Show that $Fix(ϕ)$ is ether {0}, a subgroup of the form $L = {tv : tR}$ for some $v ∈ R^2$ \ {0} or the full space $R^2$

Question

i have a problem:

The fixed point set of $\varphi ∈ Aut(R^2)$ is defined to be

$Fix(\varphi) = {v ∈ R^2: \varphi(v) = v}$

Show that $Fix(\varphi)$ is ether {0}, a subgroup of the form $L =$ {$tv : tR$} for some $v ∈ R^2$ \ {0} or the full space $R^2$

Here $\varphi$ represents 2x2 matrices

$$\varphi \begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} a & c \\ b & d \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix}= \begin{pmatrix} ax+cy \\ bx+dy \\ \end{pmatrix} $$

I have no idea how to prove that. Any suggestions?

Thank you

Answer 1

Hint : Show that $Fix(\varphi)$ is a vectorial subspace of $R^2$ then look at $dim(Fix(\varphi))$.