Down with Determinants! Sheldon Axler

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det 3 Theorem 2.1 Every linear operator on a finite-dimensional complex vector space has an eigenvalue. Proof. To show that T (our linear operator on V) has an eigenvalue, fix any non- zero vector v ∈ V.The vectors v,Tv,T2v,...,Tnv cannot be linearly independent, because V has dimension n and we have n + 1 vectors. Thus there exist complex numbers a0,...,an, not all 0, such that

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