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Mathematik-Online problems:

Problem 28: Matrix Representation of Linear Maps

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 $ V$ be a $ n$-dimensional vector space with basis $ b_1,\,\ldots , b_n$, and $ \varphi$ and $ \psi$ are the maps defined by

$\displaystyle \begin{array}{rllcll} \varphi: V\longrightarrow V, \ & b_1\longma... ...psto b_3, & \ldots , & b_{n-1}\longmapsto b_n, & b_n\longmapsto 0 \end{array}. $

a)
Find the matrix representation $ A$ of $ \varphi$ with respect to the basis $ b_1,\,\ldots , b_n$.
b)
Find the matrix representation $ B$ of $ \psi$ with respect to the basis $ b_1,\,\ldots , b_n$.
c)
Determine $ {\mathrm{Rg}}\, B^{\mathit k}$, for all $ k\in\mathbb{N}$.

(Authors: Kimmerle/Höfert)

see also:

[Solutions]
  automatisch erstellt am 14. 10. 2004