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

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In summary, the problem asks to show 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 1v, L, ..., L^(k-1). If L is invertible, the same result must hold for all values of a less than 0. The concept of minimal polynomial and linear combination are needed to understand the problem. The confusion lies in how to deal with the expressions L^a and 1v, L, ..., L^(k-1).
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catsarebad
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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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1. What does "a is greater than or equal to the degree of minimal polyn" mean?

This statement refers to a mathematical relationship between a variable, represented by "a", and the degree of a polynomial equation. It means that the value of "a" must be equal to or greater than the highest exponent in the polynomial equation.

2. How is this statement relevant in the field of science?

This statement is relevant because polynomial equations are commonly used in scientific research and analysis. Understanding this relationship between "a" and the degree of the minimal polynomial can help with solving equations and analyzing data.

3. What is a minimal polynomial?

A minimal polynomial is the smallest degree polynomial that can be used to express all the roots of a given equation. It is also known as the irreducible polynomial.

4. Why is it important to show that "a is greater than or equal to the degree of minimal polyn"?

Showing this relationship helps to determine the complexity of a polynomial equation and can provide insight into the behavior of the equation. It can also help with finding the roots and solving the equation.

5. Can this statement be applied to all polynomial equations?

Yes, this statement can be applied to all polynomial equations, as it is a fundamental property of polynomials. However, the specific value of "a" and the degree of the minimal polynomial may vary depending on the equation.

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