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## Homework Statement

Let V be a finite dimensional complex vector space and T be the linear operator of V. Prove that the following are equivalent

a V has a basis consisting of eigenvectors of T.

b T can be represented by a diagonal matrix.

c all the eigenvalues of T have multiplicity one.

d. the Characteristic polynomial of T equals the minimal polynomial of T.

## Homework Equations

Not really applicable

## The Attempt at a Solution

Ok so I proved A implies B, However, I feel that B does not imply C.

I just want to see if my argument is valid. because the identity matrix, is definitely diaganol however, it does not have a multiplicity of One. Can I assume wlog that T has distinct eigenvalues along the main diaganol?