Math 4010 Section 101 Homepage
News & Announcements
04/13 Final Project & Presentation Guidelines and Suggested Topics:
Instead of a Final Exam, we will have a final "project" of sorts. I would like everyone to
select a topic somehow related to this class (I have suggestions listed below). This needs
to be something that we didn't cover or at least didn't cover in detail.
*) Pick and topic and clear it with me (in person or via email).
*) Study your topic.
*) Create a handout. This should be at least one page front and back.
-> Your classmates are your audience. Build off of background from class.
-> Let us know where to go for further reading somewhere in your handout.
[Cite any valuable resources - like websites and textbooks - that you found.]
-> You might want to give a brief historical blurb.
-> Do worked out examples help us understand your topic? If so, include a few.
-> Is your topic about a "big theorem"? Include its proof or a sketch of its proof.
*) During the final exam period, everyone will give a brief presentation of their topic.
11 people x 10 minutes = about 2 hrs. So 10 minutes a piece should be good. A little
longer or shorter should be ok too.
*) The presentation could use slides (like tex or powerpoint) or just use the whiteboard
and handout.
*) Several people can work on related topics. If so, you'll need to coordinate presentations
and handouts. However, everyone should be creating their own unique handout (that work
should be individual not group work).
My Fall 2020 MAT 2110 page has some sample slides.
Or Slides (.pdf) [Source: (.tex)] from a recent talk I gave.
Handout? Why not model after one of my monstrosities.
For example:
Differential Algebra and Liouville's Theorem (.pdf) [Source: (.tex)]
Some differential Galois Theory (.pdf) [Source: (.tex)]
Suggested topics/ideas:
===============================================================================
Inner Product Space Stuff
[Many possibilities from from Spence, Insel, Friedberg Chapter 6]
Rational Canonical Form
[Stuff from Spence, Insel, Friedberg Chapter 7 that we skipped]
Fourier Series
Laplace Transforms or other Integral Transforms
The dimension of a infinite dimensional space's dual space
Quaternions, Hurwitz's theorem, Octonions, or other weird algebras
What is a Lie group?
What is a Jordan algebra?
What is a Lie algebra?
More about Wronskians
Differential forms (extending exterior algebra) and/or DeRham cohomology
Group representation theory/character theory
Jacobson's density theorem
Ring module theory (What if our scalars aren't drawn from a field?)
This semester...
===============================================================================
Gavin: Factorial of a matrix.
Eddie: Ring modules.
Max: Tensors in Geology
Alyssa: Laplace transforms
Natalie: Quaternions
Alex: Rational Canonical Form
Sam: Inner product spaces and Hermite polynomials
Aiden: Automating computations
Jacob: Hurwitz's theorem
Abby: ?Fourier series?
04/08 Solving a 3rd order non-homogeneous linear differential equation with
constant coefficients 5 ways: Maple (.mw) [PDF print: (.pdf)]
We cooked up an example and solved it using undetermined coefficients, variation
of parameters, our partial fraction trick, converting to a first order linear
system, and finally by cheating and just using Maple's dsolve command.
04/06 Homework #8 [Source: (.tex)] is due Wednesday, April 13th.
04/05 A handout to help clarify material covered last class:
Variation of Parameters [Source: (.tex)]
03/30 Homework #7 [Source: (.tex)] is due Wednesday, April 6th.
03/22 Homework #6 [Source: (.tex)] is due Wednesday, March 30th.
03/16 Homework #5 [Source: (.tex)] is due Wednesday, March 23rd.
03/14 Handout: Jordan Form [Source: (.tex)]
Maple Examples (for next class):
Jordan and Functions of Matrices Examples (.mw) [Pdf print: (.pdf)]
More Jordan and Functions of Matrices Examples (.mw) [Pdf print: (.pdf)]
More Jordan Examples (.mw) [Pdf print: (.pdf)]
02/23 Test #1 (Take Home Portion) is due Wednesday, March 2nd
at the beginning of our in class test.
We will have the in class portion of Test #1 on Wednesday, March 2nd.
Here's what to know / what's on (in class) Test #1 as currently written:
*) linear correspondence and using that to answer questions about sets
of vectors being linearly independent or spanning things. This includes
knowing how to take sets and extract bases or extend bases.
*) Know what dual bases do.
*) Know how change of coordinate matrices work including how to get change of
coordinate matrices from other change of coordinate matrices. Similarly,
for coordinate matrices of linear transformations. Also, changing coordinates
relative to dual bases and transpose maps.
*) Know basics about quotient spaces.
*) Know how to show something is a subspace both using the subspace test and
by recognizing it as a kernel, span, etc.
*) Know linear transformation basics (like how to show something is a linear
transformation).
*) Know how to prove statements about dimensions using our big theorem that
every linearly independent set extends to a basis and every spanning set
contains a basis.
02/11 Seeing several people have work pile up, I've changed due dates for Homeworks #3 & #4.
We'll be a little less ambitious.
02/09 Handout:
Dual Spaces [Source: (.tex)] -- some notes.
Homework #4 [Source: (.tex)] is due Wednesday, February 16th Monday, February 21th.
02/02 Handout:
Coordinate Matrix vs. Coordinates [Source: (.tex)]
01/31 Homework #3 [Source: (.tex)] is due Wednesday, February 9th Monday, February 14th.
01/28 Notes [Source: (.tex)] about independence, spanning, dimension, coordinates, & direct sums.
For stuff we're heading into...
Computing Bases (.pdf) [Source: (.tex)]
Coordinate Matrix Example (.pdf) [Source: (.tex)]
Kernel, Range, & Composition Example (.pdf) [Source: (.tex)]
01/24 Homework #2 [Source: (.tex)] is due Monday, January 31st.
01/12 Homework #1 [Source: (.tex)] is due Friday, January 21st Monday, January 24th.
01/10 Handouts:
Characterizing the Reals [Source: (.tex)]
Axioms for the Reals [Source: (.tex)]
Spence, Insel, & Friedberg = SIF
SIF suggested reading: Read the appendices especially C & D (on fields and complex numbers).
We should (after reviewing) be able to...
*) Translate between equations and augmented matrices.
*) Use Gauss-Jordan elimination to get a matrix into RREF.
*) Write the solution(s) of a linear system in vector/parametric form.
*) Use the linear correspondence between columns to recontruct a matrix from
its RREF and pivot columns.
Gauss-Jordan elimination and the Linear Correspondence material can be found
in the RREF handout [Source: (.tex)].
Note: This handout covers much more information than we need.
TECHNOLOGY:
We will eventually have homework assignments to be completed using technology.
I plan to write them assuming the use of Maple (you can download for free and
activate using the code I provided in AsULearn). Here is a quick intro focused on
linear algebra:
Introduction to Linear Algebra in Maple (.mw) [PDF print of Maple worksheet: (.pdf)]
I have a video introduction, but it's taylored for Calculus 3: Maple Intro. (38mins).
My Maple examples can be found here but most of them aren't for linear algebra.
Mathematica is Maple's main competator. It is the engine that lies behind Wolfram Alpha.
You probably can get Alpha to do linear algebra computations for you, but the
interface really isn't built for it. So I'll pass.
If you just want to compute the RREF of a matrix of integers (and show the steps),
there are many choices but Bogacki's webpage is pretty easy to use.
Symbolab is a propular choice for computing
symbolic integrals and derivatives. It can also do linear algebra stuff.
If you want industrial strength numerical calculations, you either code directly in
Python or use Matlab (matrix laboratory). Octave is an open source version of Matlab.
If you are a science major, (eventually) learning Python and Matlab are good ideas.
Finally, SAGE is open source mathematical software built on top of Python. It gives
us the easiest access to (free) non-trivial calculations. I have created a sandbox
for doing Linear Algebra in SAGE.
Note: Linear Algebra in SAGE Tutorial.
01/07 Syllabus and Tentative Schedule posted.
10/20 Texts:
Spence, Insel, & Friedberg - "Linear Algebra" 4th edition. ISBN: 9780130084514
Bookfinder.com lists used copies for about $15.
Rabenstein - "Elementary Differential Equations with Linear Algebra" 3rd edition. ISBN: 9780125739450
Abebooks.com has copies for about $6.50.
10/11 Course Data
MAT 4010 Section 101
ABST LIN ALG W APPL TO DIFF EQ
MWF 11:00am-11:50am
WA 304
Final Exam:
TBD
Course Title & Description:
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Abstract Linear Algebra with Applications to Differential Equations
I plan to spend approximately half of the semester exploring topics
from "abstract" linear algebra. Then the remainder of the semester
will be devoted to using algebraic methods to solve differential
equations and systems of differential equations.
Although I will assume familiarity with introductory linear algebra,
I will not assume any background from differential equations. Some
topics may intersect with group or ring theory (modern algebra), but
any background will be covered as needed.
Possible topics (in more or less detail according to interest):
Linear Algebra: vector spaces over different fields, infinite
dimensional spaces, dual spaces, quotient spaces,
Jordan canonical form, the matrix exponential,
tensor products, and exterior algebras.
Differential Equations: solving DEs by factoring linear operators, the
methods of undetermined coefficients, variation of
parameters, and a partial fraction decomposition
technique for solving non-homogeneous linear DEs, using
the matrix exponential and variation of parameters for
solving systems of DEs, symmetry methods for solving DEs
(finding integrating factors and very basic, very applied
Lie theory), and (if there is interest), some differential
algebra which would including a touch of differential
Galois theory which attempts to answer the question, "When
can we solve a differential equation?"
Note: Although having courses in modern algebra or differential equations
would be helpful, I will *not* be assuming that anyone has this
background! Any DEs or modern algebra background will be covered in
class as needed.
Prerequisites: MAT 1120, MAT 2110, & MAT 2240
(or permission of the instructor)
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Any questions about this class?
Send me an email at cookwj@appstate.edu