Exercise Sheet no. 3

Linear Algebra II

Exercise Sheet no. 3

SS 2011

Prof. Dr. Otto April 29, 2011

Dr. Le Roux Dr. Linshaw

Exercise 1 (Warm-up: Multiple Zeroes) For a polynomialp=Pn

i=0a_iXⁱ∈F[X]define itsformal derivativep⁰by p⁰:=

n

X

i=1

ia_iXⁱ⁻¹.

(a) Check that the usual product rule for differentiaton applies to the formal derivative of polynomials considered here!

(b) Letαbe a zero ofP. Show the equivalence of the following:

i. αis a multiple zero ofp. (In other words,(X−α)²dividesp.) ii. αis a zero ofp⁰.

iii. αis a zero of gcd(p,p⁰).

Exercise 2 (Commutative subrings of matrix rings)

LetA∈F^(n,n) be ann×nmatrix over a fieldF. LetR_A⊆F^(n,n)be the subring generated byA, which consists of all linear combinations of powers ofA.

(a) Prove thatR_Ais a commutative subring ofF^(n,n).

(b) Consider the evaluation map˜:F[X]→R_Adefined by ˜p=Pn

i a_iAⁱ forp=Pn

i a_iXⁱ. Show that this map is a ring homomorphism. Is it surjective? Injective?

Hint: By forgetting about the multiplicative structure, we may regardF[X]andR_Aas vector spaces overF, and we may regard˜as a vector space homomorphism. DoF[X]andR_Ahave the same dimension asF-vector spaces?

Exercise 3 (The Euclidean algorithm revisited)

Recall the Euclidean algorithm from Exercise Sheet 2. In particular, given natural numbersa,b, we normalise so that d₁=min{a,b}andd₀=max{a,b}. In each step, we divide with remainder, obtainingd_k−1=q_kd_k+d_k+1. At the end of this procedured_k+1=0, andd_k=gcd(a,b).

(a) Letkbe the number of steps needed to compute gcd(a₀,b₀)in this way. Consider the matrixM∈Z^(2,2)given by M=

0 1 1 q₁

0 1

1 q₂

· · · 0 1

1 q_k

.

Show thatMis regular and thatM⁻¹is again a matrix overZ. ComputeM⁻¹ d₁

d₀

.

(b) Interpret the entries in second row ofM⁻¹in terms of gcd(d₀,d₁).

(c) Recall that theleast common multiplelcm(d₀,d₁)is an integerzcharacterized by the following properties:

i. d₀|zandd₁|z.

ii. Ifais any integer for whichd₀|aandd₁|a, thenz|a.

Interpret the entries in the first row ofM⁻¹in terms of lcm(d₀,d₁). Exercise 4 (Polynomial factorisation and diagonalisation)

Consider the following polynomials inF[X]forF=Q,RandC:

p₁=X³−2, p₂=X³+4X²+2X, p₃=X³−X²−2X+2.

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(a) Which of these polynomials are irreducible inF[X]?

(b) Which of these polynomials decompose into linear factors overF[X]?

(c) Supposep_iis the characteristic polynomial of a matrixA_i∈F^(3,3). Which of theA_iis diagonalisable overF?

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