Here is a question i got for one of my exercises and I'm not too sure how i would approach such a problem.
Is it possible to span the space of n×n matrices M(n,n) using the powers
of a single matrix A, i.e. I, A, A^2, . . ., A^n, . . .?
First thing i thought about was if there were matrices with the property such that as u keep multiplying it by itself you would get another matrix independent of the previous ones but that turned out to be a confusing. I also tried thinking about the opposite where if all matrices eventually become linearly dependent at some point therefore disproving that there exists n*n linearly independent matrices hence proving that it is not possible to span the space. It does seem a bit complex though doing it the way i was thinking about, so any insight to how i can solve this problem simpler would be great.