  . 12   
 


         .       .       :         ,  ,  ,  , ,   .  12    32    ,    .         ;      ,    .     ,      .





 

  . 12   



  . 12   

 

         .       .        :       -  ,  ,  ,  , ,   .  12    32     ,    .         ;      ,    .   -  ,      .   ,          .




   





   


        : , , ,  .      ,     .        ?           ?         ?

       ,       .     ,     .     ,         .       ,       .

    .      ;         -  .  12    32   ,    .       ,         .

       .  ,    .  ,  ,     .      :   ,           .

    ,   .       ,      .       ;   ,      .




  


 ,      .  (x; y)     ;           .         : (x; y) + (u; v) = (x + u; y + v),  c(x; y) = (cx; cy).

  m * n  m   n :    .  ,  A   (2; -1; 0)  (1; 3; 4),    2 * 3.  a_ij     i   j;    .    a_12 = -1, a_21 = 1.

         .        :  (2; -1 | 5)  2x - y = 5   (x; y).    .     0,    .

  x  n   Ax   m .      A:       x,  .   (2; -1)   (3; 4)  2 * 3 - 1 * 4 = 2.   ,            .

R1, R2  R3 -  .   R2 <- R2 - 3R1    :         ,   ,   .    .      ,    .

 I    :     ,   - .  0   ,     ;      .   A^(-1) -       A.      9.

R^n -      n .        ,       .  "", ""  ""     ;     .   : 1/3     0,33.




 


     , ,      .         .      ,    ,      .

       .  ,  ,  ,  ,         .              .

   ,     ,   .     ,   . ,        .         ,   ;           .




 





 1


1.    (x; y; z)      2y - z = 3, x + z = -1. ,    (-2; 2; 1).        ,       ?

 -  1

 -  1




 2


2.   3x - 2 = 7.   2/3 - 5/6  .       .

 -  2

 -  2




 3


3.        t: 0t = 0; 0t = 4; 2t = 4.             t?

 -  3

 -  3




 4


4. A   (1; 3), (0; -2), (-1; 1).   A   A(1; -2),   "   ". ,     ,   .

 -  4

 -  4




  


1.      

2.    

3.     

4.    

5.     

6.      

7.     

8.    

9.      

10.     

11.       

12.  ,   




     





 1.      


:      ,              .       ,      .

         .

 ,         .

   .   (x; y; z)      x,   y,   z,        .

    0.      :       .

 A   .   m,   n,  A  m*n.   [A|b]   m*(n+1):       .

      .  (1; -1; 2)   (x; y; z)  x = 1, y = -1, z = 2.        .

 -           .     ,      .         .








 1  1:   (x; y; z)  2y - z = 3  x + z = -1   (0; 2; -1 | 3)  (1; 0; 1 | -1).     ;       .

       : x + 2z = 5; -2x + y - z = -5; 3y + z = -1.   (x; y; z)  A, b  [A|b]. ,      u = (1; -1; 2)  v = (-1; -4; 3)  .      .

1.      x + 0y + 2z = 5.      ,     0x + 3y + z = -1.   ,       .

2.  A  3*3   (1; 0; 2), (-2; 1; -1)  (0; 3; 1).  b = (5; -5; -1)   3*1.   5     A:        .

3.   [A|b]  3*4   (1; 0; 2 | 5), (-2; 1; -1 | -5)  (0; 3; 1 | -1).   -  ,    .     ,    .

4.  u    1 + 2*2 = 5.   -2*1 + (-1) - 2 = -5.   3*(-1) + 2 = -1.          ,  u   .

5.  v    -1 + 2*3 = 5,    -2*(-1) + (-4) - 3 = -5.     ,   ""  :      .

6.     v  3*(-4) + 3 = -9,     -1.    -9 = -1. , v   ,      .         .

7.     .    1x + 0y + 2z = 5,  - -2x + 1y - 1z = -5,  - 0x + 3y + 1z = -1.   ,  ,       .

: A   (1; 0; 2), (-2; 1; -1)  (0; 3; 1), b = (5; -5; -1).  [A|b]      5, -5  -1.        u = (1; -1; 2).

:      u  (5; -5; -1),  v - (5; -5; -9).    b     .           ;        .

 :      y,      (1; 2),     .

       .   ,      2           y.   x + 2z = 5    x + 2y = 5.

 :        (x; y; z),    ,  .     :    ,           ?

             .     .      ,        .

   




 1.1


   2*4   (2; -1; 0 | 4)  (0; 3; 1 | -2).      - (z; x; y).      x = 2, y = -8, z = 3.        ?

 -  1.1

 -  1.1




 1.2


  2x + ay = 1; x - y + z = 4   a.  a,    (1; -1; 2)   (x; y; z)  .  A   a.      ,   ?    (0; 1; 5).

 -  1.2

 -  1.2




 2.    


: ,        ,      ,          .

         .

         .

  :   ;      ;      ,    .     ,   .

 R2 <- R2 - 3R1    .       .           .

     .   c    1/c,  c != 0.  cR1      cR1,      .

    :            .   ,    .

  ,   ,    .               .

  2x - y = 1; 3x + 2y = 12,   (x; y).    ,   . ,    -     ,      .

1.     2*3   (2; -1 | 1)  (3; 2 | 12).      2: R1 <- (1/2)R1.   (1; -1/2 | 1/2)  (3; 2 | 12).   -     2.

2.  R2 <- R2 - 3R1.    ,   : 12 - 3*(1/2) = 21/2.      (1; -1/2 | 1/2)  (0; 7/2 | 21/2).   - R2 <- R2 + 3R1.

3.     2/7.   ,      7/2.   (1; -1/2 | 1/2)  (0; 1 | 3).       y = 3.

4.  R1 <- R1 + (1/2)R2.    ;      y,     1/2 + 3/2 = 2.    (1; 0 | 2)  (0; 1 | 3).    - R1 <- R1 - (1/2)R2.

5.      x = 2  y = 3,     .         . ,       ,     .

6.     .      L1 = b1  L2 = b2,  L1  L2 -   .     L2 + cL1 = b2 + cb1. ,  L1 = b1  L2 + cL1 = b2 + cb1  cL1 = cb1  L2 = b2.       c,      .

7.       :  L = b  cL = cb;  c != 0  cL = cb   c  L = b.       ,   .  ,          .

:   - (x; y) = (2; 3).  : R1 <- (1/2)R1; R2 <- R2 - 3R1; R2 <- (2/7)R2; R1 <- R1 + (1/2)R2.    .

:      2*2 - 3 = 1  3*2 + 2*3 = 12.    .      :      ,         .

 :    0  :     -    0 = 0.

      . ,  x = 1; y = 2    (1; 2).      0,  0 = 0; y = 2.    (7; 2),     .       0.

 :      .      .  0 = 0    ,              .

    ,       .   :     ,          .        .

   




 2.1


  2x + y = 4; x - y = -1   R2 <- R2 - (1/2)R1,    (2; 1 | 4)  (0; -3/2 | -1).   ,     ,         .

 -  2.1

 -  2.1




 2.2


  x + y = 3; x - y = 1.             ,  2x = 4; 2x = 4.    ?            .

 -  2.2

 -  2.2




 3.     


:        ,         .       ,    .

     .

         .

       ,           .     ;   -  .

     1.      1  -1,     .    ,        .

      .   R3 <- R3 - cR2,     ,    ;      .

      .      ,   .        ,   -  .

           z,   y,   x,      .   ,     .

  2x + y - z = 0; x - y + 2z = 9; 3x + 2y + z = 7.   - (x; y; z).     ,        ,        .

1.     3*3.    3*4   (2; 1; -1 | 0), (1; -1; 2 | 9)  (3; 2; 1 | 7).      x  1,      .  (1; -1; 2 | 9), (2; 1; -1 | 0), (3; 2; 1 | 7).

2.  R2 <- R2 - 2R1,    ,   .  (0; 3; -5 | -18),  1 - 2*(-1) = 3, -1 - 2*2 = -5  0 - 2*9 = -18.  R3 <- R3 - 3R1    (0; 5; -5 | -20);      .

3.    (1; -1; 2 | 9), (0; 3; -5 | -18)  (0; 5; -5 | -20).    5  R3 <- R3 - (5/3)R2.    :   z  -5 + 25/3 = 10/3,     -20 + 30 = 10.    (0; 0; 10/3 | 10).

4.     x - y + 2z = 9; 3y - 5z = -18; (10/3)z = 10.     10/3 .   z = 10*(3/10) = 3.      :     .

5.  z = 3   : 3y - 15 = -18.  15     3y = -3, , y = -1.   5z     ;        .

6.     x - (-1) + 2*3 = 9,   x + 7 = 9  x = 2.   .      ,    10/3      .

7.           .       z,  y,  x.    -    .        2,         .

:    (x; y; z) = (2; -1; 3).       (1; -1; 2 | 9), (0; 3; -5 | -18)  (0; 0; 10/3 | 10).

:  ,       2*2 - 1 - 3 = 0; 2 - (-1) + 2*3 = 9; 3*2 + 2*(-1) + 3 = 7.     .    1, 3  10/3     .

 :         z,      ,       .

      . ,  x + y = 2; z = 3   (0; 2; 3),  (1; 1; 3).   z   x  y.          ,      .

 :     ,         .          ,     .

    ,      .     ,      ,    .      .

   




 3.1


      2x + y - z = 1; -3y + 2z = 8; 4z = 10.   z = 5/2,   -3y = 8 + 5 = 13.  ,      ,        .

 -  3.1

 -  3.1




 3.2


  2y + z = 5; x - y = 1; 2x + y - z = h ,    z = 1.    h  ,     . ,    x        .

 -  3.2

 -  3.2




 4.    


:      ,   0 = 0   0 = c  c != 0       ,    .

        .

 ,       .

  ,        ,  ,    .       :      .

 (0; 0; 0 | 0)     0x + 0y + 0z = 0,    0 = 0.             .

 (0; 0; 0 | c),  c != 0,    0 = c.     :      .          .

       .  ,    ,      ;      ,  .

      ,      .      :    ,       .








 2 -   ,    .  x = 1, y = 2   .  x + y = 3, 0 = 1 .  x + y = 3, 0 = 0    (3 - t; t), t in R.          :       .

  x + 2y - z = 4; 2x + 4y - 2z = 8; 3x + 6y - 3z = 13   (x; y; z).    : ,    ,  ,     .

1.  A  3*3   (1; 2; -1), (2; 4; -2), (3; 6; -3).    3*4   (1; 2; -1 | 4), (2; 4; -2 | 8), (3; 6; -3 | 13).       ,      .

2.  R2 <- R2 - 2R1.     :     ,     8 - 2*4 = 0.   (0; 0; 0 | 0).  ,         2.

3.   R3 <- R3 - 3R1.   - .     ,     13 - 3*4 = 1.    (0; 0; 0 | 1).

4.    x + 2y - z = 4; 0 = 0; 0 = 1.     ,        .    0 = 1      x, y, z:       .

5. ,     .        ,       .     ,      .

6.      .    ,    3x + 6y - 3z = 12.          13.         12  13;      .

:  .   - (0; 0; 0 | 1),   R3 - 3R1;     0 = 1.  (0; 0; 0 | 0)      .

:              13 - 12 = 1.      x, y, z      ,       .

 :  ,        ,    .

 ""     .   (0; 0; 0 | 0)  (0; 0; 0 | 1)  ,   :    ,   .          .

 :     ,       .       0 = 0;  0 = c   c     .

   -   .   : 0 = 0   ,  0 = c  c != 0   .        ,    .

   




 4.1


   h   2x - y = 3; 6x - 3y = h?       h     .         .

 -  4.1

 -  4.1




 4.2


    (x; y; z)       (1; 0; 2 | 5), (0; 1; -1 | -2)  (0; 0; 0 | c).   c = 0,   c = -4.  .       ,    "   ,  ".

 -  4.2

 -  4.2




 5.     


:              .   :           .

      .

 ,    .

          .     :       ,     .

     .  "t -   "    t = 0, 1, 2:          .

   - :              .        .

  - :   ,            .         .

        .          ,      ,      .

    x + 2y - z = 4; 2x - y + 3z = 1   (x; y; z).        .     ,  y  z  .

1. A   2*3   (1; 2; -1), (2; -1; 3).    2*4   (1; 2; -1 | 4)  (2; -1; 3 | 1).  R2 <- R2 - 2R1        (0; -5; 5 | -7).

2.      x  y.   -5y + 5z = -7  y - z = 7/5,       -5. , y = z + 7/5.  ,  z, .




  .


   .

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