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1. .

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2. .

. . ( .) .

. .

1 . - . , , . = .

2 . - . r > k.

. , - , .. . , 1, 2, , k, , , ,

= .

, ( = 0, ).

.

, . : .

= ,

= .

,

.

,

.

, .

.

 

4. b⃗ =k⋅a⃗ b→=k⋅a→, a⃗ a→ b⃗ b→ ().

S={a⃗1,a⃗ 2,,a⃗ s}S={a→1,a→2,,a→s}, k1a⃗ 1,k2a⃗ 2,,ksa⃗s - S. Z(S).

S S.

S : ∀a⃗,b⃗ ∈Z(S)⇒a⃗ +b⃗ ∈Z(S).

Z(S) . ∀λ∈P,∀a⃗ ∈Z(S)⇒λ⋅a⃗ ∈Z(S).

V (F) {\displaystyle V\left(F\right)} F {\displaystyle F} {\displaystyle (V,F,+,\cdot)}(V,F,+,*),

{\displaystyle V} V , ;

{\displaystyle F} F () , ;

{\displaystyle V\times V\to V} V x V→V, {\displaystyle \mathbf {x},\mathbf {y} } x,y {\displaystyle V} V {\displaystyle V} V, {\displaystyle \mathbf {x} +\mathbf {y} } x+y;

{\displaystyle F\times V\to V} Fx V→V, {\displaystyle \lambda } λ {\displaystyle F} F {\displaystyle \mathbf {x} }x {\displaystyle V} V {\displaystyle V} V, {\displaystyle \lambda \cdot \mathbf {x} } λ*x {\displaystyle \lambda \mathbf {x} } λx;

() :

1. {\displaystyle \mathbf {x} +\mathbf {y} =\mathbf {y} +\mathbf {x} } x+y=y+x, {\displaystyle \mathbf {x},\mathbf {y} \in V} x,y V ( );

2. {\displaystyle \mathbf {x} +(\mathbf {y} +\mathbf {z})=(\mathbf {x} +\mathbf {y})+\mathbf {z} } x+(y+z)=(x+y)+z, {\displaystyle \mathbf {x},\mathbf {y},\mathbf {z} \in V} x, y, z V ( );

3. {\displaystyle \mathbf {0} \in V} 0 V, {\displaystyle \mathbf {x} +\mathbf {0} =\mathbf {x} } x+0=x {\displaystyle \mathbf {x} \in V} x V ( ), {\displaystyle V} V;

4. {\displaystyle \mathbf {x} \in V} x V - x V {\displaystyle -\mathbf {x} \in V}, {\displaystyle \mathbf {x} +(-\mathbf {x})=\mathbf {0} } x+(-x)=0, , {\displaystyle \mathbf {x} } x;

5. {\displaystyle \alpha (\beta \mathbf {x})=(\alpha \beta)\mathbf {x} } α(βx)=(αβ)x ( );

6. {\displaystyle 1\cdot \mathbf {x} =\mathbf {x} } 1 * x =x (: ( ) F ).

7. {\displaystyle (\alpha +\beta)\mathbf {x} =\alpha \mathbf {x} +\beta \mathbf {x} } (α+β)x=αx+βx ( );

8. {\displaystyle \alpha (\mathbf {x} +\mathbf {y})=\alpha \mathbf {x} +\alpha \mathbf {y} } α(x+y)=αx+αy( ).

, {\displaystyle V}V () .

, , , (, {\displaystyle \mathbb {R} ^{2}}R2 ).

 

. {\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\ldots,\mathbf {x} _{n}}x1, x2, ,xn , , :

α 1x1+ α 2x2++ α nxn=0, Iα 1I+ Iα 2I++ Iα nI≠0. {\displaystyle \alpha _{1}\mathbf {x} _{1}+\alpha _{2}\mathbf {x} _{2}+\ldots +\alpha _{n}\mathbf {x} _{n}=\mathbf {0},\quad \ |\alpha _{1}|+|\alpha _{2}|+\ldots +|\alpha _{n}|\neq 0.}

{\displaystyle \alpha _{1}\mathbf {x} _{1}+\alpha _{2}\mathbf {x} _{2}+\ldots +\alpha _{n}\mathbf {x} _{n}=\mathbf {0},\quad \ |\alpha _{1}|+|\alpha _{2}|+\ldots +|\alpha _{n}|\neq 0.} .

: {\displaystyle V}V , , , .

, () . , , , ( ́ ). . {\displaystyle {\rm {dim}}}dim.

, ( , , ), (, ). , (, ). , , . , , , , , .

:

  • {\displaystyle n}n {\displaystyle n}n- .
  • {\displaystyle \mathbf {x} \in V} x V ( ) :{\displaystyle \mathbf {x} =\alpha _{1}\mathbf {x} _{1}+\alpha _{2}\mathbf {x} _{2}+\ldots +\alpha _{n}\mathbf {x} _{n}}

X= α 1x1+ α 2x2++ α nxn.

5. m n

 
  ì ï ï í ï ï î
a 11 x 1 + a 12 x 2 + + a 1 n xn = 0
a 21 x 1 + a 22 x 2 + + a 2 n xn = 0
am 1 x 1 + am 2 x 2 + + amnxn = 0
 
   
(1)

A X = O.

(1) , :

1. x 10 = x 20 = = xn 0 = 0, , ;

2. , , .

, .

:

, , , .

. .. `` ".

. , n n ( A ) , , (det A = 0).

, .

nr .

C X 1, X 2, , Xn r .

:

  X = C 1 X 1 + C 2 X 2 + + Cn r Xn r , (2)

X 1, X 2, , Xn r C 1, C 2, , Cn r .

:

1. C 1, C 2, , Cn r X, (2), (1).

2. X 0, C 10, , Cn r 0 ,

X 0 = C 10 X 1 + C 20 X 2 + + Cn r 0 Xn r .

6. .

: .

() .

.

 

.

: ( ) ( ).

.

7. .

  1. .

, = . i- , j- - , :

i = 1

i = 2

i

, j- , . , , .

, :

j = 1

j = 2

.

, , , , .

2. Q .

.

AQ = C

, . Q 1 ,

AQ = CQ 1

, . , .

8. e⃗ 1,e⃗ 2,e⃗ 3

, . e⃗ 1,e⃗ 2,e⃗ 3

.

a⃗ e⃗1,e⃗ 2,e⃗ 3

, .. , x1,x2,x3 .

 

x1,x2,x3 a⃗

e⃗ 1,e⃗ 2,e⃗ 3 ( x1, , x2 , x3 a⃗). , 3,2,−1 a⃗ =3e⃗ 1+2e⃗ 2−e⃗ 3 (x1=3 , x2=2 , x3=−1 a⃗ =3e⃗ 1+2e⃗ 2=e⃗ 3).

e⃗ 1,e⃗ 2,e⃗ 3, () , .

R R' , , x, yR x', y'R' , x+yR x'+y'R', α, α xR α x'R'.

R R' , .

. R R' , R R' . . , . , . : . y R, . . . . R R' , .. . ■

n - R R' .

. R R' . R : . R' : . x' R' x R. , x y R x' y' R' , 2.2 x+y R x'+y' R', α x α x'. ■

9. . L M . L :

.

. .

. g V V.

 

, M1 ∩ M2 = {0} M1 ∩ M2 . . , (, , ) . M1 ∩ M2.

L1 L2 V, ( ):

dim(L1+L2)=dimL1+dimL2−dim(L1∩L2).

10. V . A: V → V , x1, x2 ∈ V t ∈ R A (x1 + x2) = A(x1) + A(x2) A(t x1) = tA (x1). , A , A . , A V x ∈ V, A (0) = A (0 x) = 0 A (x) = 0. , : .

A V, b1, b2,..., bn . n, i- A (bi) b1, b2,..., bn ( i = 1, 2,..., n), A b1, b2,..., bn.

A y ∈ V , A(x) = y x ∈ V. A x ∈ V , A (x) = 0. A Im A, Ker A.

11. A A . Im A , - A , , , - A. , : A A. Im A, A. Im A, .

: A A. Ker A, , A. Ker A, .

( ): A f1, f2,..., fn A. B n × 2n . (. . n ) A, ( n ) . B . C, C1, C2. :

(i) C1 A;

(ii) C2, C1, A.

(i) , .

(ii). , fi f1, f2,..., fn (0,..., 0, 1, 0,..., 0), 1 i - . , , B, , f1, f2,..., fn , . , , i - B A(fi) f1, f2,..., fn. , B : - x f1, f2,..., fn, A(x) . , . C B , . x1, x2,..., xk C2, C1. A(xi) = 0, . . xi ∈ Ker A i = 1, 2,..., k. , , x1, x2,..., xk . (i) , k + dim Im A = n. k = dim Ker A. , x1, x2,..., xk Ker A, Ker A.

12. , n n (α1,, αn) , . . , . .

b1,,bn 1,,n

(1,, n) (b1,, bn)

, , , , , .

 

 





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