Find monic generators of the ideals

In summary, the conversation discusses a linear operator on a four-dimensional space represented by a matrix. The first part proves that the null space of the operator is spanned by a specific vector. In the second part, the concept of monic generators of ideals is introduced, which is related to the minimal polynomial of a matrix. The problem asks for the monic generators of four different ideals. Further explanation and resources may be needed to fully understand the problem.
  • #1
vbrasic
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3

Homework Statement


Let ##T## be the linear operator on ##F^4## represented in the standard basis by $$\begin{bmatrix}c & 0 & 0 & 0 \\ 1 & c & 0 & 0 \\ 0 & 1 & c &0 \\ 0 & 0 & 1 & c \end{bmatrix}.$$ Let ##W## be the null space of ##T-cI##.

a) Prove that ##W## is the subspace spanned by ##\epsilon_4##.

b) Find the monic generators of the ideals ##S(\epsilon_4;W),\,S(\epsilon_3;W),\,S(\epsilon_2;W)##, and ##S(\epsilon_1;W)##.

Homework Equations

The Attempt at a Solution


The first part is easy. It's trivial to see that ##T-cI## sends vectors of the form ##(0,0,0,d)## to ##0##, such that the null space is spanned by ##\epsilon_4=(0,0,0,1)##. However, I have no idea how to start the second part. I'm having some trouble understanding what is meant by ##S(\epsilon_i;W)##. Any help would be appreciated.
 
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  • #2
I don't know what they mean by ##S(\epsilon_j;W)## but usually problems about matrices and monic generators of ideals are concerned with the minimal polynomial of a matrix, and the ideals in question are ideals of the polynomial ring ##F[x]##, which will be a principal ideal domain, so that any ideal of it can be generated by a single monic polynomial. The minimal polynomial of a matrix A is the unique monic polynomial that can generate the ideal consisting of all polynomials ##p[x]## in ##F[x]## such that ##f[A]=0##. Here 'monic' means that the highest-order coefficient of the polynomial is 1.

You might find this stackexchange problem helpful.
 

Related to Find monic generators of the ideals

1. What is the definition of a monic generator?

A monic generator is an element in a polynomial ring that generates an ideal and has a leading coefficient of 1.

2. How do you find monic generators of an ideal?

To find monic generators of an ideal, you can use the division algorithm to express all elements in the ideal as a polynomial combination of the generators. Then, the monic generators are the polynomials with the highest degree in each of these combinations.

3. Can there be more than one monic generator for an ideal?

Yes, there can be multiple monic generators for an ideal. This is because the division algorithm allows for different choices of monic generators that can generate the same ideal.

4. How can you prove that a polynomial is a monic generator of an ideal?

To prove that a polynomial is a monic generator of an ideal, you can use the division algorithm to show that the polynomial can generate all other elements in the ideal and that it has a leading coefficient of 1.

5. Can a non-monic polynomial be a generator of an ideal?

Yes, a non-monic polynomial can be a generator of an ideal. However, it is not considered a monic generator because it does not have a leading coefficient of 1. Monic generators are preferred because they simplify calculations and make it easier to find other generators of the ideal.

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