How to prove this? The existence of solutions to linear inequalitiesA system of real homogeneous linear inequalities λ i &gt; 0, i = 1 , 2 , … , m, has a solution if and only if there is no nontrivial linear dependence with nonnegative coefficients among the λ i . For example, λ i = ∑ a i j x j .

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How to prove this? The existence of solutions to linear inequalitiesA system of real homogeneous linear inequalities       λ    i    &amp;gt;  0,   i  =  1  ,  2  ,  …  ,  m, has a solution if and only if there is no nontrivial linear dependence with nonnegative coefficients among the       λ    i  . For example,       λ    i    =  ∑      a          i      j            x    j  .

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Consider the convex hull   C of   {      v    i    }. Then,   C is compact. Thus, the real valued function   v  ↦  ‖  v  ‖ on   C attains its minimum on   C. Say   x  :=      x      min   is the minimizer. Then,   ‖  x  ‖  &amp;gt;  0 (by hypothesis,   0  ∉  C).Then, one verifies directly that       v    ′    x  &amp;gt;  0 for all   v  ∈  C: this is clear for   v  =  x; now, if   v  ≠  x, by convexity of   C, we have   t  v  +  (  1  −  t  )  x  ∈  C for all   t  ∈  [  0  ,  1  ] whence   ‖  t  v  +  (  1  −  t  )  x  ‖  ≥  ‖  x  ‖ for all   t  ∈  [  0  ,  1  ]; it now follows that      t    2    ‖  v  −  x      ‖    2    +  2  t  ⟨  v  −  x  ,  x  ⟩  ⩾  0   for all   t  ∈  [  0  ,  1  ]  .This suggests that we look at the function  f  (  t  )  =      t    2    ‖  v  −  x      ‖    2    +  2  t  ⟨  v  −  x  ,  x  ⟩  ,    t  ∈  Rfor further analysis. The graph of   f is a parabola open upwards and for   t  ∈  [  0  ,  1  ],   f  (  t  )  ≥  0. Now, setting derivative equal to   0, we obtainwhere   f attains the minimum. Clearly,       t    0    ≤  0 (try sketching the graph of   f reminding yourself that   f  (  t  )  ≥  0 for all   t  ∈  [  0  ,  1  ]. In particular,  ⟨  v  ,  x  ⟩  ≥  ⟨  x  ,  x  ⟩  &amp;gt;  0.Thus, in particular, we have found an   x  ∈        R    n   such that       v    i    ′    x  &amp;gt;  0.

How to prove this? The existence of solutions to linear inequalities A system of real homogeneous l

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