Quick question
We know that if a matrix/linear transformation in a space has dimension n and its minimal polynomial has k different roots with algebraic multiplicity 1, that the matrix/linear transformation is diagonalizable.
Now can k be different than n? Or do we need to have n different roots for the minimal polynomial (I know it's sufficient (but not necessary) n different roots in the characteristic polynomial but I read that that's not needed for the minimal polynomial I just want to confirm)
Yes, $k$ can very well be different than $n$. For example, if $A = I_n$ then the minimal polynomial of $A$ is $(x-1)$ while the characteristic polynomial is $(x - 1)^n$.