## Homework Help Abstract Algebra Proofs

My first pass at Abstract Algebra was actually from that book. It is not great for covering the theory of Abstract Algebra as heavily as other texts but it is excellent at giving many simple clear examples that make it very easy to approach the subject. I'll phrase my advice using two excellent quotes from his own book!

As you cover material, construct the groups you are talking about. Play around with them! Get good at group operations, mappings, and getting a feel for what groups/rings look like and how they work. They are meant to generalize $\mathbb{Z}$ and $\mathbb{R}$. So as you learn new theorems, try to see how it builds on what you just learned. Carefully think about how the properties help groups and rings mimic the appearance of things like $\mathbb{Z}$ and $\mathbb{R}$. What doesn't hold in groups and rings? A useful thing here is going as many exercises from Gallian as possible. For example, he asks for many such examples where a property may hold in $\mathbb{R}$, but not under the group/ring operation.

This stock of examples not only helps you get a feel for the subject but often helps gives you a stock of counterexamples for such exercises. Groups/rings you especially will want to focus on are the symmetric group $S_n$, the alternating group $A_n$, the rings $\mathbb{Z}$ and $\mathbb{Z}_n$, the dihedral group $D_n$, cyclic groups, the basic matrix groups/rings introduced by Gallian. Moreover, don't just limit exercises to those in Gallian! Many of the exercises from the famous Dummit and Foote's Abstract Algebra are great for focusing on more proof intensive exercises than Gallians more 'example' exercises. You can get the questions from his text along with their 'solutions' (don't trust them all and moreover, DO THEM FOR YOURSELF!) here

Finally, a last piece of advice to really keep to heart and practice through this (and more importantly) and Mathematics class you will ever take:

This is highly important to learning Mathematics. The only way to learn Mathematics is to do it. Sadly, though learning the theorems and proofs often takes away vital experience discovering and proving them for yourself through lab-like exercises! So as you read through the book, think why they came to the conclusions they did. Try the proofs before you read them or at least think about how you might show it! (It helps especially with your confidence if before you read it you outline how you would try it and it turns out that that's how it's done!). Find examples where the theorem fails if you take away one of the assumptions. Check the converse. Et cetera.

But most of all, good luck! Abstract Algebra is a fascinating subject and I hope you love it as much as I do!

__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!

- You Tube Playlist for Math 421 of Fall 2017
- Course Plan for Math 421 of Fall 2017
- 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.
- 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:

- What am I asked to prove ?
- Can I define the words used ?

__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. This was based on Gallian's classic friendly text. Here are links to 2016's Lecture Notes and Lectures:

__Tests and some Solutions from Fall 2016:__

__Homework Solutions from Fall 2016:__

__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:

__Spring 2017 Linear Algebra Lectures:__

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

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

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Last Modified: 8-8-2017

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