Show that if a is greater than or equal to the degree of minimal polyn

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The discussion centers on proving that if \( a \) is greater than or equal to the degree of the minimal polynomial \( k \), then \( L^a \) can be expressed as a linear combination of the vectors \( 1_v, L, \ldots, L^{k-1} \). Additionally, it is established that if \( L \) is invertible, the same conclusion holds for all \( a < 0 \). The participants express confusion regarding the manipulation of \( L^a \) and the associated vectors, indicating a need for clarity in applying the concepts of minimal polynomials and linear combinations.

PREREQUISITES
  • Understanding of minimal polynomials in linear algebra
  • Knowledge of linear combinations and their properties
  • Familiarity with the concept of invertible linear transformations
  • Basic proficiency in manipulating powers of linear operators
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  • Study the properties of minimal polynomials in linear algebra
  • Learn how to express linear combinations of vectors in vector spaces
  • Explore the implications of invertibility in linear transformations
  • Investigate the behavior of powers of linear operators, particularly for negative exponents
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Students and educators in linear algebra, particularly those focusing on the properties of linear transformations and minimal polynomials. This discussion is beneficial for anyone looking to deepen their understanding of linear combinations and operator theory.

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Homework Statement


Show that if a is greater than or equal to the degree of minimal polynomial (say k), then L^a is a linear combination of 1v,L,…,Lk−1

If L is invertible, show the same for all a<0


Homework Equations


about minimal polynomial
http://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra )


The Attempt at a Solution


i know a few things about minimal polynomial, i know what linear combination, invertible means but i have no clue how to start this problem.

i think the most confusing part i find is that i don't know how to deal with
L^a and the 1_v,L,...,L^k-1
 
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catsarebad said:

Homework Statement


Show that if a is greater than or equal to the degree of minimal polynomial (say k), then L^a is a linear combination of 1v,L,…,Lk−1

If L is invertible, show the same for all a<0


Homework Equations


about minimal polynomial
http://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra )


The Attempt at a Solution


i know a few things about minimal polynomial, i know what linear combination, invertible means but i have no clue how to start this problem.

i think the most confusing part i find is that i don't know how to deal with
L^a and the 1_v,L,...,L^k-1


Read my response to your other question about minimal polynomials.
 
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