We have been going through how to solve the system of equations known as Ax=b, where A is a matrix, x is a vector and b is a vector. I understand that if we have A and b we must find out what x is, this happens via Gauss-Jordan elimination, back substitution, etc. What does solving the linear system of equations actually mean though?

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garbhaighzf · Accepted Answer

A system                     (1)                    A        x        =        b            of equations in unknowns       x    1  ,       x    2  ,   …,       x    n   implicitly defines the subset  S  :=  {  x  ∈                    R              n          |      A  x  =  b  }    ⊂                    R              n       .&quot;Implicitly&quot; means that for any given   x  ∈                    R              n   it is easy to test whether it is an element of   S or not (just compute   A  x and check whether this is   =  b); but you don&#039;t have a-priori a complete overview over this set   S.Solving the system   (  1  ) (or a similar system containing equations of a more complicated nature) means obtaining such an overview. If   S is in fact the empty set you want a proof of this fact; if   S contains just finitely many points you want an explicit list of these points, etc.In linear algebra the favorite case is when   S is a one-element set   {  a  }; we then call a &quot;the&quot; solution of   (  1  ). But it very often happens that   S is an infinite set; say, a two-dimensional plane embedded in                     R              n  . In such a case you want an explicit &quot;production scheme&quot; with a certain number of free variables, in other words: a parametric representation of   S, which generates every point of   S exactly once. In the case where   S is a two-dimensional plane such a parametric representation looks like                    (2)                    S        :                (        u        ,        v        )        ↦        x        :=        a        +        u        p        +        v        q                                      (                          (        u        ,        v        )        ∈                                            R                                2                                      )                                   ,            whereby the vectors   a,   p,   q have to be computed from the data   A and   b. (Note that the same   S has many different representations   (  2  ).)

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