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Mathematik-Online problems:

Problem 41: True-Flase: Linear Maps, Matrices and LSE

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$ and $ \beta$ be linear maps of $ \mathbb{R}^{\mathit n}$ into $ \mathbb{R}^{\mathit n}$ with matrix representation $ A$ respectively $ B$, and let $ b$ be a vector in $ \mathbb{R}^{\mathit n}$. Mark which of the following statements are always true, and give reasons for your answer.
$ A^2=A$ implies $ A=0_n$ or $ A=E_n$  true $ \bigcirc $  false $ \bigcirc $
$ B$ invertible $ \Longrightarrow{\mathrm{ker}}(\beta\circ\alpha)={\mathrm{ker}}(\alpha)$  true $ \bigcirc $  false $ \bigcirc $
$ A^{\mathit m}=0_n$, for an $ m\in\mathbb{N} \ \Longrightarrow \ E_n+A$ invertible  true $ \bigcirc $  false $ \bigcirc $
$ B$ invertible and $ Ax=b$ solvable $ \Longrightarrow \ (B^{-1}AB)\, x=b$ solvable  true $ \bigcirc $  false $ \bigcirc $
(Authors: Kimmerle/Roggenkamp/Höfert)

see also:

[Solutions]
  automatisch erstellt am 14. 10. 2004