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Mathematics-Online problems:

Interactive Problem 38: Matrix Representation of a Linear Map with respect to different Bases

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
Let $ \alpha: \mathbb{R}^4 \longrightarrow \mathbb{R}^4$ be the linear map defined via

$\displaystyle (4, 9, 0, 8)^{\operatorname t}$ $\displaystyle \longmapsto (2, 5, -5, 1)^{\operatorname t}$    
$\displaystyle (-2, 7, 3, 4)^{\operatorname t}$ $\displaystyle \longmapsto (-7, -8,-6, 7)^{\operatorname t}$    
$\displaystyle (-4, 2, -7, 1)^{\operatorname t}$ $\displaystyle \longmapsto (-4, -5, -3, -7)^{\operatorname t}$    
$\displaystyle (9, -2, -2, 2)^{\operatorname t}$ $\displaystyle \longmapsto (-4, -8, 9, 1)^{\operatorname t}\ .$    

Find the matrix representation $ A$ of $ \alpha$ with respect to the canonical basis $ e_1, e_2, e_3, e_4$.

$ A= \left(\rule{0pt}{8ex}\right.$
$ \left.\rule{0pt}{8ex}\right)$

Find the matrix representation $ B$ of $ \alpha$ with respect to the basis

$\displaystyle b_1$ $\displaystyle =(-1, 0, 0, 0)^{\operatorname t}$    
$\displaystyle b_2$ $\displaystyle =(1, -1, 0, 0)^{\operatorname t}$    
$\displaystyle b_3$ $\displaystyle =(0, 1, -1, 0)^{\operatorname t}$    
$\displaystyle b_4$ $\displaystyle =(0, 0, 1, -1)^{\operatorname t}\ .$    

$ B= \left(\rule{0pt}{8ex}\right.$
$ \left.\rule{0pt}{8ex}\right)$


   

(Authors: App/Höfert)

see also:

  automatically generated: 8/11/2017