Further Maths question

Hello,

I was wondering if someone knew about why if a matrix transformation is singular, the transformation has only 1 invariant line. A question I came across said that it being singular means there cannot be a second invariant line. Thank you!

Assuming it's a 2*2 nonzero matrix. What do you understand about singularity and invariant lines?

Another interpretation of singularity is that there must exist some nonzero x such that
Ax = 0

If there was two (different) invariant lines, would this be possible?

Is A a matrix? If there were two different invariant lines then I guess it wouldn’t equal zero anymore so there can only be 1 invariant line- that’s quite helpful, thank you!

Yes A is the matrix and x a vector. If you had two invariant lines, you could imagine they were axes. Each non zero vector x could be represented in terms of these two axes and multiplying x by A would produce another non zero vector. So two invariant lines would mean A was nonsingular.

Yes A is the matrix and x a vector. If you had two invariant lines, you could imagine they were axes. Each non zero vector x could be represented in terms of these two axes and multiplying x by A would produce another non zero vector. So two invariant lines would mean A was nonsingular.