.
r
−→c = r −→a + s−→b −→a = (5, 1) −→b = (3, 5) −→c = (5, 4) −→a = (2, −1) −→b = (3, 2) −→c = (5, 2)
s
A = (2, 5) B = (9, 2) C = ( 3, 4) ABCD
−
D
−a −→b → −→d = −→b + −→c −→c
−→a −→b −→d
−→c −→b //−→c −→a
−→a
−→ −−→ 1 −→ AB + BD = AC
ABC
3
−→ −→ DS//AB
−→ −−→ −→ −→ SD = CB + BA − CA −→ −→ DS = AB −BD −→ = 1 −→ 3 −→ AC − DS 3 2 −→ B 1 3 1 3
1 3
S
−→ A
D
−→ C
L M N
B C CA
Q −→ − − → −→ −→ −−→ −−→ QA + QB + QC = QL + QM + QN −AL → + −BM −→ + −CN −→ = −→0
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AB
.
√ − → −a = r−→ → → − − → p b = t q c = (−3, 2 3) → − →q p + t− b −→c = r −→ Y
−→ p 45◦
−→q
X
−b (−→a .−→c ) − −→c (−→a . −→b ) → −→b + ( →−−→−→ )−→a
−→a −→a
a. b
a
2
−→a −→b −→b −→a ≤
−→b −→a ABC A B C
−AA −→ + −BB −→ + −CC −→ = 3−GG −→
N
−→ −→ −−→ P T = s QR + tP N Q
G
G
s
t
R N
P
S
−→a = (2x − 5, 2 − x) −→b = (x − 5, 4 − x) −→a − −→b ⊥ = √ 10 2−→a + −→b − (−→a ⊥ + 3−→b ⊥ ) u = (1, 2, 3) v = (3, 2, 0) w = (2, 0, 0) α β ω αu + βv + ωw = (1, 1, 1)
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.
g (x) = Cos2(x) 1,Cos(2x)
{
C 0( 8, +8)
−
}
R3 R R3
U = (x1, 0, x3)/x1, x3 R U = (x1, 0, x3)/x1, x3 R
{ {
∈ } ∈ }
W = (x1, x2, 0)/x1, x2 R W = (0, x2, 0)/x2 R
{ {
∈ } ∈ }
R3
v1, v2, v3,
··· , v
E
n
{v , v v , ·· · , v } 1
n
2 3
v1, v2, v3,
· ·· , v
E
n
n
E
(1, x, 5)
{(1, 2, 3), (1, 1, 1)} {
| −
S = (x1, x2, x3, x4) x1
R4 S T S + T S T x2 = 0 T = (1, 1, 2, 1), (2, 3, 1, 1)
}
{
∩
−
}
S =
{(1, 2, −1, 3), (2, 1, 0, −2), (0, 1, 2, 1), (3, 4, 1, 2)} V
Q
v3 = u1 + u2 v1, v2, v3, v4 S
{
}
−
B = u1 , u2, u3, u4 4 v1 = 2u1 + u2 u3 v2 = 2u1 + u 3 + 2u4 u3 v4 = u1 + 2u3 + 3u4 S = V v B (1, 2, 0, 1)
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−
{
−
}
.
n
≥
v 3
v
x2 = x 3 (1, 0, 0,
E E
(x1, x2, x3,
··· , x ) n
··· , 0)?
E U, W V dim(U ) = dim (W ) = n
≤
V
−1
dim(V ) = n U W
∩
S = v1, , vn H = v1,...,vm S H E H
{ ··· } { }⊂
E m = n m = n
(2, 0, 0), (0, 1, 0), (0, 0, 1) R3
(1, 1, 1), (1, 1, 0), (1, 0, 0) R3
(a, 0, 0), (0, b, 0), (0, 0, c) R3
E = U W dim(E ) = dim (U ) + dim(W )
⊕
dim(W )
−
E = U + W dim(U W )
dim(E ) = dim(U ) +
∩
{(1, 0), (0, 1)} {(2, 3), (1, 1)}
R2 2
2
{x , x, 1} {x , 3x + 4, 4} u ∗ v 1 1 u ∗ v = u + v 2 2
P 2 [x] u
∗ ∗
∗ ∗
(u v ) w = u (v w)
E u = w
E = F (R; R)
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v
.
X, Y F =
⊂R
f : R
→R g : R → R
X G = Y F G E = F + G F G = 0
∩
E = F (R; R)
∩ ∪
{}
E = F (R; R) f : R [0, 1] g:R [2, 3]
F 1 = F 2 = F 1
→R →R
F 2
u = (1, 2) R2 =
∅
X Y = X Y = R
v = ( 1, 2)
−
L L 1
R2
2
u
v
L ⊕L 1
2
E
G G
x y
∈ G
y 2 = x O(3)
S (3)
A
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x2 = x, x A
∀ ∈
A
∈ G
.
∈
D
a=0 f (x) = a x
A D f : D
→D
F (R) (f + g )(x) = f (x) + g(x) (F (R), +, )
→
R f : R (g f )(x) = g (f (x))
·
·
·
Z A = Z[i] = a + bi/a, b i2 = 1 a + bi = c + di a = c b = d (a + bi) + (c + di) = (a + c)+(b + d)i (a + bi) (c + di) = (ac bd)+(ad + bc)i Z[i] = (Z[i], +, ) m p A = Q : M CD( p.n) = 1 n A
⇔
∧
{B } ∈ B= B i
i
∈
i
{
i
a
A x a = 0
a
· }
} { }∈ B= i
i
B = x
∈ A ∈ A
{ ∈ A :
A
{ ∈ A :
B = x A
0
i
∈
i
0
{ }∈ ⊂ ⊂ ⊂ ·· · i
i
0
0
1
B=
2
i
∈
0
A A
{I } ∈ I ⊂ I ⊂ I ⊂ · ·· i
0
1
i
∈
A
A x a = a x
·
− −
0
0
·
∈ } · ·
A
0
2
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I =
∈ I i
i
0
i
.
I J I
·
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A
J
A
A