Mathematik-Online problems: Problem 66: True-False: Linear Algebra

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	Mathematik-Online problems:
	Problem 66: True-False: Linear Algebra

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

a)

Proof the following formulas for all $n\in\mathbb{N}$ :

i) ${\displaystyle{\sum_{k=1}^n (2k-1)^2 = \frac{1}{3}\,n\,(4n^2-1)}}$ ii) ${\displaystyle{\sum_{k=0}^n \left({n\atop k}\right) = 2^n}}$

b)

Let be $z\in\mathbb{C}$ , $A\in\mathbb{C}^{n\times n}$ and $B\in\mathbb{C}^{m\times n}$ . Let $\beta$ be the linear map defined by $x\longmapsto Bx$ and

the quadric given by

$\displaystyle Q:\, 3x_1^2-2x_2^2+3x_3^2+6x_1x_3-1=0 \ .$

Mark which statements are always true respectively false, and give reasons for your answers.

$z+\overline{z}=0$ $\Longleftrightarrow$ $z\in\mathbb{R}$	true $\bigcirc$	false $\bigcirc$
The map $f: \mathbb{C}\longrightarrow \mathbb{R}$ , $z\longmapsto \vert z\vert$ , is surjective	true $\bigcirc$	false $\bigcirc$
${\rm {Ker}}\,(\beta)\subset\mathbb{C}^n$	true $\bigcirc$	false $\bigcirc$
${\rm {det}}\,(A+A^{\rm {t}})=2\,{\rm {det}}\,(A)$	true $\bigcirc$	false $\bigcirc$
0 is -fold eigenvalue of $\Longleftrightarrow$ ${\rm {Rg}}\,(A)=n-k$	true $\bigcirc$	false $\bigcirc$
describes a hyperbolic cylinder	true $\bigcirc$	false $\bigcirc$

(Authors: Walk/Blind/Kimmerle/Apprich/Höfert)

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automatisch erstellt am 18. 10. 2004