Let's say I have a system of inequalities: A x ≤ g for some A ∈ R 4 × 4 , x ∈ R 4 , g ∈ R 4 , and A is full rank. Here, the ≤ denotes element-wise inequality.Specifically, I know that x lies in a two-dimensional subspace of R 4 (determined by the null space of some matrix N). What I'm interested in is the dimension of the solution to the above system of inequalities. More succinctly, I'm interested in the dimension of the set { x ∈ R 4 | x ∈ N u l l ( N ) , A x ≤ g }

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Let&#039;s say I have a system of inequalities:   A  x  ≤  g for some   A  ∈        R          4      ×      4      ,   x  ∈        R    4  ,   g  ∈        R    4  , and   A is full rank. Here, the   ≤ denotes element-wise inequality.Specifically, I know that   x lies in a two-dimensional subspace of         R    4   (determined by the null space of some matrix   N). What I&#039;m interested in is the dimension of the solution to the above system of inequalities. More succinctly, I&#039;m interested in the dimension of the set      {    x    ∈              R      4             |         x    ∈          N      u      l      l        (    N    )    ,         A    x    ≤    g    }

graffus1hb30 · Accepted Answer

If   A is an invertible matrix,   S  =  {  x  :  A  x  ≤  g  } is the image of   V  =  {  y  :  y  ≤  g  } under the linear transformation       A          −      1      , and in particular is a convex cone in           R    4   with nonempty interior. If you intersect this with a two-dimensional linear subspace, the intersection will be two-dimensional if the subspace intersects the interior of   S. However, it is also possible that the subspace only intersects the boundary of   S, in which case the dimension could be   1 or   0, or that the subspace does not intersect the boundary. For example, with   A  =  I and   g  =  (  0  ,  0  ,  0  ,  0  ), your intersection would have dimension   2 if   Null  (  N  ) contains a vector with all entries   &amp;gt;  0, but it would have dimension   1 if   Null  (  N  ) is spanned by   (  1  ,  0  ,  0  ,  0  ) and   (  0  ,  1  ,  −  1  ,  0  ) and dimension   0 (consisting only of the origin) if   Null  (  N  ) is spanned by   (  1  ,  −  1  ,  0  ,  0  ) and   (  0  ,  0  ,  1  ,  −  1  ), or be empty if       g    1    &amp;lt;  0 and   Null  (  N  ) contains only vectors   v with       v    1    =  0

Let's say I have a system of inequalities: A x ≤ g for some A

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