FALL '00

Note: NO MAT 535 (Algebra II) CLASS on TH January 25, 2001. We first meet on TU, January 30, 2001. I will be teaching a winter school in algebraic geometry in Trento, Italy, and will be back only on January 28, 2001.


9225: B- ; 1808: A- ; 8958: A ; 8270: F ; 0395: B- ; 9277: C;

1908: A; 4533: I; 1579: A- ; 2197: A ; 5489: B+ ; 5708 : B+ .

Final with sketches of solutions: pdf file ps file

Nobody has solved problem n.3. I have graded the test using the remaining problems. I have assigned a number grade to the final using the following scheme:

1.a 50pts; 1.b 50pts; 2 100pts; 4.a 75 pts; 4.b 50 pts; 5.a 75 pts; 5.b 75 pts; 5.c 50 pts; 6.a 75 pts; 6.b 50 pts; 7.a 75 pts; 7.b 50 pts; 7.c 75 pts; 8 150 pts.

The total maximum for the final is 1000pts. I have added to that the 10 best hmk scores. The total maximum for the hmk is 1000pts. I took the average and curved. In some cases, the homework helped a lot! However, if you did better in the final than in the hmk I used that fact to your advantage. This has greatly helped a couple of you. Next semester there will also be a midterm. Therefore I may use a different grading scheme.

FINAL SCORES (FINAL NUMERICAL GRADE) AND REMARKS ON INDIVIDUAL PERFORMANCES IN THE FINAL. Feel free to discuss with me your opinions. In what follows the first 4 digits are the SSN, the second number is your score in the final and the third one in ()s is the average of the final and the hmk.

9225: 590pts (690); I think you scored below your expectations. You have some trouble with outlining and wording solutions. 1808: 840pts (805); you did well in all problems except the last one. There is space for improvement in the way in which you write solutions. 8958: 900pts (890); good skills in abstract things. Need to improve the computational side. 2870: 390pts (255 ); the problems you solved were solved in the rigth way. You solved too few. My impression is that you did not invest enough time in this course. 0395: 700pts (700); you struggled but worked hard; you did ok in the computational part, not as well in the more conceptual part. 9277: 460pts (625); you should improve your proof-writing skills, it is very hard to follow the tread of your arguments. I can see that you have intuition, but overall the work of this final should be improved. The homework helped. 1908: 1000pts (895); Good! It is the best final. You tried n.3. You made an interesting analysis, but your final argument is at least incomplete. The homework is not as good. 1579: 845 pts (837.5); There is space for improvement. You should reflect especially on n.8. 2197: 905pts (907.5); It is a good performance. Perhaps you could have done a bit better in the final. 5489: 720pts (815) pts. Something did not click well in this final. You have a good understanding of the material, but you should probably work on writing more proofs. The homework helped your grade. My feeling is that you can reach the A range in the immediate future. 5708: 770pts (830); you lost a lot of points in a computational problem that, looking at further work, I think you knew how to do. That is unfortunate, because, you have shown a good understanding of the material. For future reference: your work on the first part of n.8 is on the right track, but keep in mind that the details you offered were not enough. The homework helped.

There are just few points between the two A- and the two B+. One reason is that the two B+ became so with a big help from the hmk, but their performance in the final was inferior to the two A-.

Last year's final with solutions: (ignore number 6) pdf file ps file

Some of last year's hmk problems with solutions. pdf file, ps file. pdf file, ps file. (A matrix A is nilpotent, if a power of it is zero)

Syllabus: pdf file, ps file, dvi file, tex file.

Homework sets:

Due 9/12: I.1 : 1,4,7,9, 12,13,14 AND I.2: 1,2,3,5,6. Due 9/19: I.2 : 10, 11, 12, 13, 15 AND I.3: 1,2,4,5,9,10. Due 9/26: I.4 : 1,2,3,5,6 AND I.5 : 1,3,5,6,7. Due 10/3: I.5 : 8, 9, 10, 13, 14, 16, 17 AND I.6 : 1, 7, 13. Due 10/10: II.4 : 1, 2, 4, 5, 6, 7, 8, 9, AND IV.1 : 1, 2, 7, 9. DUE 10/17: NO HMK THIS WEEK!!! Due 10/24 : Prove directly that (iv) is equivalent to (iii) for Theorem IV.2.1 (page 181). Prove Theorem IV.2.5 (using Zorn's) in detail. IV.2 : 1, 5, 6, 8, 9 (for free modules over a division ring only), 14. Due 10/31 : Let F:V->W be a linear transformation of vector spaces over a field. Let F*:W*->V* be the dual map. Prove that F* is injective iff F is surjective. Prove that F* is surjective iff F is injective. IV.4 : 6, 7. VII.1: 1 (a) and b) only), 2 (only the subring part), 4, 5, 7, 8. Due 11/7 : Hungerford: VII.2: 1,3,4 , VII.3: 1,2,7 AND Artin (on reserve in the library: M.Artin, Algebra) : Ch1, Section 2 ex. 2, 18. Due 11/14: Hungerford VII.4: 1, 2, 3, 4, 5, 8, 9, 10, 13. Due 11/21 : Hungerford VII.5 : 1,3,4,5,6,7,8 AND Artin Ch7, Section 1 ex: 1,2,3,4,5. Due 11/28 : From Artin's book CH.7: Prove Proposition 2.7, Section 2 Ex.: 1,2,3,13,15,17,21. Due 12/5 : From Artin's book CH.7 : Section 4 Ex.: 3,4,5,10,11,12,15 AND Section 5 Ex.: 1,2,3,4,5,7 AND Section 8 EX.: 3,4,7. Due 12/12: Hungerford IV.5. ex: 1,2,3,4,5,7,11. Write complete proofs for THM 5.5 and 5.10 (IV.5). ADDITION: PROVE COROLLARY 5.12 for vector spaces. The sample test I will give you soon contains other questions dealing with tensor products of vector spaces.

The material covered in the last two lectures is not in Hungerford. Here are some references: 1) Finite dimensinal multilinear algebra,I, by M. Marcus, Dekker; especially: 1.2, 1.3 and 1.4. 2) Algebra, vol.2, by P.M. Cohn; John WIley and Sons; especially ch. 3. 3) Commutative Algebra with a view ...., by D. Eisenbud, GTM 150 Springer-Verlag; especially A2.1-A2.5. The following book is an old, but still nice, reference for learning tensors in differential geometry (without ever using the symbol): Differential Geometry, by P. Eisenhart.

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