16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 7.2.4

The Rational and Jordan Forms

Mathematics — Linear Algebra · t-admissible-subspaces-conductors

Problem

Let

T

be the linear operator on

F^4

which is represented in the standard ordered basis by the matrix

\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)

S(\epsilon_1; W)