# Linear independent eigenvectors and eigenvalues

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!

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If there are $n$ linearly independent vectors in $n$-dimensional space, then they must form a basis. To see what $T$ looks like, consider what $T x_k$ looks like in the basis of eigenvectors. –  copper.hat Nov 20 '13 at 3:19
@Akaichan Do you still need help with this or is Copper.Hat's comment enough. –  Git Gud Nov 25 '13 at 21:41