Let's say that we have a system of linear inequalities: [ c 1 , 1 c 1 , 2 … c 1 , n c 2 , 1 c 2 , 2 … c 2 , n ⋮ ⋮ ⋱ ⋮ c m , 1 c m , 2 … c m , n ] × [ x 1 x 2 ⋮ x n ] ≥ [ b 1 b 2 ⋮ b m ] It can be represented in a matrix form: C x ≥ bDoes it hold that: x ≥ C − 1 band that C is invertible if and only if the whole system is solvable?P.S. All the numbers x i , b i , c i , j are real. Would restricting them to be integers change the answer?EDIT 1: for matrices x and y it holds that x ≥ y if and only if every element of x is ≥ to corresponding element in y.EDIT 2: the x i are bounded to [ − 2 , 2 ].

Question

Let&#039;s say that we have a system of linear inequalities:      [                                        c                          1              ,              1                                                            c                          1              ,              2                                                …                                      c                          1              ,              n                                                                        c                          2              ,              1                                                            c                          2              ,              2                                                …                                      c                          2              ,              n                                                            ⋮                          ⋮                          ⋱                          ⋮                                                  c                          m              ,              1                                                            c                          m              ,              2                                                …                                      c                          m              ,              n                                            ]    ×      [                                        x            1                                                            x            2                                                ⋮                                                  x            n                                ]    ≥      [                                        b            1                                                            b            2                                                ⋮                                                  b            m                                ]  It can be represented in a matrix form:  C  x  ≥  bDoes it hold that:  x  ≥        C          −      1        band that   C is invertible if and only if the whole system is solvable?P.S. All the numbers       x    i    ,      b    i    ,      c          i      ,      j       are real. Would restricting them to be integers change the answer?EDIT 1: for matrices   x and   y it holds that   x  ≥  y if and only if every element of   x is   ≥ to corresponding element in   y.EDIT 2: the       x    i   are bounded to   [  −  2  ,  2  ].

Daniel Valdez · Accepted Answer

This feasibility problem   C  x  ≤  b is equivalent to the following optimization problem                    m        a        x        .                            t                                      c          i                x        +        t        ≤                              b          i                ,                 i        =        1        ,        …        ,        m            where       c    i   are the rows of   C.

Averi Mitchell · Accepted Answer

I assume you mean to define   a  ≥  b if and only if each component of   a is   ≥ than its corresponding component of   b, correct?

Let's say that we have a system of linear inequalities: [ <mtable rowspacing="4pt" colu

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New Questions in Algebra I