Let T be a linear operator on a finite-dimensional vector space V, and suppose that the distinct eigenvalues of T are λ1, λ2, . . . , λk. Prove that span({x ∈V: x is an eigenvector of T}) = Eλ1⊕Eλ2⊕· · ·⊕Eλk.

Elementary Linear Algebra (MindTap Course List)
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Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 44E: Prove that in a given vector space V, the additive inverse of a vector is unique.
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Let T be a linear operator on a finite-dimensional vector space V, and suppose that the distinct eigenvalues of T are λ1, λ2, . . . , λk. Prove that span({x ∈V: x is an eigenvector of T}) = Eλ1⊕Eλ2⊕· · ·⊕Eλk.

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