vector_spaces - MathStructures

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=====Vector spaces=====

Abbreviation: **FVec**
====Definition====
A \emph{vector space} over a [[fields|field]] $\mathbf{F}$ 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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