  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  
  [1X1.1 [33X[0;0YPhilosophy[133X[101X
  
  [33X[0;0Y[5XForms[105X  is  a  package for computing with sesquilinear and quadratic forms on
  finite  vector  spaces.  It provides users with the basic algebraic tools to
  work  with classical groups and polar geometries, and enables one to specify
  a  form  and  its  corresponding  geometry. The functionality of the package
  includes:[133X
  
  [30X    [33X[0;6Ythe construction of sesquilinear and quadratic forms;[133X
  
  [30X    [33X[0;6Yoperations  which  allow  a  user  to  change coordinates, that is, to
        ``change  form''  and  work in an isometric (or similar) formed vector
        space; and[133X
  
  [30X    [33X[0;6Ya way to determine the form(s) left invariant by a matrix group (up to
        a scalar).[133X
  
  
  [1X1.2 [33X[0;0YOverview over this manual[133X[101X
  
  [33X[0;0YThe  next  chapter ([14X2[114X) gives some basic examples of the use of this package.
  In  "Background  Theory of Forms" (Chapter [14X3[114X) we revise the basic notions of
  the  theory  of  sesquilinear  and  quadratic  forms,  where we also set the
  notation and conventions adopted by this package. In "Constructing forms and
  basic  functionality"  (Chapter  [14X4[114X), we describe all operations to construct
  sesquilinear and quadratic forms and basic attributes and properties that do
  not  require  morphisms.  In  "Morphims  of forms" (Chapter [14X5[114X) we revise the
  basic  notions of morphisms of forms, and the classification of sesquilinear
  and  quadratic  forms  on  vector  spaces  over  finite  fields. Operations,
  attributes  and  properties that are related to the computation of morphisms
  of forms, are also described in this chapter.[133X
  
  
  [1X1.3 [33X[0;0YHow to read this manual[133X[101X
  
  [33X[0;0YWe  have  tried to make this manual pleasant to read for the general reader.
  So  it  is inevitable that we will use Greek symbols and simple mathematical
  formulas.  To  make these visible in the HTML version of this documentation,
  you may have to change the default character set of your browser to UTF-8.[133X
  
  
  [1X1.4 [33X[0;0YRelease notes[133X[101X
  
  [33X[0;0YVersion  1.2.1  of  [5XForms[105X contains some changed and extra functionality with
  relation  to  trivial  forms. The changed and new functionality is described
  completely  in Section [14X4.9[114X. We gratefully acknowledge the useful feedback of
  Alice Niemeyer.[133X
  
  [33X[0;0YIn  version  1.2.2 of [5XForms[105X a minor bug, pointed out by John Bamberg, in the
  code  of  [11XIsTotallyIsotropicSubspace[111X  is  repaired.  On  the occasion of the
  release  of  the first beta versions of GAP4r5, we changed the names of some
  global  functions  such that a name clash becomes unlikely. Version 1.2.2 of
  [5XForms[105X is compatible with GAP4r4 and GAP4r5.[133X
  
  [33X[0;0YVersion  1.2.3  contains  a  new  operation  [11XTypeOfForm[111X.  Together with this
  addition,  some parts of the documentation, especially concerning degenerate
  and  singular  forms,  have  been  edited. A bug found in the methods for [11X\^[111X
  applicable on a pair of vectors and a hermitian form, and a pair of matrices
  and  a hermitian form has been fixed. A series of test files is now included
  in  the  tst  directory.  Alexander Konovalov pointed out the the init.g and
  read.g  files  had  windows  line  breaks,  this is also fixed. Finally, the
  documentation has been recompiled with the MathJax option.[133X
  
