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Mathematik-Online problems:

Problem 25:

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 $ S: Ax=b$ be a linear system of equations with matrix $ A\in\mathbb{C}^{\mathit{n\times n}}$ and right-hand side $ b\in\mathbb{C}^{\mathit n}$. Moreover let $ \varphi: \mathbb{C}^{\mathit n}\longrightarrow \mathbb{C}^{\mathit n}$ be the linear map defined by $ x\longmapsto Ax$. Mark which statements are always true, and give reasons for your answer.

$ \det(A) \ne 0 \ \Longleftrightarrow \ S$ has a unique solution  true $ \bigcirc $  false $ \bigcirc $
$ {\mathrm{rank}}(A) < n \ \Longleftrightarrow \ S$ is insolvable  true $ \bigcirc $  false $ \bigcirc $
$ b\in {\mathrm{Im}}(\varphi) \ \Longleftrightarrow \ S$ is solvable  true $ \bigcirc $  false $ \bigcirc $
$ {\mathrm{ker}}(\varphi)=\{0\} \ \Longrightarrow \ \varphi$ is bijective  true $ \bigcirc $  false $ \bigcirc $
$ A^{{\operatorname t}}A$ is symmetric  true $ \bigcirc $  false $ \bigcirc $
$ \overline{A}^{{\operatorname t}}=\overline{A^{{\operatorname t}}}$  true $ \bigcirc $  false $ \bigcirc $

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