James Cook's Abstract Algebra I Homepage:
Welcome. This webpage contains some resources I have created for Abstract Algebra I. The current semester work is found in Blackboard, thanks!

1. Course Plan for Math 421 of Fall 2018
2. You Tube Playlist for Math 421 of Fall 2018
3. My Lecture Notes for Fall 2018
4. I'm here to help. Please make wise use of my office hours when you get stuck. Also, in lecture, if I write something obviously wrong. Please politely interupt me before I burn 5 minutes of class on a bogus calculation. I have no patience for corrections! I want them immediately.
5. Someone will be unable to make it to my office hours. It's inevitable in a given semester. Therefore, if you are such a person, it especially important for you to work with a study group which has at least one person who can make it to office hours.

General Advice: When confronting many "proof" problems in this course (and in more advanced abstract math courses) as a starting point you ought to ask yourself:

1. What am I asked to prove ?
2. Can I define the words used ?

It is not usually the case that you will find the same proof in my notes or the text. Definitions are key, I cannot emphasize this enough. Past this, you should consider using theorems, propositions etc. which we have developed. However, beware of proof by synonym. A common beginners mistake is to simply restate the claim in slightly different words as to prove the claim by invoking an entirely equivalent theorem. I usually write an arrow pointing back to itself to warn you of the circularity of such logic. Anyway, enough about what not to do, you can learn what is a good method of proof simply by following lectures and paying mind to study the structure of our arguments in each lecture. We do solve problems in this course, but, our more over-arching goal is to understand the structure of abstract algebra.

Useful Abstract Algebra Materials and Links from Past Years:
Keep in mind the structure of this course differs a bit from the current course. So, beware, definitions may not exactly align.

• My Lecture Notes for Fall 2016
• You Tube Playlist for Math 421 of Fall 2016 I
• You Tube Playlist for Math 421 of Fall 2017

Tests and some Solutions from Fall 2016:

• statement of Test 1 and its solution
• statement of Test 2 and its solution
• statement of Test 3 and its solution
• statement of Test 3 and its solution
• statement of Final Exam and solution part 1 and part 2

Homework Solutions from Fall 2016:

• solution to Problems 1-4
• solution to Problems 5-8
• solution to Problems 9-12
• solution to Problems 13-16
• solution to Problems 17-20
• solution to Problems 21-24
• solution to Problems 25-28
• solution to Problems 29-32
• solution to Problems 33-36
• solution to Problems 37-40
• solution to Problems 41-44
• solution to Problems 45-48
• solution to Problems 49-52
• solution to Problems 53-56
• solution to Problems 57-60
• solution to Problems 61-64
• solution to Problems 65-68
• solution to Problems 69-72
• solution to select Problems 73-84
• solution to Problems 85-90
• solution to Problems 91-96
• solution to Problems 97-102
• solution to Problems 103-108
• solution to Problems 109-114
• solution to Problems 115-120
• solution to Problems 121-126
• solution to Problems 127-132
• solution to Problems 133-137

Linear Algebra Background:
Here's what I covered in the Prerequisite to Math 421 most recently. On occasion I mention a topic which you don't recall, if you are curious, the details are probably in here:

1. Math 321 Spring 2017 Playlist: used my notes with no formal text.
2. Math 321 Spring 2016 Playlist: based on Curtis' text.
3. Math 321 Spring 2015 Playlist: based on Diamano and Little's Dover ed. text.

Spring 2017 Linear Algebra Lectures:

1. components, rows, columns, add and scalar multiply, word on rings: 1-16-17
2. standard notations, matrix multplication, 1-20-17 (bad sound)
3. how to multiply with standard bases, special matrices, 1-20-17
4. Linear Algebra: supplemental examples of block multiplication, for 1-20-17
5. on the Gauss-Jordan algorithm, 1-23-17
6. on the structure of solution sets and elementary matrices, 1-25-17
7. on inverse matrices, 1-27-17
8. spanning, linear independence and CCP for column vectors, 1-30-17
9. CCP proof and its application, motivation of determinants, 2-1-17
10. on determinant calculation, 2-3-17
11. Cramer's Rule and the Adjoint Formula for the Inverse, 2-6-17
12. definition of vector space, examples, subspace test, 2-13-17
13. subspace examples, span, basis, 2-15-17, part 1
14. subspace examples, span, basis, 2-15-17, part 2
15. on coordinates and dimension, 2-17-17
16. theorems on manipulating bases, 2-20-17
17. the subspace theorem (in my office), 2-20-17
18. basic theory of linear transformations, 2-22-17
19. theorem on image and inverse image, standard matrx, 2-24-17
20. gallery of linear transformations, restriction, matrix of T, 2-27-17
21. finite dimensional isomorphism, matrix of linear transformation, 3-1-17
22. matrix of linear maps, coordinate change, 3-3-17
23. rank nullity for maps, congruence vs. similarity, 3-6-7
24. direct sum decomposition part 1, 3-8-17
25. direction sum decomposition theorem part 2, 3-10-17
26. direct sum of matrices, intro eigenvectors, 3-10-17
27. eigenbasis, algebraic and geometric multiplicity, 3-20-17
28. Jordan Form for Matrix or Transformation, 3-22-17
29. review for Test 2, 3-24-17
30. on polynomial operator theory, 3-31-17
31. examples of Jordan forms, Matrix Exponential, 4-3-17
32. differential equations, inner products, 4-5-17
33. orthogonal, GSA example, 4-7-17
34. orthogonal complements, 4-10-17
35. angles in complex inner product space, least squares, adjoint, 4-12-17
36. explicit formula for adjoint, linear isometries, dual space, 4-14-17
37. partial proof of spectral theorems, 4-19-17
38. concluding thoughts on quadratic forms, Quotient Space intro., 4-21-17
39. quotient vector space and the first isomorphism theorem, 4-24-17
40. maps on invariant subspaces and their quotients, 4-26-17
41. bilinear forms, metrics, geometry, music, 4-28-17
42. comments about Test 3 solution and final, 5-1-17

Spring 2015 Linear Algebra Lectures:
These are the lectures from Math 321 I taught in Spring of 2015.

1. Lecture 1 part 1: sets, index notation, rows and columns
2. Lecture 1 part 2: equality by components, rows or columns
3. Lecture 2 part 1: functions, Gaussian elimination
4. Lecture 2 part 2: row reduction for solving linear systems
5. Lecture 3: solution sets, some theoretical results about rref
6. Lecture 4: rref pattern, fit polynomials, matrix algebra basics
7. Lecture 5: prop of matrix algebra, all bases belong to us, inverse matrix defined
8. Lecture 6: elementary matrices, properties and calculation of inv. matrix
9. Lecture 7: block-multiplication, (anti)symmetric matrices, concatenation
10. Lecture 8: span and column calculations in Rn, intro to LI
11. Lecture 9: LI and the CCP
12. Lecture 9 bonus: basics of linear transformations on Rn
13. Lecture 10: fundamental theorem of linear algebra (no video)
14. Lecture 11: gallery of LT, injectivity and surjectivity for LT, new LT from old
15. Lecture 12: examples and applications of matrices and LTs
16. Lecture 13 part 1: solution to Quiz 1
17. Lecture 13 part 2: solution to Quiz 1
18. Lecture 14: vector space defined, examples, subspace theorem
19. Lecture 15: axiomatic proofs, subspace thm proof, Null(A) and Col(A)
20. Lecture 16: generating sets for spans, LI, basis and coordinates
21. Lecture 17: theory of dimension and theorems on LI and spanning
22. Lecture 18: basis of column and null space, solution set structure again
23. Lecture 19: subspace thms for LT and unique linear extension prop
24. Lecture 19.5: isomorphism is equivalence relation, finite dimension classifies
25. Lecture 20 part 1: coordinate maps and matrix of LT for abstract vspace
26. Lecture 20 part 2: examples of matrix of LT in abstract case (Incidentally, my intuition at the end of this about the rank of the BAB mapping is incorrect. That map does in fact have rank 4 despite being built with the rank 2 B.)
27. Lecture 21: kernel vs nullspace, coordinate change
28. Lecture 22 part 1: coordinate change for matrix of LT
29. Lecture 22 part 2:rank nullity, Identity padded zeros thm, matrix congruence comment
30. Lecture 22.5: proof of abstract rank nullity theorem, examples
31. Lecture 23: part 1: quotient of vector space by subspace
32. Lecture 23 part 2: quotient space examples,1st isomorphism theorem
33. Lecture 24: structure of subspaces, TFAE thm for direct sums
34. Lecture 25: direct sums again, gallery of 3D isomorphic vspaces
35. Review for Test 2 part 1
36. Review for Test 2 part 2
37. Lecture 26: motivation, calculation and interpretation of determinants
38. Lecture 27: determinant properties, Cramer's Rule derived
39. Lecture 28: adjoint fla for inverse, eigenvectors and values
40. Lecture 28 additional eigenvector examples
41. interesting example for Lecture 28
42. Lecture 29: basic structural theorems about eigenvectors
43. supplement to Lecture 29
44. Lecture 30: eigenspace decompositions, orthonormality
45. concerning rotations
46. Lecture 31: complex vector spaces and complexification
47. Lecture 32: rotation dilation from complex evalue, GS example
48. Lecture 33: orthonormal bases, projections, closest vector problem.
49. Lecture 34: complex inner product space, Hermitian conjugate and properties
50. Lecture 35: overview of real Jordan form, application to DEqns
51. Lecture 36: invariant subspaces, triangular forms, nilpotentence
52. Lecture 37: nilpotent proofs, diagrammatics for generalize evectors, A = D + N
53. Lecture 38: minimal polynomial, help with homework
54. Lecture 39: solution to takehome Quiz 3
55. Lecture 40: partial course overview

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Last Modified: 8-15-2018