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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>
+)]
+
+
+
+
+
+

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