Advanced Linear Algebra (Math 424/524) Assignment No. 5

Due December 11, 2002

1. Let F be a field. For P a polynomial in F [t] let V_{P} denote the quotient F [t]/P F [t]. Given P and Q in F [t] one defines a linear map

ϕ : V_{P Q} ⟶ V_{P} × V_{Q}

by

ϕ(h modP Q) = (h modP, h modQ) .

What is the dimension of the kernel of ϕ?

2. For each of the following rational matrices find the minimal and characteristic polynomials and find a direct sum of companion matrices that is similar to the given matrix:

(a) ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) (b) ( \begin{matrix}0 & 1 \\ − 1 & 0 \end{matrix} ) (c) ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & −1 \\ 1 & 0 & 0 \end{matrix} )

(d) ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} ) (e) ( \begin{matrix}1 & 2 & 0 & −1 & 3 \\ 0 & 0 & −2 & 1 & −1 \\ − 1 & 3 & 1 & 0 & 0 \\ 2 & −1 & 3 & −1 & 2 \\ 1 & −1 & −3 & −2 & 0 \end{matrix} )

3. Let M be the matrix of polynomials in F [t]

M = ( \begin{matrix}t^{3} + t^{2} − 2t & t^{3} − 2t + 1 \\ t^{3} − 1 & t^{3} − t^{2} \end{matrix} ) .

(a) Find a diagonal matrix of successively divisible polynomials that can be obtained from M by (restricted) row and column operations.

(b) What is the dimension of the quotient space

F [t]^{2}/MF [t]^{2} ?

4. Let M be the matrix of polynomials from the previous problem.

(a) Is there a 2 × 2 matrix A for which t · 1 − A is (restricted) row and column equivalent to M?

(b) Find an N × N matrix A for some N in the field F for which the endomorphism “multiplication by t” of the quotient space in part (b) of the previous problem is isomorphic to the endomorphism of F^{N} given by A.

5. For given λ in the field F let J = J(λ) be the r × r matrix

( \begin{matrix}λ & 1 & 0 & 0 & … & 0 \\ 0 & λ & 1 & 0 & … & 0 \\ 0 & 0 & λ & 1 & & ⋮ \\ ⋮ & ⋮ & & & & 0 \\ & & & & λ & 1 \\ 0 & 0 & 0 & … & 0 & λ \end{matrix} ) .

Find a basis for the vector space F [t]/(t − λ)^{r}F [t] in which the matrix J represents the endomorphism “multiplication by t”.