Exercise 17 Let V be a finite-dimensional complex vector space, and let TE L(V). Prove that if V = null(T-XI) Ⓒ range(T - XI) for all AEC, then 'T' is diagonalizable. Proof. If V is the direct sum of the null space and range of T - XI for all X = C, each vector in why? V is uniquely expressible as a sum of eigenvectors, constructing a basis for V consisting solely of eigenvectors. Therefore, T is diagonalizable since it admits a basis of eigenvectors.

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