Mathematics-Online course: Linear Algebra-Matrices-Linear Maps-Image and Kernel

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	Mathematics-Online course: Linear Algebra - Matrices - Linear Maps
	Image and Kernel

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 $\alpha$ .

(Authors: Burkhardt/W. Kimmerle)

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

(Authors: Burkhardt/Wipper)

automatically generated 4/21/2005

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

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