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    Diagonalizable Matrices, Image, Kernels and Direct Sums

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    Prove that if T Є L(V) is diagonalizable then V = im(T) + ker(T) (+ = direct sum)
    (Hint: Use a basis of eigenvectors. The eigenvectors of the eigenvalue zero are a basis for the null space, and the remaining eigenvectors are a basis for the image)

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    keywords: matrix

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    Solution Preview

    suppose dim(V)=n. Since T is in L(V), T is diagonalizable, then we
    can find a basis B={v1,v2,...,vn} of V, such that the representative
    matrix of T under this basis B is a diagonal matrix D=diag(t1,t2,...,tn).
    Then we have Tv1=t1v1, ...

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