Ok so to get to it, i was reading that for some linear transformation in 2D space, if it has determinant of 0 then that transformation essentially reduces everything to a line and that there would be no inverse matrix or function/transformation in this case that could "reverse" this transformation from a 1D space back to a 2D space. However ive been watching some videos on non square matrices and how they represent transformations across dimensions e.g a 2 by 1 matrix as a transformation from essentially the number line, to 2 dimensional space.
Isnt there a contradiction here or am i missing something??
Best Answer
The thing is that you can make a linear transformation from a line to a plane but the dimension of the image will still be $1$ , meaning you'll get a line in the plane and not the whole plane. From that you can understand that if you take a plane and map it into a line, you can't inverse the process by another mapping from the line back to the plane - because you'll only reach a single line from the plane.
You can also say that a linear transformation from a plane to a line isn't inverse-able because it isn't an injection.