this is my first post so I do apologise regarding any formatting issues!
I have a question regarding invariant lines and lines of invariant points; from what I can gather, an invariant line is one of which a point on said line will map to another point on that line under a given transformation, and a line of invariant points is a line that contains points for which any point on that line directly maps to itself.
I am familiar with utilizing eigenvectors, with an eigenvalue equal to one, however this only works for invariant lines through the origin.
I have a textbook question that I'm rather confused about, as it utilizes a different method than eigenvectors. The question is as follows:
The transformation $T$ maps $(x,y)$ to $(x',y')$ where $$ x ' = - 2 x + 2 y + 4 \\ y ' = 3 x - y - 4 $$
I am now asked to find an equation of the line of invariant points of the transformation and then display that there are an infinite number of invariant lines for the given transformation.
Would somebody be able to explain how to solve this question, with the theory underlying? I presume there will be some matrix work involved and some '$mx+c$' substitutions but I've never grasped these substitutions correctly, so any help would be appreciated.
Many Thanks!