Prove that an affine transformation maps parallel lines to…

Prove that an affine transformation maps parallel lines to parallel lines.

Step 1: Image of a line

Write the map as $\vec r\,'=A\vec r+\vec b$ with $\det A\ne0$. A line through $\vec r_0$ with direction $\vec d\ne\vec0$ is $\vec r=\vec r_0+t\vec d$. Its image is

\[ \vec r\,'=A\vec r_0+\vec b+t\,A\vec d , \] a line through $A\vec r_0+\vec b$ with direction $A\vec d$. Here $A\vec d\ne\vec0$, because $A$ is invertible.

Step 2: Parallel lines

Parallel lines $p:\ \vec r_1+t\vec d$ and $q:\ \vec r_2+t\vec d$ share the direction $\vec d$. So both images have direction $A\vec d$, and $p'\parallel q'$.

If $p\ne q$, the images are also different: $A$ is a bijection, so $p'\cap q'=\varnothing$, since a common point would come from a common point of $p$ and $q$. $\blacksquare$

\[p\parallel q\ \Longrightarrow\ A(p)\parallel A(q)\]