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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;
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() :
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 , , , .
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, () . , , , ( ́ ). . {\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
| (1) |
A X = O. |
(1) , :
1. x 10 = x 20 = = xn 0 = 0, , ;
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n − r .
C X 1, X 2, , Xn − r .
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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 .
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:
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 . |
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2. Q .
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AQ = C
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AQ = CQ 1
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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, y ∈ R x', y' ∊ R' , x+y ∈ R x'+y' ∈ R', α, α x ∈ R α x' ∈ R'.
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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 :
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. 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, .
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( ): 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)
, , , , , .