1, 0) , e3 = (3, 1, 2) .
2 1 2 2 1 2
e1 = , , , e2 = , , -
3 3 3 3 3 3
R3
C [-1, 1]
1
f, g = f (t) g (t) dt, f, g " C [-1, 1] .
-1
fi, fj fi (t) = ti i " {0, 1, 2}
(f0, f1, f2)
L
x = (1, 2, 4) y = (-1, 0, 2)
x = (-2, -1, 3) y = (1, 1, 2)
x = (2, -2, 6) y = (-1, 1, -3)
A = (1, 3, 1) B = (0, 1, 2) C = (2, 7, -4) " R3
A = (1, 1, 0, 0) B = (0, 1, 1, 0) C = (0, 1, 0, 1) " R4
a, b, c " R3
a × b 2 + a, b 2 = a 2 b 2
a, c a, d
a × b, c × d = det
b, c b, d
(a × b) × c = a, c b - b, c a
a × (b × c) = a, c b - a, b c
a = 2005, -2005, e2005 b = (2, 0, 0) c = (0, 0, 5)
b × (b + c) + (a + c) × (c + b) + (b + c) × (c + a)
x = (1, 1, 4) y = (-1, 2, 1) z = (0, 1, 2)
x = (2, 3, -2) y = (1, 2, -2) z = (1, -2, 2)
x = (1, -3, 6) y = (2, 1, 3) z = (1, 4, -3)
a, b, c " R3
a + 2b - c, (a - b) × (a - b - c) = 3 (a · b · c)
p q r
a = p + q - r b = 2p - q + r
c = p + 2q - 3r
A(2, 0, 0) B(0, 4, 0) C(0, 0, 2)
D(2, 3, 8) D
AB = b
AC = c AD = d A(3, 4, 3) B(9, 5, -1) C(1, 7, 0) D(3, 2, 5)
AB = b AC = c AD = d A(1, 1, 2) B(-1, 3, 2)
C(2, -1, 4) D(2, 3, 0)
A B C D
b c d
D
A = (1, 0, 0) B = (1, 1, 0) C = (0, 1, 0) D = (1, 1, 1)
A (-1, 3) B (0, 2)
A (-1, 5, 1) B (1, -2, 0)
A (1, 2, 3, -5) B (-1, 0, 3, 2)
A (2, -1)
2x + 4y - 1 = 0
x = 1 + t y = 1 - t
x + 2y - 1 = 0
x = t y = 2 - t
Ä„
Ox
3
A (-1, 2, 3) B (1, 0, 1) C (1, 0, 3)
A (0, -2, 1) x = 1 - t y = 2 + t z = 2t
A (1, 1, 1) x = -t y = 1 + t z = 2
A (-2, 1, 4) x + y + z + 1 = 0
A (-2, 1, 4) x = 1 + 2t + s y = 1 - t
z = t - s
A (-2, 1, 4) x = t y = 1 + 2t z = 2 - t
x = 2t y = 1 z = 3 + t
x = 1+t y = 2-t z = -t x = 2+t y = -t z = 3-t
x = 2t y = 2-t z = 1-t x = 2 y = t z = -1+t
(4, 0, 1) x = 2 + t y = -1 z = t x = 6 - t
y = 1 + 2t z = 2
x = 7 + t y = 3 + 2t z = 9 - t x = 3 - 7t y = 1 + 2t
z = 1 + 3t
Å„Å‚ Å„Å‚
ôÅ‚ x = 2 + 2t ôÅ‚ x = -3 + 3t
òÅ‚ òÅ‚
y = 2 + t y = t
ôÅ‚ ôÅ‚
ół ół
z = -1 - t z = -1 + t
x = y = z x - y + z = 0
A (1, 2) " R2 x + y + 3 = 0
B (2, 3, -6) " R3 x + 2y + z + 4 = 0
A (2, 3) " R2 2x + 4y - 5 = 0
A (-1, -1, -3) " R3 x + 2y - 5z + 1 = 0
x - y + 2 = 0 x + 2y + 5 = 0 R2
x = 7 + t y = 3 + 2t z = 9 - t x = 3 - 7t y = 1 + 2t z = 1 + 3t R3
x = 2 - t y = t z = 1 + t x = 3 - 7t y = 1 + 2t z = 1 + 3t R3
Å„Å‚
ôÅ‚ x = t
òÅ‚
x + y - z = 0
y = -8 - 4t R3
ôÅ‚
2x - y + 2z - 17 = 0,
ół
z = -3 - 3t
x + z - 1 = 0 x - 2y + 3 = 0
R3
3x + y - z + 13 = 0 y + 2z + 8 = 0,
Ä„
A (3, 5)
4
2x - 3y - 7 = 0
M = (15, 6)
5x - 2y - 5 = 0 2x + 5y - 2 = 0
29
2x = y + 7 = z - 1,
A (3, 2, 6)
A (6, -1) B (0, 3) C (2, 1)
Å„Å‚
ôÅ‚ x = 2x + 3y - z + 1
òÅ‚
x = 2x + 3y - 1
f : g : y = x - 2z + 2
ôÅ‚
y = x - 2y + 2,
ół
z = 2x - y + 3z - 2.
f-1 g-1
x = x - y + 2
f :
y = 2x + y - 1
A (1, -3) B (-1, 0) C (1, 1)
A B
x = 1 + t y = 2 - 3t
x + y - 2 = 0
A B C
Å„Å‚
ôÅ‚ x = x - 2y + z - 2
òÅ‚
f : y = 3x - 2y + 1
ôÅ‚
ół
z = x + y + z - 3
A (0, 1, 3) B (2, 1, -1, ) C (0, 0, 0)
A B
x = 2 - t y = 1 + 2t z = t
x = 1 + 2s - 3t y = 2t - s z = 3 - t + s
x + 2y - z + 1 = 0
x + y + z + 1 = 0
x - 2y + z - 1 = 0,
A B C
y1 = x1 + 2x2 + 2, y2 = x1 - x2 - 1,
"
1 3
x = x - y + 1
2 2
"
3 1
y = x + y - 2
2 2
Oxy 120ć%
" "
O x y A = 2 3, -4 B = 3, 0
x + 2y + 3 = 0
x = a11x + a12y + a1
y = a21x + a22y + a2
" "
ABC A = (0, 0) B = 2, 0 C = 0, 2
A B C
x2 + y2 + 4x + 6y - 12 = 0
x2 + y2 - 2x + 8y - 11 = 0
x2 - 3y2 + 2x - 6y - 8 = 0
y2 - 4y + 6x - 2 = 0
x2 + 2y2 - 2x + 8y + 9 = 0
x2 + y2 - 6x + 4y + 15 = 0
4x2 - xy = 0
îÅ‚ Å‚Å‚
1 2 0
ïÅ‚ śł
A = 0 2 0 ûÅ‚
ðÅ‚
-2 -2 -1
A : R3 R3
A (x1, x2, x3) = (7x1 - 12x2 + 6x3, 10x1 - 19x2 + 10x3, 12x1 - 24x2 + 13x3) .
P : R3 R3
îÅ‚ Å‚Å‚
1 -3 1
ïÅ‚ śł
3 ûÅ‚
ðÅ‚ -3 -2 .
3 -5 1
A : R3 R3
A (x, y, z) = (2x + z, 3y + z, 6y + 2z) ,
A : R2 R2
AB B = (e1, e2)
5 -1
AB = .
-2 4
A : R3 R3
îÅ‚ Å‚Å‚
1 3 0
ïÅ‚ śł
0 2 ûÅ‚
ðÅ‚ -1 .
0 0 4
10
f (x1, x2) = -10x1 - 7x2, -7x1 - x2
13
A (x) = 6x2 + 5x2 + 7x2 - 4x1x2 + 4x1x3,
1 2 3
x = (x1, x2, x3)
A : R4 R4
A (x1, x2, x3, x4) = (x1 + 2x2 + 4x3 - 3x4, 3x1 + 5x2 + 6x3 - 4x4,
4x1 + 5x2 - 2x3 + 3x4, 3x1 + 8x2 - 24x3 + 19x4)
dim Im A ker A
A : R3 R3
A (x1, x2, x3) = (2x1 + x2 - 4x3, 3x1 + 5x2 - 7x3, 4x1 - 5x2 - 6x3) .
A : R3 R3
A (x) = (2x1 - x2 - x3, x1 - 2x2 + x3, x1 + x2 - 2x3) ,
x = x1e1 + x2e2 + x3e3 {e1, e2, e3} R3
ker A Im A
R3
B1 = {f1, f2, f3} B2 = {g1, g2, g3} B3 = {h1, h2, h3}
v = -2h1 - h2 + 3h3
g1 = f1 - f3, h1 = -7f1 + f3,
g2 = 2f1 + f2, h2 = -f1 + 3f2 + f3,
g3 = f2 + f3, h3 = f1 + f2
v B2
B1 B2 B3 R3
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 2 0 -7 -1 1
ïÅ‚ śł ïÅ‚ śł
TB1B2 = 0 1 1 , TB1B3 = 0 3 1 .
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
-1 0 1 2 1 0
v " R3 B3 (-2, -1, 3)
B2
A : R3 R3
A (x1e1 + x2e2 + x3e3) = (x1 - x2 + x3) f1 + (x1 + x2 - x3) f2 + (-x1 + x2 + x3) f3.
(e1, e2, e3) (f1, f2, f3)
e1 = (1, 0, -1) , f1 = (0, 2, 0) ,
e2 = (0, 1, -1) , f2 = (1, 2, 0) ,
e3 = (1, 1, 0) , f3 = (0, 0, 2) .
A (e1, e2, e3)
B : R3 R3 {e1, e2, e3}
îÅ‚ Å‚Å‚
15 -11 5
ïÅ‚ śł
20 ûÅ‚
ðÅ‚ -15 8 .
8 -7 6
{f1, f2, f3}
f1 = 2e1 + 3e3 + e3, f2 = 3e1 + 4e2 + e3, f3 = e1 + 2e2 + 3e3.
B = (e1, e2, e3) f : R3
R3 a1 a2 a3 b1 b2 b3
a1 = 2e1 + 3e2 + 5e3, b1 = e1 + e2 + e3,
a2 = e1 + 2e3, b2 = e1 + e2 - e3,
a3 = e1, b3 = 2e1 + e2 + 2e3.
(R, V, +, ·) B = (e1, e2)
0 -1
AB =
-1 3
B = (e1 + 2e2, 2e1 + e2)
B = (4e1 - 3e2, 5e1 - 3e2)
A B = {e1, e2, e3}
îÅ‚ Å‚Å‚
15 -11 5
ïÅ‚ śł
AB = 20 -15 8 .
ðÅ‚ ûÅ‚
8 -7 6
B = {2e1 + 3e2 + e3, 3e1 + 4e2 + e3, e1 + 2e2 + 2e3}
A B = {-3e1 + 7e2, e1 - 2e2}
2 -1
AB = ,
5 -3
B B = {6e1 - 7e2, -5e1 + 6e2}
1 3
AB = .
2 7
A ć% B B = {e1, e2}
R2 B = {e , e } e = e1 + 2e2 e = 2e1 + 3e2
1 2 1 2
A
3 5
AB = .
4 3
B B = {e , e } e = 3e1 + e2 e = 4e1 + 2e2
1 2 1 2
4 6
AB = .
6 9
A + B B
R2 E = {e1, e2} A
1 3
AE = ,
2 4
B G = {g1, g2} g1 = 3e1 + e2 g2 = 4e1 + 2e2
4 6
AG = .
6 9
C H = {h1, h2} h1 = e1 + 2e2 h2 = 2e1 + 3e2
3 5
AH = .
4 3
A + B + C G
F (x) = 2x1x2 + 4x1x3 - x2 - 8x2,
2 3
x = (x1, x2, x3)
F (x) = x1x2 + x1x3 + x2x3.
F (x, y) = 0
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