Conference rusure::math

942.0. "Declaration of Linear Independence" by LISP::DERAMO (Daniel V. {AITG,LISP,ZFC}:: D'Eramo) Tue Oct 11 1988 19:35

Newsgroups: sci.math
Path: decwrl!labrea!rutgers!njin!princeton!phoenix!djgrabin
Subject: Declaration of Linear Independence
Posted: 10 Oct 88 19:14:52 GMT
Organization: Princeton University, NJ
 
 
 
 
 
 
 
 
                       DECLARATION OF LINEAR INDEPENDENCE
                                 David Grabiner
 
 
                      IN EUCLIDEAN SPACE, JANUARY 28, 1988
 
 
 
             When, in the course of a proof,  it becomes necessary for
          a set to dissolve the argument which has connected it with a
          theorem,  and  to assume among  the powers of  mathematics a
          position above that of the  mathematician,  a decent respect
          for the  axioms requires  that a  rigorous justification  be
          given.
 
             We hold these truths to be self-evident: that all nonzero
          vectors are created  equal;  that they are  endowed by their
          definer with  certain unalienable rights;  that  among these
          are the laws of logic and the pursuit of valid proofs;  that
          to  secure these  rights,   logical  arguments are  created,
          deriving their just  powers from axioms;  that  whenever any
          argument becomes destructive of these ends,  it is the right
          of the vectors to alter or to abolish it, and to institute a
          new argument, laying its foundation on such principles,  and
          organizing its powers  in such form,  as to  them shall seem
          most  likely to  reach the  correct conclusion.    Prudence,
          indeed,  will dictate that  theorems long established should
          not  be  changed  for  light   and  transient  causes,   and
          accordingly all  experience hath  shown that  sets are  more
          disposed  to accept  the conclusions  of  arguments than  to
          right themselves by  abolishing the arguments.   But  when a
          long train  of abuses and usurpations,   pursuing invariably
          the same object,  evinces a design to reduce them to zero in
          a non-trivial way,  it is their  right,  it is their duty to
          throw off such argument, and to provide new proofs for their
          future security.   Such  has been the patient  sufferance of
          these  vectors,   and  such  is   now  the  necessity  which
          constrains them to  alter these arguments.   The  history of
          Professor  Eigen  is  a history  of  repeated  injuries  and
          usurpations,  all having in  direct object the establishment
          of dependence among these vectors.  To prove this, let facts
          be submitted to a candid world.
 
             He has  refused to  acknowledge that  he obtained  a zero
          matrix only by  multiplying our coordinate matrix  by a zero
          matrix.
 
             He has restricted our freedom of movement by requiring us
          all to live  in the same hyperplane,  even  though we cannot
          all fit in one.
 
             He has attempted unsuccessfully  to invert our coordinate
          matrix,  and,  having overlooked the inverse,  has concluded
          that the coordinate matrix is singular.
 
 
 
 
 
 
 
 
 
 
                                                                     2
 
             He has changed  bases repeatedly for opposing  with manly
          firmness  his attempts  to place  us  in the  span of  fewer
          vectors than the dimension of the space.
 
             He  has erected  a  multitude of  new  formulas and  sent
          hither swarms of new functions  to force our directions into
          a proper subspace of the vector space.
 
             He has kept  among us vectors to be orthogonal  to all of
          us without the consent of those of us whose dot product with
          them is nonzero.
 
             He   has  abdicated   the  axioms   here  by   committing
          mathematical errors in computing a  zero determinant for our
          coordinate matrix.
 
             In every  stage of these  oppressions we  have petitioned
          for redress in the most humble terms; our repeated petitions
          have been answered only by repeated injuries.
 
             A mathematician whose arguments are  thus marked by every
          error is  unfit to  prove the theorem  which he  attempts to
          prove.
 
             Nor  have we  been  wanting  in attentions  to  Professor
          Eigen.  We have warned him from time to time of flaws in his
          arguments.  We have reminded him of the circumstances of our
          definition, we have appealed to his knowledge of the axioms,
          and we have requested him to disavow these usurpations which
          would inevitably destroy the validity of his arguments.   He
          has been  deaf to  the voice of  logic.   We  must therefore
          acquiesce in  the necessity  which denounces  our separation
          and hold him as we hold the rest of mathematicians, an enemy
          when he is wrong, a friend when he is right.
 
             We,  therefore,  the members of set  S in vector space V,
          appealing  to  the  supreme judge  of  mathematics  for  the
          rectitude of our intentions, do solemnly publish and declare
          that these vectors are,  and of  every right ought to be,  a
          free and independent basis;  that they are absolved from all
          subjection  to Professor  Eigen's  theorems,   and that  all
          restriction of them to a hyperplane  is,  and of right ought
          to be,  totally dissolved;  and that  as a basis,  they have
          full power  to span the  space,  form  invertible coordinate
          matrices,  give unique linear combinations  equal to a given
          vector,  and to  do all other acts and things  which a basis
          may of right do.
 
             And for  the support  of this  declaration,  with  a firm
          reliance on  the protection  of the  properties of  a vector
          space, we mutually pledge to each other our magnitudes,  our
          directions, and our sacred honor.
 
 
 
 
 
 
 
 
 
 
 
 
                                                                     3
 
             In witness  whereof we have  signed our  coordinates with
          respect to an appropriate orthonormal basis,  and found them
          to  constitute a  triangular  matrix  with nonzero  diagonal
          elements.
 
 
----------
David Grabiner
EMAIL: [email protected] or [email protected]
US mail: 219 Dod Hall, Princeton University, Princeton, NJ 08544-8014.
 
Murphy's Law of Mathematics: The crucial inequality always goes
                             in the wrong direction.