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annihilator annuls assume basis z1 characteristic polynomial commutes complementary completely reducible completes the proof complex numbers complex vector space define denote dim U1 dim VT dimension dimensional direct sum eigenvalue eigenvector Euclidean n-space exists f(zu finite finite-dimensional Furthermore Hence Hint i-th row induction integer irreducible irreducible polynomial Jordan Lecture 3-5 Let z1 linear combination linearly independent main diagonal mapping minimum polynomial monic polynomial n-form nilpotent operator nonnegative nonsingular nontrivial nonzero vector null space operator algebra ordered basis partial isometry permutation perpendicular projection Problems for Lecture projection with range proof of Lemma proof of Theorem quotient space real numbers real vector space relative to z1 satisfying scalar multiple scalar operator self-adjoint set of vectors similar subset subspace invariant Suppose transpositions unitary space vectors in Euclidean vectors z1 y-th entry zi+1 zk+1