Does pv=0 Imply pv₁=pv₂=0 in Monic Polynomial Problems?

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SUMMARY

The discussion centers on the implications of the equation pv=0 in the context of monic polynomials and finite-dimensional vector spaces. Specifically, it questions whether pv₁=0 and pv₂=0 must hold true when pv=0, given that v is expressed as a direct sum of invariant subspaces. The polynomial p(z) is defined as p(z)=a_0+a_1z+...+a_mz^m, where not all coefficients a_0, a_1 are zero. The conclusion drawn is that if pv=0, then the application of the polynomial to the individual components of v must also yield zero.

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  • Basic concepts of polynomial algebra and vector application.
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This discussion is beneficial for students and professionals in mathematics, particularly those studying linear algebra, polynomial theory, and vector space applications.

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Homework Statement



Let us say that p(c) is a monic polynomial such that when applied to a particular v, we have pv=0. Let V be a finite dimensional vector space. Let V be the direct sum of k invariant subspaces. Then v = v_1+...+v_k.

When I apply pv=0 does this imply that pv_1=pv_2=0.

Homework Equations





The Attempt at a Solution


p(z)=a_0+a_1z+...a_mz^m
p(T)v=a_0+a_1(T)(v_1+v_2..v_m)+...a_m(T^m)(v_1+v_2..v_m)
We know that a_0,a_1 are not all 0s.If they are then pv = 0 for all v. So we must have that a_0 = -(the rest) or the terms must cancel out each other.
or we have (T)(v_1+v_2..v_m)=0 hence T^m(v_1+v_2..v_m)=0.

This is where I am stuck
 
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What do you mean by "applying a polynomial to a vector"?
 


I mean pv=0 for a particular v
 

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