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vector_spaces_over_a_field [2010/07/29 15:46] (current)
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 +=====Vector spaces over a field=====
 +
 +Abbreviation: **FVec**
 +====Definition====
 +A \emph{vector space over a [[fields]]} $\mathbf{F}$ is a structure $\mathbf{V}=\langle V,+,-,0,f_a\ (a\in F)\rangle$ such that
 +
 +
 +$\langle V,+,-,0\rangle $ is an [[abelian groups]]
 +
 +
 +scalar product $f_a$ distributes over vector addition:  
 +$a(x+y)=ax+ay$
 +
 +
 +$f_{1}$ is the identity map:  $1x=x$
 +
 +
 +scalar product distributes over scalar addition:  $(a+b)x=ax+bx$
 +
 +
 +scalar product associates:  $(a\cdot b)x=a(bx)$
 +
 +Remark:
 +$f_a(x)=ax$ is called \emph{scalar multiplication by $a$}.
 +
 +==Morphisms==
 +Let $\mathbf{V}$ and $\mathbf{W}$ be vector spaces over a field $\mathbf{F}$.
 +A morphism from $\mathbf{V}$ to $\mathbf{W}$ is a function $h:V\rightarrow W$ that is \emph{linear}:
 +
 +$h(x+y)=h(x)+h(y)$, $h(ax)=ah(x)$ for all $a\in F$
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +
 +====Properties====
 +^[[Classtype]]  |variety |
 +^[[Equational theory]]  | |
 +^[[Quasiequational theory]]  | |
 +^[[First-order theory]]  | |
 +^[[Locally finite]]  |no |
 +^[[Residual size]]  |unbounded |
 +^[[Congruence distributive]]  |no |
 +^[[Congruence modular]]  |yes |
 +^[[Congruence n-permutable]]  |yes, $n=2$ |
 +^[[Congruence regular]]  |yes |
 +^[[Congruence uniform]]  |yes |
 +^[[Congruence extension property]]  |yes |
 +^[[Definable principal congruences]]  |no |
 +^[[Equationally def. pr. cong.]]  |no |
 +^[[Amalgamation property]]  | |
 +^[[Strong amalgamation property]]  | |
 +^[[Epimorphisms are surjective]]  | |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &\\
 +f(3)= &\\
 +f(4)= &\\
 +f(5)= &\\
 +f(6)= &\\
 +\end{array}$
 +
 +====Subclasses====
 +====Superclasses====
 +[[Abelian groups]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]
 +
 +
 +
 +
 +
 +