## MAT 534 ALGEBRA I

## 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.

## FINAL LETTER GRADES FOR MAT 534 :

##
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)

##
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.

Mark Andrea de Cataldo's homepage.