Mathematics-Online lexicon: Annotation toImage and Kernel

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	Mathematik-Online lexicon: Annotation to
	Image and Kernel

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Given a linear map $\alpha: V\to W$ . The set

$\displaystyle \operatorname{Ker}(\alpha) := \{v \in V:\ \alpha(v) = 0\}$

is called kernel of $\alpha$ , and the set

$\displaystyle \operatorname{Im}(\alpha ) := \{w\in W:\ \exists v \in V\$ mit $\displaystyle \ \alpha(v)= w \}$

is called image of $\alpha$ .

We have to show that the sets are closed with respect to the linear operations.

(Authors: Burkhardt/Wipper)

automatisch erstellt am 15. 3. 2005

$\displaystyle \alpha(\lambda\, u)$	$\displaystyle =$	$\displaystyle \lambda \,\alpha(u) = \lambda \, 0 = 0$ and
$\displaystyle \alpha(u+v)$	$\displaystyle =$	$\displaystyle \alpha(u)+ \alpha(v) = 0+ 0 = 0 \; ,$

$\displaystyle \lambda \, u$	$\displaystyle =$	$\displaystyle \lambda \, \alpha(x) = \alpha(\lambda \, x)$
$\displaystyle u+v$	$\displaystyle =$	$\displaystyle \alpha(x)+ \alpha(y) = \alpha(x+y) \; .$