, , .
:
, , , .
.
8) n- ( )
( ).
D=det A Ax=b , x1, x2,..., xn,
xi = D i / D, i=1,2,..., n,
D i - n - , A i - b.
2. .
c . , n n x1, x2,..., xn
:
xn =dn, xi = di -S nk=i+1 cik xk, i=n-1, n-2,...,1.
.
1. :
( :
o ;
o , ;
o , c).
2. : , n
,
, (n+1) -, .
9) () . .
.
, m×n:
, ≤ min(n;m). . . . 2 . , . , :
;
:
, 0 .
, , . . r(A), rang(A)
:
1) ,
2) ,
3)
. .
|
|
10)
, , . .
. - ( ) (
= 0. , .. .
, . , i - . i > r ( ). , .
. j - :
(r + 1) - ( M 0 - . aj , i : α1,...,α r. k: ( k - )
. r , , rA = r.
. ( - rA ) . ( - r ) - .
11)
12) . . .
m n :
.
. , a 11 ( , x 1 ), : , x 1 .
,
, ( , ), , , , x 2.
, x 3.
, :
1) , , ;
2) , ;
3) ( 1), .
13) . .
, , . , , :
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, () . . .
. , .
: , , , . , . . , . , .
1: , , .
: , , .. . , , , .
2: , .
: , , , . , , , .. .
, - , , , .. , .
AX = 0 . () , r = rank A < n.
( ) :
n - r - :
. - .
n - r; - .
14) .
AX=0 (1)
1 n , , .
1 = 0 2 = 0 AXn = 0
A( 1 X1 + 2 X2 + + n Xn)= 1 A X1+ 2 A X2 ++ n A Xn = 0
( )
: r = rank A<n, (n-r) (1). , .
: - 6 (1).
=0 1
=...
m x n n
r = rank A<n
:
1. (n r) (1)
2. (n r)
:
:
U1 U2... Un
X1U1 + x2U2 +... + xnUn = 0
Uk, .
Uk = 1U1 + 2U2 +... + nUn
Uk 1U1 2U2 ... nUn = 0
1, 1. (- 1 2 r, 0... 1, 0)
1 k
Xk = - 1
- 2
- r -
0 , .
1 : r+1, r+2... n,
0 r+1 Xr+1 + r+2 Xr+2 +... + n Xn = 0 (2)
|
|
r+1 = 0 => r+1 = r+2 =... = n = 0
r+2
(2) , = 0,
2: z (1)
Z = Z1
Z2
Z r+1
...
Zn
Y = Z Z r+1 X r+1 Z r+2 X r+2 -... Zn Xn
Z , ,
Y= Y1
Y2
Yn Y1U1 +Y2 U2 +... + Yr Ur = 0 (3)
0 r
0 r+1
Y1, Y2 , , (3) 1 , y =0 => =0 => Z = Zr+1 Xr+1 + Zr+2 Xr+2 +... +ZnXn (6)
(6) , z X
(1), : x = k Xk
.
, .
15) . -
11 1 + 12 x2 +... +a1n x1n = b1
A21 x1 + a22 x2 +... +a2n x2n = b2
... 1 AX = B
An1 x1 + an2 x2 +... + amn xn = bm
A = {aik}
B = (b1) -
(bm)
= (1) -
(n)
-.
(1) , . ., :
Rank A = rank A
A
(1) , .
:
1) ,
2) ,
3)
16) .
.
, , :
:
1, 2,..., n − r −1, n − r , r .
17) . .
.
, , .
b , 0 = = b
A
a+b
b
O B
.
, .
b
a+b
a2
a1
a3
an-1
a1+..+an
an
b , b + c = a
a a b = a + (-b)
a
c
b
, , b, b, . : b = a + (-b), b.
.
, λ, │λ│∙││, Q, , λ 0 , λ 0.
|
|
:
b
2a -3b
:
1) b = λa, b
b ≠ 0, λ, λb = a
2) = ││∙ -0 , -0 - .
.
:
1) + b = b + a
2) (a + b) + c = a + (b + c)
3) λ1(λ2a) = λ1λ2a
4) (λ1+ λ2)a = λ1a + λ2a
5) λ(a + b) = λa + λa
, : , , , , , .
18) .
19) .
A, - B, S - ,
20) .
1 fs = cks ln
U.
, { l1, l2ln}
U = k ln
U {e } -
2. {f1, f2 fn}
U = s fs
: ?
{ 1 2 n} {n1nr}?
: , .
: , u= 1 lk = Mk lk
(Mk lk) lk = 0
Mk = k k
U= s fs = s ( cks lk) = lk ( lks ns)=
=> = lns ns
.
21) .
22) .
, , , .
. A B , Rn : .
. A , Rn :
. AB A B , Rn :
, , .
: , , , .
23) .
, , , : ? .
. , T , ,
24)
, ( ), , ( ) f, .
f X,
- , - f, - .
, , .
- .
25) , .
26) .
|
|
.
. :
1. aii . , ( );
2. , , ( ).
:
, , . n-1 . .
, .
27)
k (x) , Rn.
Rn , , .
k (x) = λ1 x 12 + λ2 x 22 +... + λ nxn 2.
λ1, λ2,..., λ n .
: , , .
. . . .
28) . . . .
.
b , .
(1)
φ b
(1) , ││cosφ = ba │b│cosφ = b, :
(2)
φ
b
, .
:
1) .
│││b│= │b││a│, cos(a,b) = cos(b,a), ab = ba
2) , .
(λa)b = λ (ba)
(λa)b = │b│bλa = λ│b│ba = λ (ab)
3) :
a(b + ) = ab + ac
a(b + c) = │a│(b+c) = │a│(ab+ac) = │a│ab+│a│a = ab + ac
4) .
2 = ││2
2 = ∙ = ││∙││cos1 = ││∙││= │a│2
i2 = j2 = k2 = 1
, -
, .
:
=3-4b, ││=2;│b│=3; (a,b)=π\3
:
││=
5) a b , , 0.
: b 0, .
: ij = jk = ki = 0
.
= i + yj + zk b = bi + byj + bzk. , , i, j, k
i | j | k | |
i | |||
J | |||
k |
a∙b = (i + yj + zk)(bi + byj + bzk)= b + yby + zbz
a∙b = b + yby + zbz
: , (
-4;-4;4), (-3;2;2), (2;5;1) D(3;-2;2), .
:
D, , :
(2-(-4);5-(-4);1-4)=(6;9;3) BD(6;-4;0)
:
∙BD = 6∙6+9(-4)+0(-3) = 36-36 = 0
BD, ABCD .
.
l α
M
M1 L
l 1 . 1 l , . l, .
- . ││≠ 0. 1 1 l 11
l │11│, 11 l , - │11│, 11 l .
1 1 l (
)
1 1 (│11│=0), =0. l : l. = 0 l, l=0.
φ l :
A
φ l
:
1) l cosφ
l = ││∙ cosφ
1. (), () 0, .
2. .
2) . d = a + b + c; l(a+b+c) = la + lb + lc
b
a
d
a b l
d c
3) λ, : l(λa) = λ la
λ0 : l(λa) = │λa │cosφ = λ│a│ cosφ = λ la
λ0 l(λa) = │λa │cos(π-φ) = -λ│a│(-cosφ) = λcosφ = λ la
λ = 0
.
φ (; y; z) b(b; by; bz)
a b:
b + yby + zbz = 0
29) , . .
.
b ,
1) b
2) , b.
│c│=│a││b│sinφ
B S
φ
b
3) a,b,c
× b; [a,b]
i, j,k
i × j = k; j × k = i; k × i = j
, , i × j = k:
1)k i; k j
2)│k│=1, │i × j │=│i│×│j│sin900 = 1
3) i, j, k
a, b, , b.
b b
:
1) , .
a×b a×b = - b×a
S a×b b×a , (S
),
.
b×a
2) , .
λ(a×b) = (λ a)×b = (a×λ b)
:
λ0; λ(a×b) b, (λ a)×b b. λ a . λ(a×b) (λ a)×b . , , .
│λ(a×b) │= λ│a×b│=λ│││b│sin(a,b)
│(λ a)×b│= │λ a││b│sin(λ a,b) = λ│a││b│sin(a,b)
λ(a×b) = λ a×b
λ0
3) b , , :
|| b <=> a×