# Differences

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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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