189-251B: Algebra 2 Assignment 6

Due: Wednesday, February 19.

1. Let T : V ⟶ V be a nilpotent linear transformation, and let

d_{j} := dim(ker(T^{j})), d′_{j} := d_{j+1} − d_{j}.

(a) Show that the sequence d_{j} is increasing, so that d′_{j} ≥ 0 for all j.

(b) Show that the sequence d′_{j} is (non-strictly) decreasing and tends to zero.

(c) Show that T^{j} = 0 if and only if d′_{j} = 0.

2. Compute the minimal and characteristic polynomials of the linear transformation T (f) = f′ acting on the following finite-dimensional vector spaces V of continuous functions on the interval [0, 1].

(a) V = the real vector space of polynomials of degree at most d with real coefficients (with d some fixed integer).

(b) The real vector space spanned by the real-valued functions sin(x), cos(x), sin(2x), cos(2x).

(c) The complex vector space spanned by the functions sin(x), cos(x), sin(2x), cos(2x) (viewed here as complex valued functions on [0, 1].)

(d) The real vector space spanned by the functions e^{x}, xe^{x}, x^{2}e^{x}, e^{2x}.

3. Compute the characteristic and minimal polynomials of the following matrices with entries in the field Z_{2} with two elements.

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

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

4. Write down the eigenspaces and generalised eigenspaces in Z_{2}^{4} for the matrices in question 3.

5. An linear involution on a vector space V over a field F is a linear transformation whose square is the identity. Show that any involution is diagonalisable when 2 ≠ 0 in F. Give an example of a non-diagonalisable involution on a vector space over Z_{2}.