Project description
see the later attachment

Question 1 ( 5 +5 points).
Let K be a field , V a finite vector space of K, and f , g: V – ? V be two endomorphisms which are commutators of each other,

g ? f = f ? g
a) Let E ? V be an eigenspace of f , that is, there is a ? ? K with E = Eig (f, ? ) .
Prove that E is invariant under g , that is, g ( E ) ? E.
b ) Let U ? V be a generalized eigenspace of f , there is a ? ? K with U = V (f, ? ) .
Prove that E is invariant under g.

Question 2 (9 +6 points).
a)    Let f ? R [ t] a real polynomial and ?? C is a complex root of f. Prove fu r the multiplicities of the roots havethe

equation:

b) Write the following real polynomials as products of real linear factors and
real polynomials of degree 2:

As always the solution must be recognizable .

Problem 3 ( 10 +5 points).

a)    Compute a real basis for the vector space defined over
? ??
b)    Use this result and the calculations presented in the lecture , to determine a matrix  so that the matrix

is in normal form. ( The inverse matrix T -1 need not be calculated