Suppose you find n different eigenvalues for an nxn matrix. Are you then guaranteed, that the eigenvectors corresponding to those eigenvalues are linearly independent?
If so, from what does that follow?
Yes. Suppose you have n different eigenvalues of an endomorfism corresponding to the eigenvectors then the set is linearly independent.
Suppose the eigenvectors are linearly dependent, then we know there's a first vector in the row which can be written as a linear combination of the previous vectors . Thus :
and are linearly independent.
But because are linearly independent alle the coefficients are equal to zero, so:
But because all the eigenvalues are different and so , this is a contradiction because wa supposed to be an eigenvector so our statement is wrong and the eigenvectors are indeed linearly independent.