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Find monic generators of the ideals

  • #1
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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.
 

Answers and Replies

  • #2
andrewkirk
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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.
 

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