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Vector space

Posted November 30th, 2007 by Isoscel

Here is the definition of a vector space over a field $ \mathcal{K} $.
Let $ (\mathcal{K},+,.) $ be a field and ($ V $,+) a commutative group.
$ (i)\forall x,y,z \in V, x+(y+z)=(x+y)+z $
$ (ii)\exists \:0_V\in V $ such as $ \forall x\in V, x+0_V=0_V+x=x $
$ (iii)\forall x\in V, \exists -x\in V, $ such as $ x+(-x)=(-x)+x=0_V $
$ (iv)\forall x,y \in V, x+y=y+x=0_V $
Let * be a binary operation $ *:\mathcal{K}\times V\rightarrow V $ such as
$ (v)\forall \alpha ,\beta \in \mathcal{K} x\in V, (\alpha +\beta )*x= \alpha *x+\beta* x $
$ (vi)\forall \alpha \in \mathcal{K},\: x,y \in V, \alpha *(x+y)= \alpha *x+\alpha *y $
$ (vii)\forall \alpha ,\betha \in \mathcal{K} x\in V, \alpha *(\beta *x)= (\alpha \beta) *x $
$ (viii)\forall x\in V, 1_{\mathcal{K}}*x=x $
Then $ (V,+,*) $ is called a vector space over field $ \mathcal{K} $ or simply a $ \mathcal{K} $-vector space.

Morphism of Vector Space

Let $ (V_1,+,*) $ and $ (V_2,+,*) $ two $ \mathcal{K} $-vector spaces.
A function $ f:V_1\rightarrow V_2 $ is a $ \mathcal{K}- $ morphism of $ \mathcal{K} $-vector spaces if
$ \forall \alpha ,\beta \in \mathcal{K} ,\:x,y \in V, $,

$$ f(\alpha *x+\beta *y)=\alpha *f(x)+\beta *f(y)$$

Subspace of a vector space

Let $ (V,+,*) $ be a $ \mathcal{K} $-vector space. A subset $ \mathcal{W}\subseteq V $ is called a vector subspace of $ V $ if
$ \forall \alpha ,\beta \in \mathcal{K} ,\:x,y \in \mathcal{W} $,

$$ \alpha *x+\beta *y\in \mathcal{W}$$
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