Exercise 24 — Linear Algebra

Show that a function $f\colon\mathbb{R}^{1}\to\mathbb{R}^{1}$ is an automorphism if and only if it has the form $x\mapsto kx$ for some $k\neq 0$ .
Let $f$ be an automorphism of $\mathbb{R}^{1}$ such that $f(3)=7$ . Find $f(-2)$ .
Show that a function $f\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ is an automorphism if and only if it has the form
$\begin{pmatrix}x\\ y\end{pmatrix}\mapsto\begin{pmatrix}ax+by\\ cx+dy\end{pmatrix}$
for some $a,b,c,d\in\mathbb{R}$ with $ad-bc\neq 0$ . Hint. Exercises in prior subsections have shown that
$\begin{pmatrix}b\\ d\end{pmatrix}\text{\ is not a multiple of\ }\begin{pmatrix}a\\ c\end{pmatrix}$
if and only if $ad-bc\neq 0$ .
Let $f$ be an automorphism of $\mathbb{R}^{2}$ with
$f(\begin{pmatrix}1\\ 3\end{pmatrix})=\begin{pmatrix}2\\ -1\end{pmatrix}\quad\text{and}\quad f(\begin{pmatrix}1\\ 4\end{pmatrix})=\begin{pmatrix}0\\ 1\end{pmatrix}.$
Find
$f(\begin{pmatrix}0\\ -1\end{pmatrix}).$

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