I have T as a linear transformation from V to V over the field F, V has dimension n. T has the maximum number n distinct eigenvalues, then show that there exists a basis of V consisting of eigenvectors.
I know that if I let $v_1,...,v_r$ be eigenvectors belonging to distinct eigenvalues, then those vectors are linearly independent. Can I make a basis from these linearly independent vector and prove that it spans V?
Also, what will the matrix of T be in respect to this basis?
Thank you for any input!