<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <!-- %% --> <!-- %A vspc.xml GAP documentation Willem de Graaf --> <!-- %A Thomas Breuer --> <!-- %% --> <!-- %% --> <!-- %Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland --> <!-- %Y Copyright (C) 2002 The GAP Group --> <!-- %% -->
<Chapter Label="Vector Spaces">
<Heading>Vector Spaces</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Operations and Attributes for Vector Spaces">
<Heading>Operations and Attributes for Vector Spaces</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Domains of Subspaces of Vector Spaces">
<Heading>Domains of Subspaces of Vector Spaces</Heading>
<!-- %T The domains of subspaces are useful for example because groups and algebras --> <!-- %T act on their sets of elements. --> <!-- %T (show an example) -->
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Bases of Vector Spaces">
<Heading>Bases of Vector Spaces</Heading>
<#Include Label="[1]{basis}">
<#Include Label="IsBasis">
<#Include Label="Basis"> <!-- %T show the use of the <Q>no check</Q> version -->
<#Include Label="CanonicalBasis">
<#Include Label="RelativeBasis">
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Operations for Vector Space Bases">
<Heading>Operations for Vector Space Bases</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Operations for Special Kinds of Bases">
<Heading>Operations for Special Kinds of Bases</Heading>
<#Include Label="IsCanonicalBasis">
<#Include Label="IsIntegralBasis">
<#Include Label="IsNormalBasis"> <!-- % add an example of a non-integral basis; --> <!-- % for that, add a method that takes an integral basis, --> <!-- % and inspects the basechange matrix -->
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Vector Space Homomorphisms">
<Heading>Vector Space Homomorphisms</Heading>
<#Include Label="[1]{vspchom}">
<#Include Label="LeftModuleGeneralMappingByImages">
<#Include Label="LeftModuleHomomorphismByImages"> <!-- % add an example for fin. fields! -->
<#Include Label="LeftModuleHomomorphismByMatrix"> <!-- % show images! -->
<#Include Label="NaturalHomomorphismBySubspace"> <!-- % show the computation of images etc.! -->
<#Include Label="Hom">
<#Include Label="End">
<#Include Label="IsFullHomModule">
<#Include Label="IsPseudoCanonicalBasisFullHomModule">
<#Include Label="IsLinearMappingsModule"> <!-- % example: Create a space, show some computations! -->
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Vector Spaces Handled By Nice Bases">
<Heading>Vector Spaces Handled By Nice Bases</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="How to Implement New Kinds of Vector Spaces">
<Heading>How to Implement New Kinds of Vector Spaces</Heading>
<!-- % put this into the <C>prg</C> or <C>ext</C> manual? -->
<#Include Label="DeclareHandlingByNiceBasis">
<#Include Label="NiceBasisFiltersInfo">
<#Include Label="CheckForHandlingByNiceBasis">
</Section>
<Section Label="Tensor Products and Exterior and Symmetric Powers">
<Heading>Tensor Products and Exterior and Symmetric Powers</Heading>
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