C
2 × 2
N, Z, Q, R, C
N0
m " M m M
Ä…"
‚"
f ć% g
M × N M N
f : M N M N
{x1, x2, . . . , xn} x1 x2 xn
(x1, x2, . . . , xn) x1 x2 xn
Sn
An
Mat (K) K
Mat (K) n × m K
n×m
E
diag(1, 2, . . . , n)
det A A
-
AB A B
V(E3), V(E2)
ab
a × b
a '" b
(a, b, c)
|x|
 z
|AB| AB
" x x2 + 1 > 0
2
" x dx = 2
0
" 1000
A B
" A B
" A A
" A B
" A Ò! B A B
" A Ô! B A B A
B
Ò! Ô!
A x x2 - 3x + 2 > 0
x -1
A -1 A 3/2
A x x
A x
" x A x
" x A x
" "x : A x
" "x Ò! A x
" "
x x1 x2 xn "x : A x
A x1 A x2 . . . A xn
"x Ò! A x
A x1 A x2 . . . A xn
A
B
A B
A B
A
A Ò! B
A Ô! B
Ò! 1800
Ò! 1000
A A Ò! B B
"n Ò! S(n) = 0

"n, m(S(n) = S(m) Ò! n = m)
A x
x A 0 n
A n A S(n) A n
n
S
 A1 A2
A3, . . . Ai
Ak Ak Ò! Ai k < i
P1, P2, P3, . . . n
" P1
" n Pn Pn+1
Pn
n(n + 1)
1 + 2 + 3 + . . . + n =
2
 n = 1
n
n(n + 1) (n + 1)((n + 1) + 1)
1 + 2 + 3 + . . . + n + (n + 1) = + n + 1 =
2 2
n + 1
P1 n > 1
Pk k < n Pn Pn
n > 1
n = 2
k > 1 n n
n n
n = mk 1 < m < n 1 < k < n
m k n
m + 1 := S(m) m + (n + 1) := (m + n) + 1 n " N;
m · 1 := m m · (n + 1) := m · n + m n " N.
0! := 1 (n + 1)! := n!(n + 1) n " N.
m
Cn m n 0 d" m d" n
0 m m-1 m
Cn = 1; Cn+1 = Cn + Cn .
m n!
Cn =
m!(n-m)!
P P
B B P
P Ò! (B ( B)) (B ( B)) Ò! ( P)
B ( B) Ò! P
B B Ò! P
B B B Ò! B
P
P
P p
m = p ! + 1
B q q d" p
q p ! m B ( B)
P
( P) P =
P
H1, H2, . . . , Hn
H1 . . . Hn
Hk
P P
(A B)
A B (A B) A B)
A B
(A B) A B
A B A B ( A B)
A B (A B)
( ( A B))
P ( P) P
"k(Hk Ò! P), "k(( Hk) P)
(( H1) P) . . . (( Hn) P).
( (H1 . . . Hn)) P
H1 . . . Hn Ò! P
H1 . . . Hn
P
n(n + 1)(n + 2) n
n = 6m Ò! 6|n Ò! 6|A
n = 6m + 1 Ò! 2|n + 1 3|n + 2 Ò! 6|A
n = 6m + 2 Ò! 2|n 3|n + 1 Ò! 6|A
n = 6m + 3 Ò! 3|n 2|n + 1 Ò! 6|A
n = 6m + 4 Ò! 6|n + 2 Ò! 6|A
n = 6m + 5 Ò! 6|n + 1 Ò! 6|A
Hk
"
x2 + x - 2 > x
(-", -2]*"[1, +")
x d" -2 x e" 1
x
x2 + x - 2 > x2 x > 2 (-"; -2] *" (2, +")
x M x " M
x M
 x " M M N
/
M = N Ô! "x(x " M Ô! x " N).
"
"x Ò! x " ".
/
M P(M)
N M
N M
N Ä…" M Ô! "x (x " N Ò! x " M).
" M N M *" N
M N
" M N M )" N
M N
" M N N
M M \ N
M N
(m, n)
(m, n) = (m , n ) Ô! m = m n = n .
(m, n, k) = ((m, n), k), (m, n, k, q) = ((m, n, k), q), . . .
M × N M N
(m, n) m M n N
U x " U
A x {x " U | A x }
R M
a, b " M a
R b aRb a R
b
R M
N (M, N, G) G Ä…" M × N
G R mRn
(m, n) " G
M N = M
R aRa a " M
aRb bRc aRc
R a
R b b R a a = b
d"
a, b " M
a d" b b d" a d"
M, d"
R
aRb bRa
<"
 <"  M
m " M [m]
b " M m <" b M
[m] M = *"m"M[m] [m] [m ]
M = *"i"IXi
M
m H" m Ô! "i " I : m, m " Xi
 [m] [m ] n
m <" n m <" n n <" m <" x " [m]
m <" x
m <" n, n <" m, m <" x m <" x
x " [m ] [m] Ä…" [m ]
[m] = [m ]
m " [m]
H" m H" n n H" k
i j m, n " Xi n, k " Xj n
Xi Xj Xi = Xj
m, n, k Xi
m H" k
f M N f : M N
x " M
f(x) " N M
N x f(x)
f x x f
f : M N (M, N, G)
m " M n " N (m, n) " G
n = f(m) f
f
"x1, x2 " M (f(x1) = f(x2) Ò! x1 = x2).
f N
y " N x f(x) = y
x " M x IdM : M M
f : M N g : N U
g ć% f : M U g ć% f(x) = g(f(x)) x " M
f g
h : U T
h ć% (g ć% f) = (h ć% g) ć% f.
 m "
M h(g(f(x)))
g ć% f = g ć% f f g = g
g ć% f = g ć% f g f = f
f : M N
f-1 : N M
f ć% f-1 = IdN f-1 ć% f = IdM
f g : N M
n " N m = g(n) f(m) = n
f ć% g = IdN g ć% f = IdM g = f-1
f ć% g = IdN g ć% f = IdM g : N M
f(m) = f(m ) m = g(f(m)) = g(f(m )) = m
n " N m := g(n) N
f(m) = n f
a, b " M
a " b "
· ć% M
M2 M M2 = M ×M
M M M
{"} M
M
Z
N Ä…" M
" a, b " N
a " b N
" M
a " (b " c) = (a " b) " c
 a, b, c " M
a " b = b " a.
a, b " M e " M
"
a " e = e " a = a
a " M " e = 1
" e
R
[A, B] = AB - BA
M "
M
e
x
e, x, x2 = xx, x3 = xxx, . . .
e f ef = e f
ef = f e e = f
(M, ", e) b " M a " M
a"b = b"a = e b = a-1 a
" a-1 -a -a
(Z, +, 0)
(Z, ·, 1) Ä…1
(M, ·, e)
e e-1 = e
a-1 (a-1)-1 = a
a, b " M ab
(ab)-1 = b-1a-1
 b1, b2 a "
M b1 = b1e = b1(ab2) = (b1a)b2 = eb2 = b2,
(ab)(b-1a-1) = ((ab)b-1)a-1 = (a(bb-1))a-1 = (ae)a-1 = e.
e
(R, +, ·, 0, 1, d")
·
M N
M N M N
R M N
RN
a, b " N, aRNb Ô! aRb.
2Z
M = (Z, +, ·, 0, d" ) (Z, ·, 1)
1 + 2Z (Z, ·, 1) (Z, +)
M
M N
M × N
(m, n) " (m , n ) = (m " m , n " n ), (m, n)u = (mu, nu).
n
 M
 <"  M
M m1 <" m 1 m2 <" m 2
M m1 " m2 <" m 1 " m 2 " (m1)u <" (m 1)u
()u M/ <"
[m1] " [m2] = [m1 " m2] [m]u = [mu].
+ = + =
(Z, +, 0)
z1 <" z2 z1 z2
{ }
Z
M N
f : M N
a, b " M
f(a " b) = f(a) " f(b)
"
aRb, f(a)Rf(b)
R
f(au) = (f(u))u f
M N f aRb Ô! f(a)Rf(b)
R a, b " M
 G "
"
e " G e " g = g " e = g
g " G
g " G g " G g " g = g " g = e
" e
1G g
g-1
"
e g
g -g
(G, ·)
ax = b xa = b x = a-1b
x = ba-1 G x + a = b
x = b - a b + (-a)
G ·
(Z, +, 0)
(R, +, 0)
{e}
ee = e
{Ä…1} -1
D
SE(D) D
SE(D)
SE(D) D
-1
·
e S G
e " S a, b " S ab
a-1 S
G
-1
(G, ·, , e) a " G a
a0 = e, an = a · a · . . . · a (n , a-n = (a-1)n (n " N).
z1, z2
1 2 1 1 2 1
az · az = az +z2, (az )z = az z2.
a G
a a
a
n an = e
1 2
az = az z1 = z2

. . . , a-2, a-1, e, a, a2, a3, . . .
a
a ord(a) = "
n " N an = e n
e, a, a2, . . . , an-1
a = {e, a, a2, . . . , an-1}.
z " Z z = nq + r 0 d" r d" n - 1 az = ar
a n ord(a) = n
a n
G G |G|
"
SE(")
0, 120
240 Id, r120, r240
sa, sb, sc r120sa = sar120

 A B C H
2 3
SE(") r240 = r120, r120 = Id
r120 H
SE(")
G
(Z, +, 0)
Sn {1, 2, . . . , n} {1, 2, . . . , n}
Sn
Id
Sn n
Ä " Sn
1 2 3 . . . n
Ä =
i1 i2 i3 . . . in
Ä = (i1, i2, i3, . . . , in) i1, i2, i3, . . . , in
n Ä(j) = ij n!
Sn n!
i j
(ij) Sn n(n-1)/2
Sn
Ä
Ä t1
n n
t1, t2, . . . , tm tmtm-1 . . . t1Ä = e e Ä =
t-1t-1 . . . t-1 t-1 = t t
1 2 m
 Ä = (i1, i2, . . . , in) sgn Ä " {Ä…1}
Ä k, l 1 d" k < l d" n
ik > il Ä sgn Ä = 1
Ä
Ä sgn Ä = -1
Ä t = (jk)
|k - j|
t = (j j + 1)
Ä = (i1, i2, . . . , in)
Ät = (i1, . . . , ij+1, ij, . . . , in) j, j + 1
Ä ij > ij+1 Ät
ij < ij+1 j, j + 1 Ät Ä
j , k j < k j = j

k = j + 1 Ä

Ät
t |k - j| < N
|k - j| = N > 1 (jk) = (kj)
j < k (j, k) = (j j + 1)(j + 1 k)(j j + 1) Ät = ((Ä(j j +
1))(j + 1 k))(j j + 1)
Ä sgn(Ät) =
- sgn Ä
Ä
Ä = t1t2 . . . tk Ä
k
sgn(e) = 1 e
(. . . (e · t1)t2) . . .)tk = Ä
Ä
sgn Ä = (-1)k
n > 1 sgn : Sn {Ä…1} Sn
{Ä…1}
An An
n!/2
 Ä Á k l
ÄÁ k+l
sgn ÄÁ = (-1)k+l = (-1)k(-)l = sgn Ä · sgn Á sgn
sgn e = 1 sgn(12) = -1 sgn
t tAn
Sn = An *" tAn tAn
An Ä tÄ An tAn
An Sn n!/2
(K, +, ·)
(K, +)
(K, ·)
a(b + c) = ab +
ac (b + c)a = ba + ca a, b, c " K
K 1a = a1 = a a " K K
ab = ba a, b " K K
R
F(M) = {f : M R}
F[a, b]
[a, b] C[a, b]
C1[a, b]
R[x]
n
Z = nZ *" (1 + nZ) *" . . . *" (n - 1 + nZ).
0, 1, 2, . . . , n - 1 n m " Z
m = nq + r m + nZ = r + nZ m
n Zn = {0, 1, . . . , n - 1}
a + b = a + b; ab = ab
a, b " Z
Zn 0 1
2100 Z7 23 = 1 2100 = 299 · 2 = 2 2100
k K kb = 0 bk = 0
b " K k kn = 0
n u " K u
(K, ·, 1) d " K ud =
du = 1
R(x)
Zn n
n = k · m k m n
k · m = 0 k m Zn
n m " Zn
m n a, b ma+nb = 1
ma = 1 m
Zn Zn
 T P P
T 1P " T a, b " T a + b, a · b, -a a-1
a = 0 T

x2 = 2
"
" Q R Q[ 2] =
Q + 2Q x2 + 1 = 0
-1
" R
R -1
z = x + iy x y
i
"
i = -1 i2 = -1
x z
Re z y z
Im z z
x + iy = x + iy x = x
y = y x + i0 x
0 + iy iy
z
(x, y)
C
z1 = x1 + iy1 z2 = x2 + iy2
z1 + z2 = (x1 + x2) + i(y1 + y2);
z1z2 = (x1x2 - y1y2) + i(x1y2 + x2y1).
(C, +)
0 = 0+i0 1 = 1+i0
z = x+iy -z = (-x)+i(-y)
x
z z-1 = + ix -y
x2+y2 2+y2
(C, +)
(x + iy)(1 + i0) = (x · 1 - y · 0) + i(x · 0 + y · 1) = x + iy
 1 + i0
z = x + iy
x = 0 y = 0 x2 + y2

x -y
(x + iy)( + i ) =
x2 + y2 x2 + y2
x2 y(-y) x(-y) yx
= - + i + = 1 + 0i
x2 + y2 x2 + y2 x2 + y2 x2 + y2
R C x
x + i0
(x1 + i0) + (x2 + i0) = (x1 + x2) + i0; (x1 + i0)(x2 + i0) = x1x2 + i0.
x1 x2
x x+i0
iy 0 + iy
i z2 + 1 = 0
i2 = (0 + 1i)(0 + 1i) = (0 · 0 - 1 · 1) + (0 · 1 + 1 · 0)i = -1 + 0i = -1
C C z = x + iy
z = x - iy
z = x + iy, z1, z2
z = z Ô! z " R
 z1 + z2 = z1 + z2 z1z2 = z1z2
z = z
zz = x2 + y2
C
z = x+iy
Õ
z z arg z
2Ä„ (0, 2Ä„]
x2 + y2 z
|z|
z
"
z = x + 0i |z| = x2 = |x|
z z1 z2
|z1z2| = |z1||z2|
|z1/z2| = |z1|/|z2|
|z1 + z2| d" |z1| + |z2|
zz = |z|2
|z| = 0 z = 0
z = x + iy r Õ
x = r cos Õ
z = r(cos Õ + i sin Õ)
y = r sin Õ
z.
cos Õ + i sin Õ eiÕ. z = reiÕ.
ez exp(z)
ex(cos y+i sin y) z = x
 ez = ex
z z1 z2
Õ, Õ1, Õ2 m
1 1 2 1 1 2
ez +z2 = ez ez ei(Õ +Õ2) = eiÕ eiÕ
1-z2 1 2
1-Õ2)
1 2
ez = ez /ez ei(Õ = eiÕ /eiÕ
(ez)m = emz
1
e-z =
ez
|eiÕ| = 1 eiÕ
ei(Õ+2Ä„) = eiÕ eiÕ
Õ 2Ä„
1 2
eiÕ eiÕ = (cos Õ1 + i sin Õ1)(cos Õ2 + i sin Õ2) =
= (cos Õ1 cos Õ - sin Õ1 sin Õ2) + i(sin Õ1 cos Õ2 + cos Õ1 sin Õ2) =
1
= cos(Õ1 + Õ2) + i sin(Õ1 + Õ2) = ei(Õ +Õ2).
1 1 2 2 1-z2
ez +z2 = ez ez ez · ez =
2 1
ez +z1-z2 = ez
cos x sin x
Õ " R n " Z
(cos Õ + i sin Õ)n = cos nÕ + i sin nÕ.
(cos Õ + i sin Õ)n = (eiÕ)n = einÕ = cos nÕ + i sin nÕ
 C
"
D = reiÕ r, Õ " R; r > 0 Ä… reiÕ/2
z2 = D
2
z1, z2 z1 =
2
z2 = D z2-D = (z-z1)(z+z1) z1,2
z2 = D
az2 + bz + c = 0 (a, b, c " C ; a = 0)

D = b2 - 4ac
2
b D
z + =
2a 4a2
"
Ä… D D
"
-b Ä… D
z1,2 = .
2a
 K
K = R
y(x) = kx k " K
x1, x2 
y(x1 + x2) = y(x1) + y(x2)
y(x) = y(x)
y(x)
y(x) = kx k " K
R y = |x|
y = x/Ä„ y = ex y = 0
x
ax = b
a b K
K
x
a = 0

ax = b Ô! a-1(ax) = a-1b Ô! (a-1a)x = a-1b Ô! 1 · x = a-1b Ô! x = a-1b
x = b/a
a-1
n n
a = 0 b = 0

a = b = 0 K
a = 0, b = 0

a = 0

K a = b = 0
x y
ax + by = c
K = R
Oxy
a = b = 0 c = 0 a = b = c = 0

K2
a b
K = R
b = 0

c
y = -ax-
b b
c
b = 0 a = 0 x = -a

Oy
c
b = 0 (x, -ax - ) | x " K x

b b
c
K b = 0 (-a, y) | y " K
y
2 × 2
x y
a1x + b1y = c1
a2x + b2y = c2
(x, y)
y
b2 b1
(a1b2 - a2b1)x = c1b2 - c2b1
(x, y)
a1b2 - a2b1 2 × 2
a1 b1
a1b2 - a2b1 = .
a2 b2
"
c1 b1
"x = c1b2 - c2b1 = .
c2 b2
a1 a2
"y = c2a1 - c1a2 = .
c2 c2
y "·y = "y
"y = "y
 " = 0 " · y = "y

"x "y
x = , y = .
" "
2 × 2
"x "y 1
a1x + b1y = a1 + b1 = (a1(c1b2 - c2b1) + b1(c2a1 - c1a2)) =
" " "
1 1
= (a1c1b2 - b1c1a2) = (a1b2 - b1a2)c1 = c1.
" "
"x "y
,
" "
" = 0 "x = 0 "y = 0

"x = 0 "·y = "y "y = 0

" = "x = "y = 0
"y = "y
a1 = b1 = 0 c1 = 0

a1 = b1 = c1 = 0
a1 b1 a2 b2
a2 b2 b2 c2
" = 0 "x = 0 = =
a1 b1 b1 c1
a2 b2 c2
a1 = 0 b1 = 0  = = =

a1 b1 c1
 a2 = a1, b2 = b1 c2 = c1
-
a1x + b1y = c1;
.
0 · x + 0 · y = 0
1

 a1x+
b1y = c1
2×2
" = 0 " = 0

K2
K2
2 × 2
a1 b1 a2 b2
K = R
(x, y) P (x, y) Oxy
a1x + b1y = c1 Oxy
: a1x+b1y = c1 : a2x+b2y =
1 2
c2
1 2
Ô! " = 0

1 2
" = 0 "y = 0 "x = 0

1 2
m n x1, x2, . . . , xn
Å„Å‚
= b1
ôÅ‚
ôÅ‚a11x1 + a12x2 + . . . + a1nxn
ôÅ‚
òÅ‚a x1 + a22x2 + . . . + a2nxn
= b2
21
ôÅ‚. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ôÅ‚
ôÅ‚
ół
am1x1 + am2x2 + . . . + amnxn = bm
aij " K i j
b1, b2, . . . , bm
n
x1, x2, . . . , xn
 i
j 
i j 
i j -
k k-1
1 2
 n
n
a11, a21, . . . , am1
aij = 0 bj = 0
a1j = 0 j

a11 = 0 j
(1, 1)
a11 = 0

21
-a
a11
(2, 1)
(3, 1), . . . , (m, 1)
Å„Å‚
ôÅ‚
ôÅ‚a11x1 + a12x2 + a13x3 + . . . + a1nxn = b1
ôÅ‚
òÅ‚
a 22x2 + a 23x3 + . . . + a 2nxn = b 2
ôÅ‚
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ôÅ‚
ôÅ‚
ół
a m2x2 + a m3x3 + . . . + a mnxn = b m
C k
n×n
d" Cnk
aij
Å„Å‚
ôÅ‚
ôÅ‚a11x1 + akx2 + . . . + a1nxn = b1
ôÅ‚
òÅ‚
a2kxk + . . . + a2nxn = b2
ôÅ‚
. . . . . . . . . . . . . . . . . . . . . . . .
ôÅ‚
ôÅ‚
ół
ampxp + . . . = bm
x1, xk, . . . , xp
 1 < k < . . . < p
0 · x1 + 0 · x2 + . . . + 0 · xn = 0
0 · x1 + 0 · x2 + . . . + 0 · xn = b,
b = 0

m d" n n
e" 1 e" m
m n
m d" n
xp
xp = bm/amp - amp+1/ampxp+1 - . . . - amn/ampxn
p = n xn = bn/ann
amp
3 × 3
a1 = 0, b2 = 0, c3 = 0

Å„Å‚ Å„Å‚
ôÅ‚
òÅ‚a1x + by + c1z = d1 ôÅ‚ + b1y = d1 - d3c1/c3
òÅ‚a1x
Ô!
b2y + c2z = d2 b2y = d2 - d3c2/c3 Ô!
ôÅ‚ ôÅ‚
ółc z
= d3 ółz = d3/c3
3
Å„Å‚
ôÅ‚
òÅ‚a1x = d1 - d3c1/c3 - d2b1/b2 - d3c2b1/(c3b2)
Ô! y = d2/b2 - d3c2/(c3b2)
ôÅ‚
ółz = d3/c3
Å„Å‚
ôÅ‚
òÅ‚x = d1/a1 - d3c1/(c3a1) - d2b1/(b2a1) - d3c2b1/(c3b2a1)
y = d2/b2 - d3c2/(c3b2)
ôÅ‚
ółz = d3/c3
m = n
K
K
3×3
Oxyz Å‚
A(1, 0, 0), B(0, 1, 0) C(0, 0, 1) 2 × 3
Å‚
(0, 0, . . . , 0)
 x1, x2, . . . , xn
aij b1, b2, . . . , bn
m × n m × n m × n
K
ëÅ‚ öÅ‚
a11 a12 a13 . . . a1n
ìÅ‚
a21 a22 a23 . . . a2n ÷Å‚
ìÅ‚ ÷Å‚
ìÅ‚
a31 a32 a33 . . . a3n ÷Å‚ ,
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚
am1 am2 am3 . . . amn
aij " K m×n
{1, 2, . . . , m} × {1, 2, . . . , n} K aij
(i, j) A, B, C
A = (aij)m×n i
m j n i
m × n A A = (aij)
n × n aij = (-1)i+j aij = min{i; j}
aij = max{i; j}
m × n k
aij
(i, j) aij (i, j) A
ëÅ‚ öÅ‚
a1j
ìÅ‚ ÷Å‚
a2j
ìÅ‚ ÷Å‚
(ai1, ai2, . . . , ain);
ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚
amj
i j
A
n m
 1×1
K
K Mat (K)
Mat (Z)
m × n Mat (K)
m×n
"
Mat (K) = Mat (K).
m×n
m,n=1
Mat (K)
A = (aij) B = (bij) m × n r " K
A B m × n A + B (i, j)
aij + bij
A r " K
Ar = rA = (raij) Ar rA
m × n (i, j) raij
Ar rA
A + (B + C) = A + (B + C)
A, B, C
A + B = B + A A, B " Mat (K)
m×n
A +0 = 0+A = A
A
A
-A = (-aij) = (-1) · A
A A + (-A) = 0.
 (rs)A = r(sA) r, s " K
A " Mat (K) A(sr) a" (As)r
r(A + B) a" rA + rB (r + s)A a" rA + sA
Mat (K)
m×n
rA + sX = C
A, C m × n r, s " K K
X " Mat (K)
m×n
A + (-B) A - B
A B
A, B - A - B
m × n A m × n
A (i, j) (j, i) A
ëÅ‚ öÅ‚
a11 a21 a31 . . . am1
ìÅ‚a12 a22 a32 . . . am2÷Å‚
ìÅ‚ ÷Å‚
A = .
ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚
a1n a2n a3n . . . amn
A A
(A + B) = A + B ; (rA) = rA ; (A ) = A
A, B " Mat (K) r " K
m×n
 a11, a22, a33, . . .
A A = A
A = -A
Mat (R) n2
n×n
n × n
OX ax OY ay
OZ az
OX OY OZ
Fx Fy Fz Fx Fy Fz
ax ay az
ëÅ‚ öÅ‚
ax
íÅ‚ayÅ‚Å‚
W = Fxax + Fyay + Fzaz = (Fx, Fy, Fz)
az
ëÅ‚ öÅ‚
b1
ìÅ‚b2÷Å‚
ìÅ‚ ÷Å‚
(a1, a2, . . . , an) = a1b1 + a2b2 + . . . + anbn
ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚
bn
n n
m × n A = (aij) n × k
B = (bij) = n A
AB D = (dij) m×k (i, j)
i A j B
ëÅ‚ öÅ‚
b1j
ìÅ‚ ÷Å‚
dij = ai1b1j + ai2b2j + . . . + ainbnj = (ai1 + . . . + ain) .
íÅ‚ Å‚Å‚
bnj
A
B C m × n n × k k × l
A(BC) = (AB)C.
BC AB
m × l
D = (dij) m × k AB F = (fij) m × l (AB)C
Ä…=k Ä…=k ²=n
fij = diÄ…cÄ…j = ( ai²b²Ä…)cÄ…j = ai²b²Ä…cÄ…j
1d"²d"n
Ä…=1 Ä…=1 ²=1
1d"Ä…d"k
M = (mij) n × l BC Q = (qij) m × l
A(BC)
²=n ²=n Ä…=k
gij = ai²m²j = ai² bÄ…²cÄ…j = ai²b²Ä…cÄ…j
1d"Ä…d"k
²=1 ²=1 Ä…=1
1d"²d"n
fij = gij (i, j) F = Q
A(BC) = (AB)C
m × n A B C
n × k
A(B + C) = AB + AC.
A k × m B C
(B + C)A = BA + CA.
AB BA
AB = BA
b1 b1 b1a1 b1a2
(a1, a2) = a1b1 + a2b2; · (a1a2) =
b2 b2 b2a1 b2a2
AB BA
AB = BA AB = BA

 AB BA
2 1 1 0 3 2 1 0 2 1 2 1
= = = .

0 1 1 2 1 2 1 2 0 1 2 3
Mat (K) n × n
n×n
Z
n × n
1 · z = z · 1 = z
Mat (K) n×n En
n×n
EmA = AEn = A A "
Mat (K)
m×n
(i, j) ´ij
1, i = j;
´ij =
0, i = j.

(i, j) EmA cij
cij = ´i1a1j + ´i2a2j + . . . + ´imamj = 0 · a1j + . . . + 1 · aij + . . . + 0 · amj = aij
EmA = A AEn = A
A = (aij) " Mat (R)
n×n
A A
A A
n × n
ëÅ‚ öÅ‚
1 0 . . . 0
ìÅ‚ ÷Å‚
0 2 . . . 0
ìÅ‚ ÷Å‚
diag(1, 2, . . . , n) =
íÅ‚. . . . . . . . . . . .Å‚Å‚
0 0 . . . n
 n×n
Mat (K)
n×n
A B
(AB) = B A
B A
A An = 0
n n
n × n
ëÅ‚ öÅ‚
0 a12 a13 . . . a1n
ìÅ‚0 0 a23 . . . a2n÷Å‚
ìÅ‚ ÷Å‚
A = ,
ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚
0 0 0 . . . 0
An = 0
2 × 2 3 × 3
Å„Å‚
ôÅ‚
òÅ‚a1x + b1y + c1z = d1;
a2x + b2y + c2z = d2;
ôÅ‚
óła x + b2y + c2z = d2.
2
b2 c2 b1 c1 b1 c1
"11 = ; "21 = ; "31 = .
b3 c3 b3 c3 b2 c2
"11 -"21
"31
(a1"11 - a2"21 + a3"31)x + (b1"11 - b2"21 + b3"31)y +
+(c1"11 - c2"21 + c3"31)z = (d1"11 - d2"21 + d3"31).
b1"11-b2"21+b3"31 = 0 c1"11-c2"21+c3"31 =
0
(a1"11 - a2"21 + a3"31)x = (d1"11 - d2"21 + d3"31)
a1 b1 c1
b2 c2 b1 c1 b1 c1
a2 b2 c2 = a1 - b1 + c1 =
b3 c3 b3 c3 b2 c2
a3 b3 c3
= a1b2c3 + a3b2c2 + a2b3c1 - a1b3c2 - a2b1c3 + a3b2c1.
3 × 3
3 × 3
a1 b1 c1 d1 b1 c1
a2 b2 c2 x = d2 b2 c2 .
a3 b3 c3 d3 b3 c3
a1 b1 c1 a1 d1 c1 a1 b1 c1 a1 b1 d1
a2 b2 c2 y = a2 d2 c2 , a2 b2 c2 z = a2 b2 d2 .
a3 b3 c3 a3 d3 c3 a3 b3 c3 a3 b3 d3
a1 b1 c1
" = a2 b2 c2 ,
a3 b3 c3
d1 b1 c1 a1 d1 c1 a1 b1 d1
"x = d2 b2 c2 , "y = a2 d2 c2 , "z = a2 b2 d2 .
d3 b3 c3 a3 d3 c3 a3 b3 d3
" = 0

"x "y "z
x = , y = , z = .
" " "
3 × 3
3 × 3 (aij)
a11 a12 a13
a21 a22 a23 = a11a22a33 + a12a23a31 + a13a21a32-
a31 a32 a33
-a13a22a31 - a11a23a32 - a12a21a33.
S3 S3
2 × 2
a11 a12
= a11a22 - a12a21.
a21 a22
A = (aij) n × n
A K
det A = sgn à a1Ã(1)a2Ã(2) . . . anÃ(n).
Ã"Sn
F (x1, x2, . . . , xn) n xi " Kn F
, µ " K i
F (. . . x i + µx i . . .) = F (. . . x i . . .) + µF (. . . x i . . .)
= i

 f(x) = f(x1, . . . , xn)
x = (x1, x2, . . . , xn) B1, B2, . . . , Bn
f(x) = B1x1 + B2x2 + . . . + Bnxn
F
F (. . . xi . . . xj . . .) = -F (. . . xj . . . xi . . .)
K
F (. . . xi . . . xj . . .) = 0 xi = xj K
K = Z2
0 = F (. . . xi + xj . . . xi + xj . . .) =
= F (. . . xi . . . xi . . .) + F (. . . xi . . . xj . . .) + F (. . . xj . . . xi . . .)+
+F (. . . xj . . . xj . . .) = F (. . . xi . . . xj . . .) + F (. . . xj . . . xi . . .)
a 11 + µa 11 a 12 + µa 12 . . . a 1n + µa 1n
a21 a22 . . . a2n
=
. . . . . . . . . . . .
an1 an2 . . . ann
= sgn à (a 1Ã(1) + µa 1Ã(1))a2Ã(2) . . . anÃ(n) =
Ã"Sn
=  sgn à a 1Ã(1)a2Ã(2) . . . anÃ(n) + µ sgn à a 1Ã(1))a2Ã(2) . . . anÃ(n) =
Ã"Sn Ã"Sn
a 11 a 12 . . . a 1n a 11 a 12 . . . a 1n
a21 a22 . . . a2n a21 a22 . . . a2n
=  + µ .
. . . . . . . . . . . . . . . . . .
an1 an2 . . . ann an1 an2 . . . ann
A = (aij) t s
B = (bij)
Å„Å‚
ôÅ‚
òÅ‚aij, i = t, i = s;
bij = asj, i = t;
ôÅ‚
óła , i = s.
tj
Ä (ts)
t < s
det B = sgn à b1Ã(1) . . . btÃ(t) . . . bsÃ(s) . . . bnÃ(n) =
Ã"Sn
= sgn(ÃÄ) b1ÃÄ (1) . . . btÃÄ(t) . . . bsÃÄ (s) . . . bnÃÄ (n) =
Ã"Sn
= sgn à sgn Ä b1Ã(1) . . . btÃ(s) . . . bsÃ(t) . . . bnÃ(n) =
Ã"Sn
= - sgn à a1Ã(1) . . . asÃ(s) . . . atÃ(t) . . . anÃ(n) = - det A,
Ã"Sn
det E = 1
A = (aij) n×n a1Ã(1)a2Ã(2) . . . anÃ(n) =
0 i Ã(i) < i
i Ã(i) e" i Ã(n) = n
Ã(n - 1) = n - 1 Ã(1) = 1
det A = a11a22 . . . ann
D : Mat (K) K D(A) = D(E) det A
n×n
A D(E) = 1 D(A) = det A
A
n × n eij (i, j)
n × n A = (aij)
A = aijeij
1d"i,jd"n
D(A) = a1j . . . anj D(e1j + . . . + enj ).
1 n 1 n
1d"j1,...,jnd"n
 D(e1j + . . . + enj ) = 0 i = i ji = ji

1 n
j1, j2, . . . , jn
D(e1j + . . . enj ) = sgn(j1, j2, . . . , jn)D(e11 + . . . + enn) =
1 n
= sgn(j1, j2, . . . , jn)D(E),
(j1, j2, . . . , jn) Ã
D(A) = D(E) sgn à a1Ã(1) . . . anÃ(n) = D(E) det A.
Ã"Sn
sgn à = sgn Ã-1 Ã
B = (bij)
n × n A
det A = sgn à a1Ã(1)a2Ã(2) . . . anÃ(n) =
Ã"Sn
= sgn à aÃ-1 aÃ-1 . . . aÃ-1 =
(1)1 (2)2 (n)n
Ã"Sn
= sgn µ aµ(1)1aµ(2)2 . . . aµ(n)n = sgn µ b1µ(1)b2µ(2) . . . bnµ(n) = det B
µ"Sn µ"Sn
det A = det A
F (. . . xi + xj . . . xj . . .) = F (. . . xi . . . xj . . .) + F (. . . xj . . . xj . . .) =
= F (. . . xi . . . xj . . .)
F
(i, j) A
A i j
Mij (i, j) A
Aij = (-1)i+jMij
j i
det A = a1jA1j + a2jA2j + . . . + anjAnj;
det A = ai1Ai1 + ai2Ai2 + . . . + ainAin.
D(A) = a1jA1j + a2jA2j + . . . + anjAnj
D(E) = 1
r r r = i

r = j

0 = ar1Ai1 + ar2Ai2 + . . . + arnAin,
0 = a1rA1j + a2rA2j + . . . + anrAnj.
A det A = 0

det(AB) = det A det B n × n A B
 B D(A) = det AB
det AB = D(A) = det A · D(E) = det A · det B
A, B
A C
= det A · det B
0 B
C
A 0
= det A · det B
D B
D
A B
A C E C
= det A · det B = det A · det B.
0 B 0 E
x1, . . . , xn
1 x1 x2 . . . xn-1
1 1
1 x2 x2 . . . xn-1
2 2
= (xj - xi)
. . . . . . . . . . . . . . .
1d"i1 xn x2 . . . xn-1
n n
(x2 - x1)(x3 - x1) . . . (xn - x1) · (x3 - x2) . . . (xn - x2) . . . (xn - xn-1)
n(n-1)
2
x1
(n - 1) × (n - 1)
x2 - x1 x2(x2 - x1) . . . xn-2(x2 - x1)
2
x3 - x1 x3(x3 - x1) . . . xn-2(x3 - x1)
3
. . . . . . . . . . . .
xn - x1 xn(xn - x1) . . . xn-2(xn - x1)
n
(x2 - x1)(x3 - x1) . . . (xn - x1)
Å„Å‚
ôÅ‚
ôÅ‚a11x1 + a12x2 + . . . + a1nxn = b1
ôÅ‚
òÅ‚a x1 + a22x2 + . . . + a2nxn = b2
21
ôÅ‚. . . . . . . . . . . . . . .
. . .
ôÅ‚
ôÅ‚
ół
an1x1 + an2x2 + . . . + annxn = bn
n n
A
det A1 det A2 det An
x1 = , x2 = , . . . , xn = ,
det A det A det A
Ai A i
(b1, b2, . . . , bn)
det A = 0

det A = 0 det A = 0

k, 1 d" k d" n i
Aik aik A
i = 1, 2, . . . , n
n n
A aijAik = 0 j, j = k aikAik =

i=1 i=1
det A
n
det A · xk = bikAik = det Ak
i=1
 det Ak k
xk = det Ak/ det A
det A = 0 det Ak = 0 k

b1 = b2 = . . . = bn = 0
det A = 0
n × n D
n × n A AD = DA = E.
A-1
A A-1 (A-1)-1 = A
A B AB (AB)-1 =
B-1A-1
n × n A
A
ëÅ‚ öÅ‚
A11 A21 . . . An1
ìÅ‚A12 A22 . . . An2÷Å‚
1
ìÅ‚ ÷Å‚
A-1 = ,
íÅ‚ Å‚Å‚
. . . . . . . . . . . .
det A
A1n A2n . . . Ann
Aij (i, j) A
A AB
 det A = 0

A
det A-1 = 1/ det A
A
A ·
A-1 = E det A det A-1 = 1
GL(n, K) n × n
det : GL(n, K) K \ {0}
K
n n
A = (aij) B = (b1, b2, . . . , bn)
X = (x1, x2, . . . , xn)
AX = B
A-1
A
X = A-1B.
ax = b
AXB = C
X A B
E - AB
E - BA
A E-A
 AB
A
- -

AB AB AB
-
|AB|
-
A = B AB
-
- -
0 AB A B
-

a A B a = AB
-

AB a A
A B A(x1, y1, z1) B(x2, y2, z2)
X = x2 - x1, Y = y2 - y1, Z = z2 - z1
-
AB
x y z
a(X, Y, Z)
"
2
|a| = X2 + Y + Z2.
Ä…, ², Å‚ a OX, OY,
OZ cos Ä…, cos ², cos Å‚
a a
ao a
(X, Y, Z) ao
X Y Z
, , ,
|a| |a| |a|
a
X Y Z
cos Ä… = , cos ² = , cos Å‚ = .
|a| |a| |a|
X = |a| cos Ä…, Y = |a| cos ², Z = |a| cos Å‚.
-
a b a A a = AB b
- -
-
B a b = BC AC
a b
A
- - -
-
AB + BC = AC.
-
a b A a = AB
- -
-
b = AD ABD ABCD a + b = AC
A
-

a = AB  " R
A B D |AD| = |||AB|  > 0
D A B  < 0
D A B  = 0
a = 0
 V(E3)
a + b = b + a
a + (b + c) = (a + b) + c
0 + a = a
a -a
a + (-a) = 0
(µ)a = (µa)
( + µ)a = a + µa (a + b) = a + b
1 · a = a
a, b, c " V(E3)
 µ
- -
-
a A a = AB b B b = BC c
-
-
C c = CD
- - - - - -
- - - -
a + (b + c) = AB + (BC + CD) = AB + BD = AD,
- - - - - -
- - - -
(a + b) + c = (AB + BC) + CD = AC + CD = AD.
a + (b + c) = (a + b) + c
- - - -
a = AB -a = BA AB + BA =
-

AA = 0
|| |µ| |a|
(µ)a (µa)
 > 0 µ > 0  > 0 µ < 0  < 0 µ > 0  < 0 µ < 0
 = 0 µ = 0 (µ)a (µa) a
-a
 V(E3)
a b (X1, Y1, Z1) (X2, Y2, Z2)
a + b a (X1 + X2, Y1 + Y2, Z1 + Z2)
(X1, Y1, Z1)
OX, OY, OZ
i, j, k
a (X, Y, Z)
a = Xi + Y j + Zk V(E2)
E2
i j a(X, Y )
m × n F(X) X
f, g " F(X)
 " R
(f + g)(x) = f(x) + g(x); ( · f)(x) = f(x) (x " X).
L
K K
 (L, +)
0 " L
a " L -a a + (-a) = 0
(µ)a = (µa)
( + µ)a = a + µa (a + b) = a + b
1K · a = a
a, b, c " L , µ " K
R
Mat (K)
n×m
Mat (K) F(X, K)
n×m
K X
F(X)
n
Kn = Mat (K) K = Mat (K)
1×n n×1
n
0·a = 0 0·a = (0+0)·a = 0·a+0·a
0 · a = 0
a " L -a = (-1) · a
a + (-1)a = 1 · a + (-1)a = (1 + (-1)) · a = 0 · a = 0 Ò! (-1)a = -a.
 · 0 = 0
 " K
 · 0 = (0 · 0) = ( · 0)0 = 0 · 0 = 0,
 H L 0 "
H H
K
a, b " H,  " K Ò! a + b " H; a " H.
L
L
R2 R3
R2
R3
R[x] ‚" C1[a, b] ‚" C[a, b] ‚" R[a, b] ‚" F[a, b]
b
R[a, b] f(x) dx
a
C[a, b] [a, b] C1[a, b]
R[x]
"
" n × n
"
F(N)
"
" [a, b]
" C1[a, b]
[a, b]
a1, a2, . . . , an L 1, 2, . . . , n
K 1a1 + · · · + nan
a1, a2, . . . , an
i
 0 b " L
a1, a2, . . . , an b
L Ka1 + Ka2 + . . . + Kan
L
ai {ai}
L {ai}
b " L {ai}
{ai} i " I
b " Ka1 + Ka2 + . . . + Kan
ai
1a1 + 2a2 + · · · + nan = 0 1 = 2 = . . . = n = 0
a1, a2, . . . , an
a1,
a2, . . . , an
a, b " V(E3)
"
"
"
" a1, a2, . . . , am
f1, f2, . . . , fn m > n a1, a2, . . . , am
 1 · 0 + 0a2 + . . . + 0am = 0
ai = a1if1+. . .+anifn ai {fj} i = 1, 2, . . . m
1aj1 + 2aj2 + . . . + majm = 0 (j = 1, 2, . . . , n)
", . . . , " n
1 m
m
m m
"a1 + . . . + " am = ( "a1i)f1 + . . . + ( "ani)fn = 0 · f1 + . . . + 0 · fn = 0,
1 m i i
i=1 i=1
" {1, x, x2, . . .} R[x]
1 2 n
" ek x, ek x, . . . ek x F(R) ki
" {1, sin x, cos x, sin 2x, cos 2x, . . .} C[0, 2Ä„]
" {Eij} (i, j)
Mat (K)
n×m
" 3 × 3
Mat (K)
3×3
1, cos 2x, sin2 x , sin 3x, sin3 x, sin x , ex, ex+2
2 × 2
L
 L
Kn
e1 = (1, 0, 0, . . . , 0), e2 = (0, 1, 0, . . . , 0), . . . , en = (0, 0, 0, . . . , 1)
e1, e2, . . . , en
(x1, x2, . . . , xn) = x1e1 + x2e2 + . . . + xnen
x1e1 + x2e2 + . . . + xnen = 0 x1 = x2 = . . . = xn = 0
{ei}
f1, f2, . . . , fn f2, f1, . . . , fn
(3, -2) " R2
f1 = e2 f2 = e1
F = (fi) L
(fi)
(fi)
F Ò! F
F F
f1 F f1 F
f1 = 2f2 + . . . + nfn f1 -
2f2 - . . . - nfn = 0 F
F F
Ò! F F
1f1 + 2f2 +
n
2
. . . + nfn = 0 1 = 0 f1 = - f2 - . . . - fn

1 1
F F f1 a = µ1f1 + . . . + µnfn
a " L
2 n
F a = (µ1 - )f2 + . . . + (µn - )fn a F
1 1
F F
F Ò! F a = 1f1 + . . . +
nfn a " L F 1f1 + . . . + nfn +
(-1)a = 0
F a
Ò! F F
a " L F a
µa + 1f1 + . . . + nfn + (-1)a = 0,
µ, 1, 2, . . . , n µ = 0
F
µ = 0

1 n
a = - f1 - . . . - fn.
µ µ
F
1, x, x2, x3, . . . R[x]
1, x, x2, x3, . . . , xn Pn
n
dim L = "
L
L
L
L dim L
dim L
L n L n = dim L
F = (f1, f2, . . . , fn) E = (e1, e2, . . . , em)
L m > n E
m d" n
E F m > n
m d" n
F E m d" n
F E n d" m m = n
 F = (f1, f2, . . . , fn) E = (e1, e2, . . . , en)
E
E
E
F = (f1, f2, . . . , fn) L E =
(e1, e2, . . . , en) F
F
H L dim L =
= n H = L H

n B = (b1, b2, . . . , bm)
H m B H
m = n B L H = biK = L
H = L dim H = m < n = dim L

F[0, 1]
b1, b2, . . . , bn b F = (f1, f2, . . . , fn)
b = b1f1 + b2f2 + . . . + bnfn.
F = (f1, . . . , fn) L
fi F
Å„Å‚
ôÅ‚ = c11f1 + . . . + cn1fn
òÅ‚f1
. . . . . . . . . . . .
ôÅ‚
ółf = c1nf1 + . . . + cnnfn
n
C = (cij)
(f1, . . . , fn) = (f1, . . . , fn)C
ëÅ‚ öÅ‚
b 1
ìÅ‚b2÷Å‚
ìÅ‚ ÷Å‚
b = b 1f1 + b 2f2 + . . . + b nfn = (f1, f2, . . . , fn)
ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚
b n
 b
(f1, f2, . . . , fn)
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
b1 b 1
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
. .
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
. = C . .
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
. .
bn b n
C b 1, . . . , b n
n
b = b ifi
i=1
n
{fi } b = bifi =
i=1
0 C
C-1
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
b 1 b1
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
. .
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚ ÷Å‚
. = C-1 ìÅ‚ . .
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
. .
b n bn
F = (f1, f2, . . . , fn) L
bi = b1if1 + . . . + bnifn (1 d" i d" n)
L det(bij) = 0

B = {b1, b2, . . . , bn} B
(bij) X n
B
B
BX = 0 Ò! X = 0
FBX = 0 Ò! X = 0
BX = 0 Ò! X = 0
det B = 0

 bi = (b1i, . . . bni) " Kn 1 d" i d" n
det(bij) = 0

A = F · a F
a F
F = Fa + N Fa a N a
|Fa||a|, Fa a ,
A =
-|Fa||a|, Fa a .
A =
|F||a| cos(F, a) F, a F a
a b
a · b = |a||b| cos(a, b)
ab = ba.
a2 = a · a
a2 e" 0 a2 = 0 Ô! a = 0.
"
|a| = a2
a · b
cos(a, b) = (a = 0, b = 0)

|a||b|
a Ä„" b Ô! a · b = 0
ij = jk = ki = 0; i2 = j2 = k2 = 1.
a, b, c O
a Ox
a Xb, Xc, Xb+c
b, c, b + c Ox
a(b + c) = |a||b + c| cos(b + c, x) = |a|Xb+c =
= |a|(Xb + Xc) = |a|Xb + |a|Xc = |a||b| cos(b, x) + |a||c| cos(c, x) = ab + ac
a(b + c) = ab + ac; (a + b)c = ac + bc; a(b) = (a)b = (ab).
 (a+b)c = ac+bc
a(b) = ab
a(b) = |a|Xb = |a|Xb = |a||b| cos(a, b) = ab
a =
X1i + Y1j + Z1k b = X2i + Y2j + Z2k
a · b = X1X2 + Y1Y2 + Z1Z2
a b
X1X2 + Y1Y2 + Z1Z2
cos (a, b) =
2 2 2 2
X1 + Y12 + Z1 X2 + Y22 + Z2
a b X1X2 +
Y1Y2 + Z1Z2 = 0
L
L × L R a, b " L
a · b
(a1 + a2)b = a1b + a2b; a(b1 + b2) = ab1 + ab2;
(ab) = (a)b = a(b).
ab = ba
a2 = aa e" 0
a2 = 0 Ð!Ò! a = 0.
 a, a1, a2, b, b1, b2 " L  " R
(f(x), g(x)) f(x)·g(x)
0 · b = (0 + 0)b = 0b + 0b,
0b = 0 b " L
Rn
(x1, . . . , xn)(y1, . . . , yn) = x1y1 + x2y2 + . . . + xnyn.
b
C[a, b] (f(x), g(x)) = f(x)g(x) dx
a
a b
(ab)2 d" a2b2
a b
a = 0 a b
a
f(t) = (ta + b)2 = t2a2 + 2tab + b2.
f(t) e" 0 t
4(ab)2 - 4a2b2 d" 0
t
f(t) = 0
ta + b = 0 a b
b a b = sa
s " R f(-s) = 0
"
|a| = a2 a " L a, b
a, b L
Õ " [0, Ä„]
ab
cos Õ =
|a||b|
Õ
ab
|a||b|
Ä„
2
a Ä„" b
" a, b " L ab = 0
|a| e" 0 |a| = 0 a = 0
|a| = |||a|  " R a " L
|a + b| d" |a| + |b|
n n
| ai| d" ai|
i=1 i=1
a, b = b, a
a, b = 0 b = a
 a, b = Ä„ b = a
 < 0
a, c d" a, b + b, c
b
a c
F = (f1, f2, . . . , fn) L
fi Ä„" fj (i, j) i = j

fi F
 f1, . . . , fn L
1f1 + . . . + nfn = 0 f1
2
1f1 = 0 f1fi = 0 i = 1 1 = 0

2 = . . . = n = 0 dim L = n
o o
f1, . . . , fn L f1 , . . . , fn
f1, . . . , fn b
b = (bf1)f1 + . . . + (bfn)fn,
i b
bfi
b = x1f1 + . . . + xnfn b (fi)
fk
bfk = x1 · 0 + . . . + xk · 1 + . . . + xn · 0 = xk,
1
sin x cos x sin 2x cos 2x sin nx cos nx
2
Ä„
1
f(x)g(x) dx
Ä„ -Ä„
H L
HÄ„" = {v " L | vw = 0 w " H}
H
0Ä„" = L LÄ„" = 0 HÄ„" = 0
H L
Ä„
1
L = C[-Ä„, Ä„] f(x)g(x) dx
Ä„ -Ä„
 (1/2, sin x, cos x,
sin 2x, cos 2x, . . .) H L
f(x) " L
Ä„ Ä„
f(x) sin nx dx = f(x) cos nx dx = 0
-Ä„ -Ä„
n f(x)
HÄ„" = 0 L
HÄ„"
v1, v2 " HÄ„" w " H
(1v1 + 2v2)w = 1(v1w) + 2(v2w) = 0
1, 2 " R
H )" HÄ„" = 0
Ä„" Ä„"
H1 Ä…" H2 H1 ‡" H2
HÄ„"Ä„" ‡" H
v " H w " HÄ„" wv = vw = 0
H L
W H )" W = 0 H + W = L
H
W
H
W = HÄ„" H + W = L
H
H
L
F = (f1, f2, . . . , fk) a " L
aH = (f1a)f1 + (f2a)f2 + . . . + (fka)fk
a H
aH " H, a - aH " HÄ„"
H + HÄ„" = L,
L H HÄ„"
b " H
|a - aH| d" |a - b|
b = aH
a " H a - aH " H (F, (a - aH)0)
/ /
H + aR
fi aHfi = afi i a - aH " HÄ„" a =
aH + (a - aH) H HÄ„"
b " H
(a - b)2 = (a - aH + aH - b)2) =
= (a - aH)2 + 2(a - aH)(aH - b) + (aH - b)2 = (a - aH)2 + (aH - b)2
|a - b| b = aH
a - aH " H a - aH " H )" HÄ„" = 0 a = aH " H
a " H a - aH " H aH " H (a - aH)0R + H = aR + H
/ /
 L
H
dim H + dim HÄ„" = dim L; H = (HÄ„")Ä„"
H
L
L dim L =
1
< n dim L = n
0 = H = L dim H < dim L

H f1, . . . , fk
HÄ„" = L f1 " HÄ„" HÄ„"
/
fk+1, . . . , fm F = (f1, f2, . . . , fm)
H +HÄ„" = L
F F L m = n
dim H + dim HÄ„" = k + (n - k) = n = dim L.
dim(HÄ„")Ä„" = n - dim HÄ„" = n - (n - dim H) = dim H
H = L HÄ„" = 0
H = 0 HÄ„" = L
H1 = f1R f1
H1
H1 H2
Ä„" Ä„" Ä„" Ä„"
(H1 + H2)Ä„" = H1 )" H2 (H1 )" H2)Ä„" = H1 + H2
 F O F
-

A a = OA
O
F M
Ä O
a F G
B O M
B Ä
-
-
G b = OB F a F
a G b
M |F| |a| sin(F, a)
L K
g = r1a1 '" b1 + r2a2 '" b2 + . . . + rmam '" bm,
ri " K ai, bi " L g
g = r1a 1 '" b 1 + r2a 2 '" b 2 + . . . + rm a m '" b m
g g
riai '" bi
(r + r )a '" b r a '" b + r a '" b r(a + a ) '" b ra '" b + ra '" b ra '" (b + b )
ra '" b + ra '" b (r)a '" b r(a) '" b ra '" (b)
a '" b -b '" a K
a '" a 0 '" 0
 g g
g g
g g g g
g g
L = V(E3)
(2i + j + k) '" (i - j) = 2i '" i + j '" i + k '" i - 2i '" j - j '" j - k '" j =
= -i '" j + k '" i - 2i '" j - j '" k = -3i '" j + j '" k + k '" i.
'" c d c '" d
(c, d) (c , d )
c d c d c'"d = c '"d
2×2
x y
z t
x y
(c, d) = (c , d ) .
z t
x y 1 t-1y x - t-1zy 0 1 0
= .
z t 0 1 0 t t-1z 1
L '" L
i=k
K g f = sici '" di
i=1
i=m i=k i=m
g + f = riai '" bi + sici '" di,  · f =  (si)ci '" di
i=1 i=1 i=1
0 '" 0 0
a '" b + 0 '" 0 = a '" b + 0 '" 0 · b = a '" b + 0 · 0 '" b = (a + 0 · 0) '" b = a '" b.
g (-1)g
a b a '" b = 0 a = b
b = a 
a '" b = (b) '" b = (b '" b)
b'"b = -b'"b 2b'"b = 0 b'"b = 0
 L F = (f1, f2, . . . , fn)
n(n-1)
2
L '" L Cn =
2
{fi '" fj} (1 d" i < j d" n)
n n
( rifi) '" ( sjfj) = ( risjfi '" fj)
i=1 j=1 1d"i,jd"n
fi '" fj = -fj '" fi {fi '" fj}
(1 d" i < j d" n)
f1 '" f2
f1 '" f2 = rijfi '" fj
ia11 a12 a21 a22 am1 am2
r1 + r2 + . . . + rm ,
b11 b12 b21 b22 bm1 bm2
n n
ai = aijfj bi = bijfj
j=1 j=1
1 0
1· = 1 rij"ij = 0
0 1
"ij
{fi '" fj} (1 d" i < j d" n)
V(E3) '" V(E3) j '" k k '" i i '" j
a = X1i + Y1j + Z1k b = X2i + Y2j + Z2k
Y1 Z1 X1 Z1 X1 Y1
a '" b = j '" k - k '" i + i '" j.
Y2 Z2 X2 Z2 X2 Y2
V(E3)
(a, b, c)
c a b
 a b
c
a, b, c
i, j, k
a = X1i + Y1j + Z1k b = X2i + Y2j + Z2k c = X3i + Y3j + Z3k
X1 Y1 Z1
X2 Y2 Z2 > 0.
X3 Y3 Z3
(a, b, c)
a, b, c
V(E3)
a b c
" c Ä„" a c Ä„" b
" |c| = |a|b| sin(a, b)
" a b c = 0 (a, b, c)
a × b
|a × b| = S a,b
a b M = F × a
F O
a =
X1i + Y1j + Z1k b = X2i + Y2j + Z2k
i j k
Y1 Z1 X1 Z1 X1 Y1
a × b = X1 Y1 Z1 = i - j + k
Y2 Z2 X2 Z2 X2 Y2
X2 Y2 Z2
c c Ä„" a
Y1 Z1 X1 Z1 X1 Y1
a · c = X1 - Y1 + Z1 = 0
Y2 Z2 X2 Z2 X2 Y2
c Ä„" b
2 2 2
Y1 Z1 X1 Z1 X1 Y1
|c|2 = + + =
Y2 Z2 X2 Z2 X2 Y2
2 2 2 2
= (X1 + Y12 + Z1)(X2 + Y22 + Z2) - (X1X2 + Y1Y2 + Z1Z2)2 =
= |a|2|b|2 - (a · b)2 = |a|2|b|2(1 - cos2(a, b)) = |a|2|b|2 sin2(a, b)
|c| = |a||b| sin(a, b)
a b
a, b, c
X1 Y1 Z1
2 2 2
X2 Y2 Z2 Y1 Z1 X1 Z1 X1 Y1
= + (-1)2 + > 0
Y1 Z1 X1 Z1 X1 Y1 Y2 Z2 X2 Z2 X2 Y2
-
Y2 Z2 X2 Z2 X2 Y2
3 × 3
a b
2 2 2
Y1 Z1 X1 Z1 X1 Y1
Sa,b = + + .
Y2 Z2 X2 Z2 X2 Y2
a = X1i + Y1j b = X2i + Y2j
a b
X1 Y1
S a,b = mod .
X2 Y2
a × (b + c) = a × b + a × c (a × b) = (a) × b = a × (b)
a × b = -b × a a b
i × j = k j × k = i k × i = j
i=n
g = riai '" bi " V(E3) '" V(E3)
i=1
i=n
g× = riai × bi g
i=1
f
f× g×
f× = g× g g×
(g + f)× = g× + f×; (g)× = g×.
g g×
g× = f× (g-f)× = g× -f× =
0 g - f = xj '" k + yk '" i + zi '" j
0 = (g - f)× = x(j '" k)× + y(k '" i)× + z(i '" j)× = xi + yj + zk
x = y = z = 0 g - f = 0 g = f
g g× V(E3) '" V(E3)
V(E3)
V(E3)
L
Ä… = r1a1 '" b1 '" c1 + r2a2 '" b2 '" c2 + . . . + rnan '" bn '" cn (ri " K; ai, bi, ci " L)
Ä… Ä… = ria i '" b i '" c i
Ä… Ä…
ra '" b '" c
a '" b '" c = -b '" a '" c = -a '" c '" b = -c '" b '" a.
K Ä… ² =
sipi '" qi '" ri
Ä… + ² = riai '" bi '" ci + sipi '" qi '" ri,
Ä… 
Ä… = (r1)a1 '" b1 '" c1 + (r2)a2 '" b2 '" c2 + . . . + (rn)an '" bn '" cn.
L '" L '" L '"3L
F = (f1, f2, . . . , fn) L {fi '" fj '" fk}
n(n-1)(n-2)
3
1 d" i < j < k d" n Cn =
3
'"3L
'"3V(E3) i '" j '" k
a = X1i + Y1j + Z1k b = X2i + Y2j + Z2k c = X3i + Y3j + Z3k
X1 Y1 Z1
a '" b '" c = X2 Y2 Z2 (i '" j '" k)
X3 Y3 Z3
(X1i + Y1j + Z1k) '" (X2i + Y2j + Z2k) '" (X3i + Y3j + Z3k)
(a, b, c) (a , b , c )
a '" b '" c = a '" b '" c
3 × 3
C (a, b, c) = (a , b , c )C.
Ä… = a '" b '" c
a b '" c
a
b '" c
Ä…
a, b, c
-
a, b, c a · (b × c)
(a, b, c)
a =
X1i + Y1j + Z1k b = X2i + Y2j + Z2k c = X3i + Y3j + Z3k
X1 Y1 Z1
(a, b, c) = X2 Y2 Z2
X3 Y3 Z3
i j k
X1i + Y1j + Z1k X2 Y2 Z2
X3 Y3 Z3
a, b, c
(a, b, c) > 0 (a, b, c) < 0 a, b, c
(i, j, k) = 1
a '" b '" c (a, b, c)
i '" j '" k i '" j '" k = 0

'"3V(E3)
(a, b, c)
a, b, c
a, b, c - a, b, c
a b × c Õ
V = H · S b,c = |a| cos Õ|b × c| = a · (b × c),
a = X1i + Y1j + Z1k b = X2i + Y2j + Z2k c = X3i + Y3j + Z3k
a, b, c
(a, b, c) = 0
X1 Y1 Z1
X2 Y2 Z2 = 0;
X3 Y3 Z3
, µ, ½ a + µb + ½c = 0
a '" b '" c = 0
1 Ò! 2 b c
b × c a
(a, b, c) = a(b × c) = 0
2 Ò! 3
3 Ò! 4 Ä…, ², Å‚
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
X1 Y1 Z1 Ä… 0
íÅ‚X2 Y2 Z2Å‚Å‚ · íÅ‚²Å‚Å‚ íÅ‚0Å‚Å‚
=
X3 Y3 Z3 Å‚ 0
Ä…a + ²b + Å‚c = 0
4 Ò! 5 Ä…a + ²b + Å‚c = 0 Ä… = 0

² Å‚
Ä… = -Ä…b - c
Ä…
² Å‚
5 Ò! 1 Ä… = -Ä…b - c a, b, c
Ä…
b c
2 Ô! 6
y = kx, y = A1x1 + A2x2 + . . . + Anxn
L M K
È : L M
È(a + b) = È(a) + È(b); È(a) = È(a)
a, b " L  " K È
È L = M
È(a) = 0 a " L
IdL : L L Id Id(a) a" a
k hk L hk(a) =
ka a " L
Ä…
rÄ…
Ä…
a"
s
L
b
f(x) f(x) dx
a
[a, b] R
n m
A m × n K K
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
x1 x1
ìÅ‚x2÷Å‚ ìÅ‚x2÷Å‚
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
n m
" K A · " K
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
xn xn
n m
K K
È È(L)
È(a) a L Ker È
È a " L È(a) = 0
È : L M
M L
È
M
È
Ker È = 0 È(L) = M È
È(a1) = È(a2) È(a1 - a2) = 0
a1 - a2 " Ker È = 0 a1 = a2
L M
Õ : L M
<"
L M
=
n
n
Kn
F = (f1, f2, . . . , fn) L
(x1, x2, . . . , xn) x1f1 + x2f2 + . . . + xnfn
Kn L
n K n
6
P6; Mat (R); Mat (R); R6; R
2×3 3×2
L =
H + T H )" T = 0
L H T prT : L L
H
h + t h " H, t " T prT (h + t) = h
H
L H T symT :
H
L L h + t h " H, t " T
symT (h + t) = h - t
H
L T
HÄ„" H prH(a) = aH
H symH(a) = 2aH -a
F = (f1, . . . , fn) L
È : L L È(fi) F
Å„Å‚
ôÅ‚
ôÅ‚È(f1) = a11f1 + a21f2 + . . . an1fn
ôÅ‚
òÅ‚È(f ) = a12f1 + a22f2 + . . . an2fn
2
ôÅ‚. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ôÅ‚
ôÅ‚
ół
È(fn) = a1nf1 + a2nf2 + . . . annfn
n×n A = (aij) È
F
A
b = b1f1 + b2f2 + . . . + bnfn " L
È(b)
ëÅ‚ öÅ‚
b1
ìÅ‚ ÷Å‚
.
ìÅ‚ ÷Å‚
A .
íÅ‚ Å‚Å‚
.
bn
n n n n n
È(b) = biÈ(fi) = bi ajifj = ( ajibi)fj
i=1 i=1 j=1 j=1 i=1
L End L
(Õ+È)(a) =
Õ(a)+È(a) F
L Õ " End L AÕ
n × n F
AÕ+È = AÕ + AÈ; AÕć%È = AÕ · AÈ
Õ AÕ
End L Mat (K)
n×n
cos Ä… - sin Ä…  0 1 0 1 0
, , ,
sin Ä… cos Ä… 0  0 -1 0 0
Ä…  OX
OX
L = H + T H T F = (f1, f2, . . . , fn)
L
H = f1K + . . . + fmK; T = fm+1K + . . . + fmK
prT symT F
H H
diag(1, . . . , 1, 0, . . . , 0), diag(1, . . . , 1, -1, . . . , -1)
m
A È : L L
F F L
C A = C-1AC È
b = b1f1 + . . . + bnfn = b 1f1 + . . . + b nfn " L
È(b) A(b1, b2, . . . , bn)
A (b 1, b 2, . . . , b n)
C-1A(b1, b2, . . . , bn) = A (b 1, b 2, . . . , b n) (b1, b2, . . . , bn) =
= C(b 1, b 2, . . . , b n)
C(b 1, b 2, . . . , b n) (b1, b2, . . . , bn)
C-1AC(b 1, b 2, . . . , b n) = A (b 1, b 2, . . . , b n)
b 1, . . . , b n " K C-1AC = A
L È
{C-1AC | C " GL(n, K)}
A
F : Mat Mat
F (diag(1, . . . , n)) = diag(F (1), . . . , F (n))
exp : Mat (R) Mat (R)
n×n n×n
A2 A3 A4
exp(A) = E + A + + + + . . .
2! 3! 4!
A = max A(u1, u2, . . . , un) | u2 + u2 + . . . + u2 = 1
1 2 n
AB d" A B
" A k
k=0
k!
A d" R
exp(diag(1, 2, . . . , n)) = diag(exp(1), . . . , exp(n))
0 Õ cos Õ sin Õ
exp( ) =
-Õ 0 - sin Õ cos Õ
Õ = 0, Ä„

H L È
L H
È È(a) " H a " H
 È H
H
F L
f1, . . . , fk H
È È F
B D
,
0 C
B k × k (n - k) × k
È(H) Ä…" H È(fi) i = 1, . . . , k
(f1, f2, . . . , fk)
L
È H T F =
(f1, f2, . . . , fn) H = f1 K +. . . + fkK T = fk+1K +. . . + fnK
B 0
È F
0 C
È
È
(aK) Ä…" aK a
a
È L
a " L 
È(a) = a
a " L È
aK = {a |  " K}
L
D
 D x2 ekx sin Éx cos Éx D2
D D2
È2 = È È L
È L
H = Ker È T = Ker(Id -È) È
H T
È2 = Id È L
È L
H = Ker(Id -È) T = Ker(Id +È) È
H T
A È F =
(f1, f2, . . . , fn)
a = a1f1 + a2f2 + . . . + anfn " L
a 

det(A - E) = 0,
(a1, a2, . . . , an)
(A - E)X = 0
È(a) = a A(a1, a2, . . . , an) =
(a1, a2, . . . , an)
det(A - E) = 0
n
(-1)n
È A
det A n-1 (-1)n-1 Tr A Tr A A
 F = (f1, f2, . . . , fn) L
C È
A = C-1AC
det(A - E) = det(c-1AC - E) = det(C-1AC - C-1C) =
= det C-1(A - E)C =
= det C-1 det(A - E) det C = det C-1 det C det(A - E) = det(A - E),
b1, . . . , bn È :
L L 1, . . . , n x1b1 + . . . + xnbn = 0
xi " K
n - 1 È Èi(bj) = i bj
j
Å„Å‚
ôÅ‚
0,
1
ôÅ‚x b1 + . . . + xnbn =
ôÅ‚
ôÅ‚
ôÅ‚1x1b1 + . . . + nxnbn =
ôÅ‚ 0,
òÅ‚
2x1b1 + . . . + 2xnbn = 0,
1 n
ôÅ‚
ôÅ‚. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
ół x1b1 + . . . + n-1xnbn = 0,
n-1
1 n
ëÅ‚ öÅ‚
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
1 1 . . . 1
x1b1 0
ìÅ‚
1 2 . . . n ÷Å‚ ìÅ‚x2b2÷Å‚ ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
0
ìÅ‚ ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
2 2 . . . 2 ÷Å‚ · =
1 2 n
ìÅ‚ ÷Å‚ íÅ‚ Å‚Å‚ íÅ‚. . .Å‚Å‚
. . .
íÅ‚ Å‚Å‚
. . . . . . . . . . . .
xnbn 0
n-1 n-1 . . . n-1
1 2 n
i
x1b1 = x2b2 =
. . . = xnbn = 0 x1 = x2 = . . . = xn = 0 bi
bi
È n
È n
È
f1, . . . , fn
n 1, . . . , n
 È {fi} diag(1, 2, . . . , n)
r e1R+e2R 900
r
e1C + e2C rC
rC
2
n2 [-R, R]n R > 0
n2 A = (aij) " Mat (R)
n×n
n

È -  Id
1, . . . k
È L n
ni = dim Ker(È - i Id) È
n1 + n2 + . . . + nk = n
k
H = Ker(È - i Id)
i=1
H
Ker(È - i Id) dim H = n1 + n2 + . . . + nk
n1 + n2 + . . . + nk = n H = L
È
Ker(È - i Id)
È
È {fi} È diag(µ1, µ2, . . . , µn)
fi
1 = µ1 = . . . = µs, 2 = µs+1 = . . . = µt, . . . , k = µq = . . . = µn.
H1 = f1K + . . . + fsK Hk = fqK + . . . + fnK Hj Ä…"
Ker(È-j Id) H1+. . .+Hk L
n1 + . . . + nk e" dim H1 + . . . + dim Hk = dim L = n
k
ni d" n
i=1
k
ni = n
i=1
 Ker(È - i Id)
k
ni = n Ker(È - i Id)
i=1
A È
K
Å„Å‚
ôÅ‚
ôÅ‚a11x1 + a12x2 + . . . + a1nxn = b1
ôÅ‚
òÅ‚a x1 + a22x2 + . . . + a2nxn = b2
21
ôÅ‚. . . . . . . . .
. . .
ôÅ‚
ôÅ‚
ół
am1x1 + am2x2 + . . . + amnxn = bm
n m
È : K K m × n A = (aij)
(b1, b2, . . . , bm) È(nK) m
m
È(nK) = K
dim È(nK) < m
(b1, b2, . . . , bm) " È(nK)
È
A
A
r1
r2 r1 r2
A
ëÅ‚ öÅ‚
1 a12 a13 . . . a1n
ìÅ‚
0 1 a23 . . . a2n÷Å‚
ìÅ‚ ÷Å‚
ìÅ‚. . . . . . . . . . . . . . .÷Å‚
ìÅ‚ ÷Å‚
ìÅ‚
A = 0 0 1 . . . akn÷Å‚
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
0 0 0 . . . 0
ìÅ‚ ÷Å‚
íÅ‚. . . . . . . . . . . . . . .Å‚Å‚
0 0 0 . . . 0
r1 r2 A
A rang A
A
A"
rang A = rang A" A"
(b1, b2, . . . , bm) A
x0 x0 x0 K
1 2 n
n
(a1i, a2i, . . . , ami) x0 = (b1, b2, . . . , bm)
i
i=1
(x0, x0, . . . , x0 )
1 2 m
(x0, x0, . . . , x0 )
1 2 m
A A"
M
A r × r r = rang A
L
È : L L
a · È(b) = È(a) · b
a, b " L
F = (f1, f2, . . . , fn)
L È A
È F A = A
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
b1 a1
ìÅ‚b2÷Å‚ ìÅ‚a2÷Å‚
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
(a1, a2, . . . , an)A = (b1, b2, . . . , bn)A
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
bn an
a1, . . . , an,
b1, . . . , bn
(a1, a2, . . . , an) (b1, b2, . . . , bn) ei ej
ei (a1, a2, . . . , an) ej
(b1, b2, . . . , bn) aij = aji
(i, j)
H
È HÄ„"
a " H b " HÄ„"
È(b)a = bÈ(a) = 0,
È(a) " H b Ä„" H È(b) Ä„" H È(b) " HÄ„"
a b
b c
2 - (a + c) + ac - b2
D = (a + c)2 - 4ac + b2 = (a - c)2 + b2
1
f1 f2
(f1R)Ä„" f2
f1, f2
1 2
 È L
(f1, f2, . . . , fn)
L dim L = 1
n dim L = n
z " C f() = 0
z " R
z
aizi = 0 Ò! aizi = 0 Ò! aizi = 0
ai " R
b = b1f1+. . .+bnfn " fiC b = b1f1+. . .+bnfn
1
z z b " L a := b + b " L c := (b - b) " L
/
i
H = aR + cR È
È(H) Ä…" ((bC + bC) )" L) = H.
È H
H (-z)(-z)
È
1 H = Ker(È - 1R)
È H = 0 H = L È

1
L = H + HÄ„"
H HÄ„"
Õ È ÕÈ =
ÈÕ
Õ È ÕÈ
È
Õ È
 Õ L
|Õ(x)| = |x|
x " L
L
2(x, y) = |x + y|2 - |x|2 - |y|2 ,
Õ Õ
Õ(x)Õ(y) xy
cos(Õ(x), Õ(y)) = = = cos(x, y)
|Õ(x)| |Õ(y)| |x| |y|
x, y " L
Ä…
R3
ax + by = 0
R3
O(L)
L
F = (f1, f2, . . . , fn) L
A Õ
x = xifi y = yifi L Õ(x)Õ(y) = xy
i i
îÅ‚ ëÅ‚ öÅ‚Å‚Å‚ ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
x1 y1 y1
ïÅ‚ ìÅ‚x2 ÷łśł ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
y2 y2
ïÅ‚A ìÅ‚ ÷łśł ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
ðÅ‚ íÅ‚. . .Å‚Å‚ûÅ‚ · A íÅ‚. . .Å‚Å‚ = (x1, x2, . . . , xn) íÅ‚. . .Å‚Å‚
xn yn yn
A A = E
A
O(n) GL(n, R)
a b
O(2) A = " O(2)
c d
Õ
a2 + b2 = 1, b2 + d2 = 1, ab + cd = 0.
Ä… " [0, 2Ä„) a = cos Ä… c = sin Ä… b = - sin Ä…
d = cos Ä… A Ä… b = sin Ä… d = - cos Ä…
A Ä…1 s1 = cos Ä…f1 + sin Ä…f2
s2 = - sin Ä…f1 + cos Ä…f2 Õ s1 s2
diag(1, -1)
R2
cos Ä… - sin Ä… 1 0
sin Ä… cos Ä… 0 -1
O(3)
x Õ

|x| = |Õ(x)| = |x| = || |x|
|x| = 0
M L
Õ : L L M
Õ M Õ
a " M b " M Õ(b)a = bÕ(a) = 0 Õ(a) "
M Õ(b) " M
Õ
ëÅ‚ öÅ‚
Ä…1 0 0
íÅ‚
0 cos Ä… - sin Ä…Å‚Å‚
0 sin Ä… cos Ä…
f1 Õ
M = f1R Õ M
Õ f1
f2 Õ/M
f1 f2 f3 Õ(f2) = f2
Õ(f3) = -f3 Õ(f1) = f1 fi
s1 = f3 s2 = f2 s3 = f1 Õ
diag(-1, 1, 1) Õ fi i = 1, 2, 3
diag(-1, 1, -1) fi s1 = f2 s2 = f1
s3 = f3 si Õ
ëÅ‚ öÅ‚
Ä…1 0 0
íÅ‚
0 cos Ä„ - sin Ä„Å‚Å‚
0 sin Ä„ cos Ä„
R R
C H
q " H
q = a0 + a1i + a2j + a3k,
a0, a1, a2, a3 " R a0 = 0
H R
(a0 + a1i + a2j + a3k) + (a 0 + a 1i + a 2j + a 3k) =
= (a0 + a 0) + (a1 + a 1)i + (a2 + a 2)j + (a3 + a 3)k;
(a0 + a1i + a2j + a3k) = a0 + a1i + a2j + a3k.
 V(E3) = iR + jR + kR
i, j, k
a, b " V(E3) a " b a · b
q = a + a t = b + b
a, b a, b
q · t = (a + a)(b + b) = (ab - a " b) + (ab + ba + a × b).
q t
(a0 + a1i + a2j + a3k)(b0 + b1i + b2j + b3k) = (a0b0 - a1b1 - a2b2 - a3b3)+
+(a0b1 + b0a1 + a2b3 - a3b2)i + (a0b2 + b0a2 + a3b1 - a1b3)j+
+(a0b3 + b0a3 + a1b2 - a2b1)k.
i2 = j2 = k2 = ijk = -1.
ij = -ji = k; jk = -kj = i; ki = -ik = j
a - a q = a + a
q
qq = qq = a2 + a2 + a2 + a2,
0 1 2 3
qq = 0 Ô! q = 0 qq q
q q
|q|
1
q q-1 = q R H
q
a a + 0i + 0j + 0k
H R
a a + 0
(a + 0) + (b + 0) = (a + b) + 0;
(a + 0)(b + 0) = (ab - 0 " 0) + (a0 + b0 + 0 × 0) = ab + 0.
1 = 1+0
(s + t)q = sq + tq; q(s + t) = qs + qt.
q = a + a s = b + b t = c + c a, b, c " R a, b, c " V(E3)
q(s + t) = (a + a)((b + c) + (b + c)) =
= a(b + c) - a " (b + c) + a(b + c) + (b + c)a + a × (b + c) =
= ab - a " b + ab + ba + a × b + ac - a " c + ca + ac = qs + qt.
(s + t)q = sq + tq
q(st) = (qs)t
R C q, s, t " {1, i, j, k}
q, s, , t q(st) = (qs)t
33 = 27
H
ij = ji H rq = qr

r " R q " H
1
q q
q
1 1 q
q · ( q) = qq = = 1
q q q
R H
q1 + q2 = q1 + q2 q1q2 = q2 · q1
q1 = a + a q2 = b + b
q1 + q2 = (a + b) + (a + b) = (a + b) - a - b == q1 + q2
q1q2 = (ab - a " b) + ab + ba + a × b = ab - a " b - ab - ba - a × b =
= ab - (-a) " (-b) - ab - ba + (-b) × (-a) = q2q1
q " V(E3) q = -q
q = q
|q1q2| = |q1| |q2|
|q1q2|2 = q1q2q1q2 = q1q2q2q1 = q1 |q2|2 q1 = |q1|2 |q2|2
|q1 + q2| d" |q1| + |q2|
q1 q2
HR
1, i, j, k
|q-1| = 1/ |q1|
S3 = {q " H | |q| = 1}
S3 u " S3
conju : V(E3) V(E3)
conju(b) = ubu-1.
b b " V(E3) ubu-1
u-1 = u u " S3
ubu-1 = ubu = ubu = u(-b)u = -ubu-1.
S3
SO(n) O(n)
conju " SO(3)
|ubu-1| = |u| |b| |u-1| = |b| conju
u(r1b1 + r2b2)u-1 = r1(ub1u-1) + r2(ub2u-1)
 r1, r2 b1, b2
conju V(E3) u = c + df f " V(E3)
|f| = 1 c, d " R |u| = 1 c2 + d2 = 1
Ä… " [0, 2Ä„) c = cos Ä… d = sin Ä… Ä… = 0 u = 1 conju = Id
det(conju) Ä…
F (Ä…) F (Ä…) 3 × 3
cos Ä… sin Ä… F (Ä…) conju
F (Ä…) Ä…1
F (0) = 1 Ä…0 F (Ä…0)
Ä…" " (0, Ä…0) F (Ä…") = 0
conju
F (Ä…) a" 1 conju
u = cos Ä… + sin Ä… f Ä… f " V(E3) |f| = 1
conju f 2Ä…
fu = uf
ufu-1 = f f conju
b " V(E3) f
conju(b) = ubu = (cos Ä… + sin Ä… f)b(cos Ä… - sin Ä… f) =
= (cos Ä… + sin Ä… f)(sin Ä… b " f + cos Ä… b - sin Ä… b × f) =
= (cos Ä… + sin Ä… f)(cos Ä… b - sin Ä… b × f) =
= cos2 Ä… - (sin Ä… f) " (cos Ä… - sin Ä… b × f) + cos2 Ä… b - cos Ä… sin Ä… b × f+
+ sin Ä… cos Ä… fb - sin2 Ä… f × (b × f) =
= cos2 Ä… b + 2 sin Ä… cos Ä… f × b - sin2 Ä… b = cos 2Ä… + sin 2Ä… f × b
b conju(b) Õ
cos Õ = b " (cos 2Ä… + sin 2Ä…f × b) = cos 2Ä…
u conju S3 SO(3)
Ä…1
conjuv = conju ć% conjv
conjuv(b) = uvb(uv)-1 = uvbv-1u-1 = u · conjv(b)u-1 =
= conju(conjv(b)) = conju ć% conjv(b).
 È V(E3)
u " S3 conju = È
conj
u conju conju = Id ubu-1 = b
b " V(E3) bu = ub bu = ub b " R
qu = uq q
u " R R )" S3 =
{Ä…1}
x2 = -1
{Ä…1, Ä…i, Ä…j, Ä…k}