BasismŁnec sto bibl—o tou G.Strang · PDF fileShmei‚seic maj€matoc M1122...

1122
G.Strang
2011

1122 , 2009, - , Rn Cn. - Gauss, Rn Cn, . - . .
G.Strang, , 2006 (Linear Algebra and its Applications, Fourth Edition,Thomson, 2006).

1 Gauss 1
2 32
3 58
4 68
5 89
6 , . 112

1
Gauss
, .
2x y = 1x + y = 5
. () .
2x y = 1
: x- 1/2 y- 1.
x + y = 5
x- 5 y- 5. , , , (x, y) . -, (2, 3).
; , . ,
2x y = 14x + 2y = 0
. ,
2x y = 14x + 2y = 2
: (x, y) .
1

2
:
x
[21
]+ y
[ 11
]=
[15
].
- , x y, . , , .
; , - . , . , .
3 .
2u + v + w = 54u 6v = 2
2u + 7v + 2w = 9(1.1)
(). (5
2, 0, 0), (0, 5, 0), (0, 0, 5).
u- v- (12, 0, 0) (0, 1
3, 0).
u = 0 v = 0 0w = 2, . w-. . , . . , (1, 1, 2).
; . :
,
, ,
, .
,
, ,
.
, .
(1.1) ,
u
24
2
+ v
16
7
+ w
102
=
52
9
. (1.2)

1 Gauss 3
u, v w . , R3 . , , , (u, v, w) = (1, 1, 2).
; -, , . .
u
123
+ v
101
+ w
134
= b .
b =
257
.
b =
256
. -
. , . - , .
1.1 x + 2y = 2x y = 1
. .
. .
. x y , .
. ( ) , .
1.2 :[
11
]x +
[2
1]
y =
[21
]
, x y 1.1, .

4
1.3 .
.2u + v + w = 54u + 2v + 2w = 6
2u + 7v + 2w = 9
.u + v + w = 2
2u + + 3w = 53u + v + 4w = 6
.u + v + w = 2
2u + + 3w = 53u + v + 4w = 7
. 1 Rn, n 0.
R0 = {0} Rn = {(x1, x2, . . . , xn) |xi R, i = 1, 2, . . . , n} .
Rn , Rn - ( ). x1, . . . , xn x = (x1, x2, . . . , xn) Rn.
, , (x1, x2, x3) - . , (0, 0, 0) (x1, x2, x3). , . , Rn n- , x = (x1, x2, . . . , xn).
, , ,
x =
x1x2...
xn
,
, , , , (x1, x2, . . . , xn).
1 6 ,
Cn = {(z1, z2, . . . , zn) | zi C, i = 1, 2, . . . , n} .

1 Gauss 5
Rn n , , .
, :
1234
+
201
5
=
324
1
.
,
x1x2...
xn
+
y1y2...
yn
=
x1 + y1x2 + y2
...xn + yn
.
, , :
2
2015
=
2
202
52
.
,
x1x2...
xn
=
x1x2...
xn
.
, (),
x1x2...
xn
+
y1y2...
yn
+
z1z2...zn
=
x1 + y1 + z1x2 + y2 + z2
...xn + yn + zn
.
, - , .( R3.) x = (x1, x2, . . . , xn) y = (y1, y2, . . . , yn) , x y
x y = x1y1 + x2y2 + + xnyn .
n n , :
-, 2 Rn, n .
2 n 1!

6
, - .
1.4 y1, y2, y3 (0, y1), (1, y2) (2, y3) ;
1.5
31
2
,
11
4
,
20
1
. 2.
. 2 2.
. 2 2.
. 1.
;
1.6 :
u + v + w + z = 6u + w + z = 4u + w = 2 .
, ; - u = 1; .
1.7 , 4 . . . . . . . . . . . . . . .., 4 - b. (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 0) (1, 1, 1, 1) b = (3, 3, 3, 2); , x, y, z, t ;
Gauss
. - Gauss (Gauss elimination), , . Gauss - (1.1), , , .

1 Gauss 7
2u + v + w = 54u 6v = 2
2u + 7v + 2w = 9 u 0. , -
, u , , 0, u .
2 . 1 (
).
u . 2 1 u - .
2u + v + w = 58v 2w = 12
8v + 3w = 14
u . - v 0. , v .
1 . Gauss.
2u + v + w = 58v 2w = 12
w = 2
v . w 0 .
.
w = 2 .
w , 8v 4 = 12, v = 1 .
v w , 2u + 1 + 2 = 5,
u = 1 .
(back substitution). Gauss u, v w -
, :

8
, - .
Gauss , , .
:
2 1 1 54 6 0 2
2 7 2 9
2 1 1 50 8 2 120 8 3 14
2 1 1 50 8 2 120 0 1 2
, , , . n n , n ( ) , , .
, .
1. -, , , .
u + v + w = a2u + 2v + 5w = b4u + 6v + 8w = c
(1.3)
1 1 1 a2 2 5 b4 6 8 c
1 1 1 a0 0 3 2a + b0 2 4 4a + c
1 1 1 a0 2 4 4a + c0 0 3 2a + b
, .
2. -, . . ,
u + v + w = a2u + 2v + 5w = b4u + 4v + 8w = c
u,
1 1 1 a2 2 5 b4 4 8 c
1 1 1 a0 0 3 2a + b0 0 4 4a + c
. , . .

1 Gauss 9
2a + b = 6 4a + c = 7,
3w = 64w = 7
. . 2a + b = 6 4a + c = 8,
3w = 64w = 8
w = 2. u v. u v . .
1.8
2u 3v = 34u 5v + w = 72u v 3w = 5
. - u, v, w. , .
. Gauss : , - u . . () () , v -. w v u.
1.9 b ; b ; , x, y, z.
x + by z = 0x 2y z = 0
y + z = 0
1.10
u + v + w = 23u + 3v w = 6u v + w = 1
. Gauss ( ).

10
. v , .
1.11 . .
. (x, y, z) (X, Y, Z) , ;
. 25 , ;
1.12 :2x + y = 0x + 2y + z = 0
y + 2z + t = 0z + 2t = 5.
1.13 Gauss;
. , 1 3.
. - 2 3.
. , - 1 2 3.
. .
-, b. (1.2),
b =
52
9
.
, , -,
x =
uvw
. (1.4)
9 , , ,
A =
2 1 14 6 0
2 7 2
. (1.5)

1 Gauss 11
3 3.
. m n , m n mn m n , [, ]. m = n . m 6= n .
, -, . n- n 1. , - . :
[2 1 14 6 0
]+
[3
6 52 7 2
]=
[5 1 +
6 4
2 1 2
].
3
[2 1 14 6 0
]=
[6 3 3
12 18 0]
.
. i j aij. m n A
A =
a11 a12 . . . a1j . . . a1na21 a22 . . . a2j . . . a2n...
......
...ai1 ai2 . . . aij . . . ain...
......
...am1 am2 . . . amj . . . amn
A = [aij]. , A = [aij] B = [bij] mn A+B = [aij +bij]. (A+B)ij ij A + B,
(A + B)ij = aij + bij ,
(B)ij = bij .
(1.2) A, 1.5, x, 1.4. , .
. m n A n- x m-Ax, i i- A x, Ax = (y1, y2, . . . , yn),
yi = ai1x1 + ai2x2 + + ainxn, i = 1, . . . , m.
m n A, n- x m- Ax.
1.1 3 3 3- 3-,
1 2 63 0 31 1 4
250
=
2 + 10 + 06 + 0 + 02 + 5 + 0
=
1267
,

12
2 3 3- 2-,[
1 2 63 0 3
]
250
=
[2 + 10 + 06 + 0 + 0
]=
[126
].
Ax A x:
1 2 63 0 31 1 4
250
= 2
131
+ 5
201
+ 0
634
=
1267
.
, -
:
(Ax)i =n
j=1
aijxj .
. , , :
[ 2 1 0 ]
2 1 14 6 0
2 7 2
= [ 0 8 2 ] .
- . 2 , (1.1).
[0 0 1
],
[0 0 1
]
2 1 14 6 0
2 7 2
= [ 2 7 2 ] .
. A E:
E A =
1 0 02 1 0
0 0 1
2 1 14 6 0
2 7 2
=
2 1 10 8 2
2 7 2
(1.6)
. mn A np B. AB m p , ij i- A j- B,
(AB)ij =n
k=1
aikbkj, i = 1, . . . , m j = 1, . . . , p.

1 Gauss 13
A B.
1.2 2 2 I 2 3 B:
I B =
[1 00 1
] [2 3 74 6 0
]=
[2 3 74 6 0
].
1.3 , AB BA:
AB =[
1 6] [ 2
1
]=
[8
],
B A =
[21
] [1 6
]=
[2 121 6
].
1.4 P B, P B:
P B =
[0 11 0
] [2 37 8
]=
[7 82 3
],
B P =
[2 37 8
] [0 11 0
]=
[3 28 7
].
, .
1.1 1. i- AB B i- A.
2. j- AB A j- B.
. i- AB
[(AB)i1 . . . (AB)ip
]=
[ nk=1 aikbk1 . . .

BasismŁnec sto bibl—o tou G.Strang · PDF fileShmei‚seic maj€matoc M1122...

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