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p(T) is invertible or zero

Source: Iran PPCE 2012-Linear Algebra exam-P5

February 16, 2012
vectoralgebrapolynomialLaTeXcomplex numberslinear algebralinear algebra unsolved

Problem Statement

Suppose that for the linear transformation T:V⟶VT:V \longrightarrow V where VV is a vector space, there is no trivial subspace W⊂VW\subset V such that T(W)⊆WT(W)\subseteq W. Prove that for every polynomial p(x)p(x), the transformation p(T)p(T) is invertible or zero.
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