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Mathematik-Online problems:

Problem 176: Matrix Representation of a Linear Map

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
Each $ t\in\mathbb{R}$ defines a linear map $ \alpha_t : \mathbb{R}^3\longrightarrow\mathbb{R}^3$ via

$\displaystyle (2, 12, 1)^{\rm {t}}\longmapsto (t, 0, -1)^{\rm {t}}, \quad (-3, ... ...0)^{\rm {t}}, \quad (5, -4, -2)^{\rm {t}}\longmapsto (-1, 2, 1-t)^{\rm {t}} \ .$

a)
Find the matrix representation $ D_t$ of $ \alpha_t$ with respect to the canonical basis of $ \mathbb{R}^3$.
b)
For which $ t\in\mathbb{R}$ is $ \alpha_t$ bijective?
c)
Determine $ t$ in such a way, that $ \alpha_t$ maps the point $ P=(1, 1, 1)$ into the plane $ E: x_1-3x_2+2x_3=0$.


(Authors: Apprich/Höfert)

see also:

[Solutions]
  automatisch erstellt am 14. 10. 2004