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**worc3247** (a) [8 marks] Let V be a real vector space and let $\displaystyle T: V\rightarrow V$ be a linear transformation.

Define the terms eigenvalue and eigenvector of T. Show that if $\displaystyle v_1, . . . , v_n$ are eigenvectors

for distinct eigenvalues $\displaystyle \lambda_1,...,\lambda_n$ then the set of $\displaystyle v_1,...,v_n$ is linearly independent.

(b) [12 marks] Now let $\displaystyle v_1,...,v_n$ be a basis of eigenvectors with distinct eigenvalues $\displaystyle \lambda_1,...,\lambda_n$. Prove that for any linear map $\displaystyle S:V\rightarrow V$ with ST = TS, $\displaystyle v_i$ 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?