(a) [8 marks] Let V be a real vector space and let

be a linear transformation.

Define the terms eigenvalue and eigenvector of T. Show that if

are eigenvectors

for distinct eigenvalues

then the set of

is linearly independent.

(b) [12 marks] Now let

be a basis of eigenvectors with distinct eigenvalues

. Prove that for any linear map

with ST = TS,

is also an eigenvector of S for each i = 1, . . . , n.

Deduce that an n×n matrix that commutes with every diagonal n×n matrix must

be diagonal itself.

So far I have managed part a), but am stuck on the first part of b). I have tried taking TS(v) and then using the eigenvalues and the fact that S and T are linear to find something, but I can't get anything of the form S(v)=av (a constant). Could someone start me off?