[Hovedside][Øvinger]

MA2201/TMA4150 - Øvinger

På denne siden vil øvingsoppgavene bli lagt ut hver uke. Ta kontakt med Dagfinn for spørsmål om øvingene. Det kreves ikke godkjente øvinger for tilgang til eksamen, men øvingene er en viktig del av kurset likevel.

Øvingstidene er De første øvingene er 15. januar og 18. januar.

Det er mange gode oppgaver i læreboka, og dere må gjerne gjøre flere enn de som står listet opp her!

Retting: Dere kan nå levere inn øvingene deres til retting, om dere ønsker det. Det er satt ut pappbokser merket med fagkode ved inngangen til Industriell Matematikk, 3. etasje i Nordre Lavblokk. (Gå opp trappa ved kortsenteret (der man får adgangskort) til andre etasje. Der finner man en mindre trapp der det står "Lavblokk Nord".) Dere kan selvsagt også levere direkte til Dagfinn på gruppa.

Oppgavene som legges ut i uke X bør være i boksen i løpet av fredag i uke X+1. (Øving 1 kan dere levere sammen med øving 2.) Markér besvarelsen med mandags- eller torsdags-gruppe.
Øving 1 Sec. I.2: 12, 13
Sec. I.4: 3, 6, 10 (see Def. 3.7 page 29), 15 (consider both the general case and the case with the set of all upper triangular nxn matrices with non-zero determinant), 18, 19, 30, 32, 34, 38
Øving 2 Sec. I.4: 41
Sec. I.5: 5, 6, 11, 13, 22, 23, 26, 33, 41, 45, 46, 51, 53
Øving 3 Sec. I.6: 17, 23, 36, 44, 45, 50, 51, 53
Sec. II.8: 3, 9, 17, 20, 44, 46, 47
Sec. II.9: 1, 9, 11
Øving 4 Sec. II.9: 5, 13, 17, 27 a-b, 29, 34
Sec. II.10: 2, 6, 7, 12, 15, 28, 32, 34
Sec. II.11: 1, 6
Øving 5 Sec. II.10: 39, 40
Sec. II.11: 22, 26, 27*, 39*, 47, 48*
Sec. III.13: 2, 6, 8, 17, 44, 45, 47*, 51*
Sec. III.14: 6, 12
Øving 6 Sec. III.13: 23, 49, 52*
Sec. III.14: 25*, 27, 30*, 31
Sec. III.15: 1, 4, 28, 35, 36, 40*
Also: Show that if ɸ: G → G' is an isomorphism, and g∈G has order n, then ɸ(g) also has order n.
Øving 7 Det blir gruppeøvinger også i uke 9, selv om det ikke er forelesninger.
Sec. III.15: 3, 29
Sec. III.16: 1, 2, 3, 6, 7, 10, 13
Sec. III.17: 2, 3, 5
28. mai 2004: 1
Midtsemester
  • The midterm exam will be arranged in room S8 in the Central Building (this is now confirmed) on monday Mars 12, 15.00-17.00.
  • The midterm exam counts for 20 percent of the total grade if you do better on the midterm than on the final. Otherwise, it does not affect your grade.
You might want to try the previous midterm exams: When you are preparing for the midterm exam, you should read the textbook and study the problem sets which have been given during the semester. There are also some relevant old exam questions you can take a look at: (Note that the midterm exam will be a multiple choice test.)
  • August 12, 2004: 1a, 3, 4 (corrected from September 1)
  • May 28, 2004: 1, 3, 5
  • December 7, 2001: 1
  • December 5, 2000: 2, 3
  • December 2, 1991: 1, 4
You find a pdf-file with the old problems here. (On the front page you find some newer exams.)
Øving 8 Sec. IV.18: 5, 6, 8, 11, 13, 23*, 28, 34*, 35, 37*, 38, 41*, 46, 49, 54*
Sec. IV.19: 1, 2, 11, 20, 26*, 29
Øving 9 Sec. IV.20: 1, 4, 6*, 8, 9, 10, 24, 27*, 28*
For MA2201: Sec. VII.36: 1, 3, 11*, 12*, 13, 17*, 22*
Øving 10 These problems are for the first week after Easter. They are only from the syllabus for the MA2201 course.
Sec. VII.36: 5, 14, 16, 18, 19
Sec. VII.37: 7
23. mai 2001: 2
28. mai 2004: 6
Also: Show that if G is a finite abelian group with n elements, and r divides n, then G has a subgroup with r elements.
Øving 11 Sec. IV.18: 25, 48, 55
Sec. IV.19: 5, 8, 23, 25
Sec. IV.20: 11, 13, 14, 15
24. august 2000: 1
28. mai 2004: 4
On RSA (see the note on the Euler phi-function):101 and 113 are prime numbers, and 7467*3=1 mod 11200. Suppose you post the (RSA-)key (11413,7467) on the web with the information "space=100, a=101, b=102, etc.". One day you receive an e-mail with the three numbers 4654, 676 and 2931. What is the sender trying to say to you? (It's in Norwegian...)
Øving 12 Sec. IV.22: 1, 2, 3, 6, 9, 14, 17, 22, 24
Sec. IV.23: 1, 2, 9, 12, 26, 36
26. november 1999: 2
Øving 13 For all:
  • Sec. IV.23: 13, 14, 34
  • Sec. V.26: 12, 13, 14, 18, 22, 24, 25, 26, 27
For TMA4150:
  • Sec. V.27: 1, 7, 18
  • Show that p(x)=x4+x3+x2+x+1 in the polynomial ring Z2[x] is irreducible, and F=Z2[x]/<p(x)>. Find a generator for the cyclic group F\{0}. (Remember Theorem 10.12 in Fraleigh.)
  • Show that q(x)=x5+x2+1 in Z2[x] is irreducible, and let E=Z2[x]/<q(x)>. Find a generator for the cyclic group E\{0}.
  • 17. august 2006:: 7
For MA2201: Also, you should work on some exam problems, especially those that Idun will present in the lectures. Try them before you go to these lectures!