Find the affine transformation that sends the line $ε:3x+2y+4=0$ of $\mathbb{R^2}$ to the line $x=0$
I am having some problems here, and I am getting confused, if anyone could help I would appreciate it.
The first issue that I have is that an affine transformation is define to be a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2 $ such that $f(x)=Ax+k$ , $x\in \mathbb{R}^2$ $,A\in Gl(n), k\in \mathbb{R}^2$ how we "feed" a line equation in the function?
Then i thought to use the fundamental theorem of affine geometry and the points $(0,0)$ , $(0,1)$ and $(1,0)$ didn't belong to the line $ε$ so I peek $3$ points on the line $a,b,c$ to qualify the equations $f(a)=(0,0)$ etc but i couldent find $A , k$
Mapping a line (or other object) to another means that it maps every point from that line to a point on the other bijectively. So we want to take the set $\varepsilon = \{(x, y) \in \mathbb{R} : 3x + 2y + 4 = 0\}$ and for each point $p \in \varepsilon$ map it to a point $q \in l = \{(x, y) : x = 0\}$. And, in this case, we want to do so using an affine transform.
The easiest way to do this would be to pick a few points from $\varepsilon$ and assign points on $l$ to send them to, and see what that means for the transformation. Since it takes two points to uniquely define a line, that seems like a reasonable starting point - maybe take the points where $x = 0$ and $y = 0$, and go from there.