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

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In summary, the conversation discusses a monic polynomial p(c) and its application to a specific vector v resulting in pv=0. It also mentions a finite dimensional vector space V and its decomposition into k invariant subspaces. The question arises whether pv=0 implies pv_1=pv_2=0. The solution attempt involves expanding p(z) and considering the cases where a_0 and a_1 are not all 0s. It is concluded that either a_0 = -(the rest) or the terms must cancel out, or (T)(v_1+v_2..v_m)=0. The conversation ends with a clarification of what is meant by "applying a polynomial to a vector".
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rapple
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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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  • #2


What do you mean by "applying a polynomial to a vector"?
 
  • #3


I mean pv=0 for a particular v
 

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

1. What is a Monic Polynomial?

A Monic Polynomial is a type of polynomial where the coefficient of the highest degree term is equal to 1. For example, x^2 + 3x + 2 is a Monic Polynomial because the coefficient of the x^2 term is 1.

2. What is the degree of a Monic Polynomial?

The degree of a Monic Polynomial is the highest exponent of the variable in the polynomial. It is also the number of terms in the polynomial. For example, the degree of x^2 + 3x + 2 is 2.

3. How do you solve Monic Polynomial problems?

To solve Monic Polynomial problems, you can use various methods such as factoring, graphing, or the quadratic formula. These methods help you find the roots, or solutions, of the polynomial.

4. What are the applications of Monic Polynomials?

Monic Polynomials are used in various areas of mathematics, such as algebra, calculus, and number theory. They are also used in physics and engineering to model real-world situations and solve equations.

5. What is the difference between a Monic Polynomial and a non-Monic Polynomial?

The main difference between a Monic Polynomial and a non-Monic Polynomial is the coefficient of the highest degree term. In a Monic Polynomial, this coefficient is always 1, whereas in a non-Monic Polynomial it can be any other number. This difference can affect the methods used to solve the polynomial and the resulting solutions.

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