Линейная алгебра. Теоремы и алгоритмы. Яцкин Н.И. - 132 стр.

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132 Линейные пространства. Базисы и размерности Гл. 1
·
W
1
, B
1
=
−1 1 1
1 1 1
−1 1 1
−1 −1 0
−1 −3 −1
1 −1 −1
, d
1
=3, A
1
=
−1 0 1 0 0 0
1 0 0 −2 1 0
1 0 0 0 0 1
, c
1
=3,
[ −x
1
+x
3
=0, x
1
−2x
4
+x
5
=0, x
1
+x
6
=0 ]
¸
·
W
2
, B
2
=
−2 1
0 0
−1 0
0 0
1 0
0 1
, d
2
=2, A
2
=
1 0 0 0 2 −1
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
, c
2
=4,
[ x
1
+2x
5
−x
6
=0, x
3
+x
5
=0, x
4
=0 ]
¸
·
W
3
, B
3
=
−1 1 1 −2
1 1 1 0
−1 1 1 −1
−1 −1 0 0
−1 −3 −1 1
1 −1 −1 0
, d
3
=4, A
3
=
"
0 0 1 −2 1 0
−1 2 0 0 0 1
#
, c
3
=2,
[ x
3
−2x
4
+x
5
=0, −x
1
+2x
2
+x
6
=0, ]
¸
·
W
0
, B
0
=
−1
0
−1
0
1
1
, d
0
=1, A
0
=
1 0 0 0 0 1
0 1 0 0 0 0
0 0 1 0 0 1
0 0 0 1 0 0
0 0 0 0 1 −1
, c
0
=5,
[ x
1
+x
6
=0, x
2
=0, x
3
+x
6
=0, x
4
=0, x
5
−x
6
=0 ]
¸
·
W
4
, B
4
=
−1 1 −2
1 1 0
−1 1 −1
−1 0 0
−1 −1 1
1 −1 0
, d
4
=3, A
4
=
1
2
1
2
−1 1 0 0
1 1 −1 0 1 0
−1 0 2 0 0 1
, c
4
=3,
[
1
2
x
1
+
1
2
x
2
−x
3
+x
4
=0, x
1
+x
2
−x
3
+x
5
=0, −x
1
+2x
3
+x
6
=0 ]
¸
132       Линейные пространства. Базисы и размерности                                                                          Гл. 1


                                   
                  −1    1      1
                                 
                1      1      1                                                                               
  ·            
               
                                  
                                                 −1                    0           1           0       0    0
                −1     1      1                                                                            
      W1 , B1 =
               
                                  , d1 =3, A1 = 1
                                                                      0           0       −2          1    0
                                                                                                              , c1 =3,
                −1    −1      0 
                                                 1                    0           0           0       0    1
                                 
                −1    −3      −1 
                  1    −1      −1
                                                                                                                                   ¸
                                                [ −x1 +x3 =0, x1 −2x4 +x5 =0, x1 +x6 =0 ]
                  −2   1
                                                                                              
                0     0               1             0   0         0           2           −1
  ·            
               
                        
                                                                                            
                −1    0             0              1   0         0           0           0 
      W2 , B2 =
               
                        , d2 =2, A2 =
                                     
                                                                                              , c2 =4,
                                                                                              
                0     0             0              0   1         0           0           0 
                       
                       
                1     0               0             0   0         1           0           0
                  0    1
                                                                                                                                   ¸
                                                            [ x1 +2x5 −x6 =0, x3 +x5 =0, x4 =0 ]
                  −1    1      1        −2
                                          
                1      1      1        0 
  ·            
               
                                           
                                                         "                                                              #
                −1     1      1        −1                 0                           0        1      −2       1   0
      W3 , B3 =
               
                                           , d3 =4, A3 =
                                                                                                                            , c3 =2,
                −1    −1      0        0                  −1                          2        0      0        0   1
                                          
                                          
                −1    −3      −1       1 
                  1    −1      −1       0
                                                                                                                                   ¸
                                                          [ x3 −2x4 +x5 =0, −x1 +2x2 +x6 =0, ]
                       
                 −1                                                                            
                                      1         0       0     0           0           1
                0      
  ·            
               
                        
                        
                                      
                                      0          1       0     0           0
                                                                                          
                                                                                        0 
                −1                                                                     
      W0 , B0 =        , d0 =1, A0 =
                                      0          0       1     0           0
                                                                                          
                                                                                        1 , c0 =5,
                                                                                       
                0                                                                      
                                    0          0       0     1           0           0 
                       
                1      
                                        0         0       0     0           1           −1
                  1
                                                                                                                                   ¸
                                            [ x1 +x6 =0, x2 =0, x3 +x6 =0, x4 =0, x5 −x6 =0 ]
                                   
                  −1    1      −2
                                 
                1      1      0                                                                               
  ·            
               
                                  
                                  
                                                   1                    1
                                                                                    −1              1   0    0
                −1     1      −1               2                     2
                                                                                                              
      W4 , B4 =
               
                                  , d4 =3, A4 = 1
                                                                      1           −1              0   1    0
                                                                                                              , c4 =3,
                −1     0      0 
                                                −1                    0               2           0   0    1
                                 
                −1    −1      1 
                  1    −1      0
                                                                                                                                   ¸
                       [ 12 x1 + 12 x2 −x3 +x4 =0, x1 +x2 −x3 +x5 =0, −x1 +2x3 +x6 =0 ]

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