Math 2240 Section 101 Homepage
News & Announcements
08/04 Final grades have been sent out.
If you ever have any questions about this class, other math classes, or whatever, my door
is open. Stop by anytime (even just to say "Hi") - it would be nice to meet in person some day!
I hope you all have a great summer (what's left of it anyway)!
07/30 Quiz #4 is tomorrow and our Final Exam is Tuesday (August 4th).
The Final Exam is cumulative, but you may skip some material. The sections of
our text covered are: 1.1-1.9, 2.1-2.3, 2.8-2.9, 3.1-3.2, 4.1-4.7, 5.1-5.3, 6.1-6.7
You may have a page of notes. Please copy down formulas, theorems, examples,
etc. that might be of use during the exam.
General notes:
*) I will ask you to do some computations by hand. You should know how to row
reduce a matrix, multiply, transpose, find inverses via row reduction, compute
determinants, find eigenvalues, and find eigenvectors. [by hand.]
*) You should know how to recognize subspaces as column and null spaces. You
should also know how to find bases for such spaces. You should be able to find
coordinates for a vector and be able to recover a vector from its coordinates.
*) You should be able to diagonalize a matrix.
*) You should be able to run the Gram-Schmidt algorithm, find orthgonal projections,
and find a least squares line of best fit.
*) While I may ask theory questions about abstract vector spaces and linear
transformations, I do not plan on asking you to do computations with abstract
spaces (e.g., spaces of polynomails). I don't plan on asking you to compute a
coordinate matrix for a linear transformation, but you should know how to use
such a matrix to tell if a transformation is one-to-one, onto, find rank, nullity,
etc.
Studying? I recommend looking over notes, homework answer keys, test answer keys,
and then looking over old exams and finals.
Here's a list of relevant old exams to consider...
[Note: These exams don't have many inner product space (i.e., Chapter 6) problems.
Also, I want you to recognize subspaces as column spaces or null spaces.
Many of these exams/keys will run a subspace test instead.]
Fall 2012: Test #1 -- all of it.
Test #2 -- problems 2-5.
Test #3 -- problems 2-3,5.
Test #4 -- problems 1,3-6.
Final Exam -- problems 1,2,4,5c,6,7,8bc,10.
Spring 2011: Test #1 -- all of it.
Test #2 -- problems 1,3-6.
Test #3 -- problems 2-5.
Test #4 -- problems 3-6.
Final Exam -- problems 1,2,4,5bc,6,8bc,9a,10.
Fall 2008: Test #1 -- problems 1-5.
Test #2 -- problems 1-3,4a,5.
Test #3 -- problems 1,4ac,5.
[Rutgers]
Spring 2008: Exam #1 -- problems 2,3a,4ab,5-8.
Exam #2 -- problems 1-5,6a.
Fall 2006: Exam #1 -- problems 2,4ab,5-8.
Exam #2 -- problems 1,3,5,6,7ab.
Summer 2006: Exam #1 -- problems 1,3ab,4,5,7-9.
Exam #2 -- problems 1,2,4,6,7a,8,9a.
Final Exam -- problems 1-3,5-7,8a,9,10cd.
07/28 Announcements for the upcoming week (7/28-8/04)...
Curved quiz/homework grades (eventually).
Final Exam score can replace lowest test score. (if it helps)
Prepare a page of notes for the final.
Notes on Grades: During a regular semester I drop a lowest quiz and homework.
However, I don't usually do this during the summer since we don't have many quizzes
and our homeworks aren't weighted evenly. Instead I tend to curve a little.
I will replace your lowest test with the final exam score (if it helps). So if
your lowest test score was a 50 and you score 80 on the final exam, the 50 will be
replaced by 80 in final grade calculations.
Notes on Final Exam: I will post final review stuff on my webpage soon. Since
we've covered so much ground in such a short period of time, I traditionally allow a
sheet of notes on the final exam. Please, prepare a page of notes for the final exam.
On your page include any definitions, formulas, and worked out examples that might be
helpful. Preparing your page will likely help you prepare for the exam. Anything you
want is fair game for this page: theorem statements, algorithms, formulas, worked out
examples, motivational poetry, doodles, etc.
Handouts: Basic inner product space definitions (.pdf) [Source: (.tex)]
Gram Schmidt (.pdf) [Source: (.tex)]
Linear Regression Examples (.mw) [(.pdf)]
Gram-Schmidt & QR Example (.mw) [(.pdf)]
Weekly announcements (7/28-8/04):
Tuesday (today): Watch up to "Inner Product Spaces: Orthogonality and Cauchy-Schwarz"
Homework 3 (our last homework) is due.
I logged in at 10:20am as usual to answer questions.
Wednesday: Watch up to "Gram-Schmidt"
I will log in at 10:20am as usual to answer questions.
Thursday: Watch up to "Projection Matrices and Least Squares"
I will log in at 10:20am as usual to answer questions.
Friday: Watch final video "Survey of Symmetric Matricies, Spectral Theorem, QR Decomp, and SVD"
Quiz #4 covers Sections 6.1-6.4.
Log in at 10:20am. We will go over questions and then have the quiz.
Monday: Final Exam REVIEW:
I will log in at 10:20am as usual. We can review for the final exam.
Tuesday (August 4th: Final Exam -- Log in at 10:20am and I will send out your test (via email).
(Course webpage for Final Exam details -- to be updated soon.)
07/22 Test #3 is Monday at 10:20am:
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Test #3 will cover Chapter 3 (on determinants) and Chapter 5 sections 5.1-5.5 (this skips the applications
at the end of the chapter). Make sure you also know about computing square roots and arbitrary powers of
matrices (there are questions about this on old Rutgers exams). Be aware that there aren't many (?any?) old
questions covering complex eigenvalues/vectors.
Relevant handouts:
An Eigenhandout [Source: (.tex)]
As with the last test: You should be using technology to find RREFs of matrices. When doing this, make sure
you write down the matrix you start with and then write its RREF. But you don't need to do/show row operations
by hand and write all of these steps unless the specific problems calls for that (like computing a determinant
using row reduction).
I recommend looking at your quiz, notes, and suggested homework when studying. In addition, I have several
old tests posted. However, these do not correspond to our current text. The following problems are relevant...
Fall 2012: Test #2 -- problems 2-5
Test #4 -- problems 4-6.
Final Exam -- problems 4, 10.
Spring 2011: Test #2 -- problems 3-6.
Test #4 -- problems 4-6.
Final Exam -- problems 4, 10.
Fall 2008: Test #1 -- problems 4b.
Test #2 -- problems 1, 2.
[Rutgers]
Spring 2008: Exam #2 -- problems 1, 2, 5, 6.
Fall 2006: Exam #2 -- problems 1, 3, 5, 7.
Summer 2006: Exam #2 -- problems 1, 4-6, 9.
Final Exam -- problem 6, 9.
07/21 Announcements for the upcoming week (7/21-7/28)...
Handout: An Eigenhandout [Source: (.tex)]
Weekly announcements (7/21-7/28):
Tuesday (today): Watch up to "Cramers Rule"
Homework #2 is due (in the afternoon - 5ish).
I will log in at 10:20am as usual to answer questions.
Note: Tests and answers were returned via email.
Wednesday: Watch up to "Eigenvector Theory and Diagonalization"
Quiz #3 covers Chapter 3 (determinants quiz)
Log in at 10:20am. We will go over questions and then have the quiz.
Thursday: Watch to "Diagonalization: Examples and Why it's Useful"
I will log in at 10:20am as usual to answer questions.
Friday: Watch up to "Application: Markov Processes"
TEST REVIEW: I will log in at 10:20am as usual. We can review for Monday's test.
Monday: Test #3 -- Log in at 10:20am and I will send out your test (via email).
(See course webpage for Test #3 details -- to be updated soon.)
Tuesday: Watch up to "Inner Product Spaces: Orthogonality and Cauchy-Schwarz"
Homework 3 is due.
I will log in at 10:20am as usual to answer questions.
07/16 Quiz #2 is today.
Test #2 is Monday at 10:20am:
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We will back up a little and go over the beginning of chapter 2 again.
Test #2 will cover Chapter 2 sections 2.1-2.3,2.8-2.9 [you may skip 2.4-2.7]. It will also
cover Chapter 4 sections 4.1-4.7 [you may skip 4.8-4.9]. Finally, it covers matrices of
linear transformations. In our textbook this is part of Chapter 5 section 5.3. This material
is better illustrated on our handouts, videos, and old tests. Relevant handouts:
Computing Bases (.pdf) [Source: (.tex)]
Coordinate Matrix Example (.pdf) [Source: (.tex)]
Kernel, Range, & Composition Example (.pdf) [Source: (.tex)]
I expect you to perform some matrix multiplications by hand. The remainder of the test can
be completed with the assistance of technology. However, all you should be using technology
to do is find RREFs of matrices. When doing this, make sure you write down the matrix you
start with and then its RREF. But you don't need to do row operations by hand and write all
of these steps.
I recommend looking at your quiz, notes, and suggested homework when studying. In addition, I have several
old tests posted. However, these do not correspond to our current text. The following problems are relevant...
Fall 2012: Test #1 -- problems 4, 5.
Test #2 -- problem 1.
Test #3 -- all of it.
Test #4 -- problems 1-3.
Final Exam -- problems 2,3, 5-9.
Spring 2011: Test #1 -- problems 5, 6, 8.
Test #2 -- problem 2.
Test #3 -- all of it.
Test #4 -- problems 1-3.
Final Exam -- problems 2, 3, 5-7, 8ab, 9.
Fall 2008: Test #1 -- problems 2cd, 3, 4a.
Test #2 -- problems 3-5.
Test #3 -- all of it.
[Rutgers]
Spring 2008: Exam #1 -- problems 2, 3a, 4, 5.
Exam #2 -- problems 3, 4
Fall 2006: Exam #1 -- problems 2, 4, 5, 8.
Exam #2 -- problems 2, 6.
Summer 2006: Exam #1 -- problems 3, 4, 7-9.
Exam #2 -- problems 2, 7, 8.
Final Exam -- problem 3.
07/14 Announcements for the upcoming week (7/14-7/21)...
Tuesday (today): Watch up to "Vector Spaces: Linear Transformations"
Homework #1 is due today.
[I logged in at 10:20am as usual to answer questions.]
Wednesday: Watch up to "Vector Spaces: Coordinates and More about Isomorphisms"
I will log in at 10:20am as usual to answer questions.
Thursday: Watch up to "Coordinate Matrices and Change of Basis"
Quiz #2 covers 4.1, 4.2, 2.8 & 2.9
Log in at 10:20am. We will go over questions and then have the quiz.
Handouts:
Coordinate Matrix Example (.pdf) [Source: (.tex)]
Computing Bases (.pdf) [Source: (.tex)]
Friday: Watch up to "Simple Determinants"
TEST REVIEW: I will log in at 10:20am as usual. We can review for Monday's test.
Handouts:
Kernel, Range, & Composition Example (.pdf) [Source: (.tex)]
Monday: Test #2 -- Log in at 10:20am and I will send out your test (via email).
(See course webpage for Test #2 details -- to be updated soon.)
Tuesday: Watch up to "Cramer's Rule"
Homework #2 is due.
I will log in at 10:20am as usual to answer questions.
Maple demo:
Visualizing Determinants (.mw)
Handouts referenced here:
Computing Bases (.pdf) [Source: (.tex)]
Coordinate Matrix Example (.pdf) [Source: (.tex)]
Kernel, Range, & Composition Example (.pdf) [Source: (.tex)]
Optional Handout:
What is the difference between vector space coordinates and a coordinate matrix?
Coordinates vs. Coordinate Matrices (.pdf) [Source: (.tex)]
07/08 Homework #1 [Source: (.tex)] is due Tuesday, July 14th.
Homework #2 [Source: (.tex)] is due Tuesday, July 21st.
Homework #3 [Source: (.tex)] is due Tuesday, July 28th.
Upcoming assignments/meetings:
Homework #1 [Source: (.tex)] is due Tuesday, July 14th.
Please turn in your via email by the end of the day Tuesday. 5ish would be great.
New to Maple?
A few helpful tips for doing linear algebra in Maple: (.mw) and (.pdf)
In case I didn't emphasize it enough: You should know about the Math Lab (free tutoring
center). This summer's math lab meets via Zoom. Its hours of operation are Monday through
Thursday from 5 to 8pm. The lab's Zoom link is: https://appstate.zoom.us/my/genmatlearninglab.
Thursday: Watch up to "Matrix Algebra: Add, Scale, Multiply, Transpose" or the next video.
I will log in at 10:20am as usual to answer questions.
Friday: Watch up to "Graphics: Homogeneous Coordinates".
I will log in at 10:20am as usual. We can review for Monday's test.
Monday: Test #1 -- Log in at 10:20am and I will send out your test (via email).
(See Test details below.)
Test #1 is Monday at 10:20am:
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Test #1 will cover chapter 1 (sections 1.1-1.9 we skipped 1.10), sections 2.1-2.2, and
the Linear Correspondence material from our handout.
I want the beginning of the test to be completed by hand (showing detailed
calculation). Technology will be helpful but not totally necessary to complete the
rest of the test.
I recommend looking at your quiz, notes, and suggested homework when studying. In addition, I have several
old tests posted. However, these do not correspond to our current text. The following problems are relevant...
Fall 2012: Test #1 -- all of it.
Test #2 -- problem 1.
Test #3 -- problem 3ab and problem 6 sort of applies.
Test #4 -- problem 1a and if 1-1 or onto.
Final Exam -- problems 1-3,5c,6c,9a (just finding the matrix and if 1-1 or onto),
and problem 8a sort of applies.
Spring 2011: Test #1 -- all of it.
Test #2 -- problems 2 and 6b.
Test #3 -- problems 3, 4a and problem 6b sort of applies.
Test #4 -- problem 1a and if 1-1 or onto.
Final Exam -- problems 1-3,5b,6c,9a (just finding the matrix and if 1-1 or onto),
and problem 8a sort of applies.
Fall 2008: Test #1 -- problems 1-4,6.
Test #2 -- the first two parts of problem 3.
Test #3 -- problems 2a and 3a sort of apply.
[Rutgers]
Spring 2008: Exam #1 -- problems 1a,2-5,6ab,7b.
Fall 2006: Exam #1 -- problems 1a,2-5,6ab,7,8.
Summer 2006: Exam #1 -- problems 1-5,7,8,10.
Exam #2 -- problem 8ab.
Final Exam -- problems 1,2,10.
07/06 Quiz Tomorrow covers 1.1-1.5 plus the linear correspondence between columns.
What I want you to be able to do tomorrow:
*) 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 via email). 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.
07/01 Met at 10:20am to introduce ourselves and discuss syllabus and logistics.
We will meet at 10:20am tomorrow (Thursday) to answer questions.
Videos? Watch up through "Linear Systems: RREF" today and then up through
"Application: Balancing Chemical Equations" on Thursday and Monday.
We will have a Quiz on Tuesday covering up to these application videos. I'll
give more details in the future (probably on Monday).
06/30 We meet tomorrow at 10:20am via Zoom to discuss the syllabus and logistics.
[You will need a password for the Zoom meeting - I will email it to you.]
Feel free to check out the first few videos: Math 2240 Videos.
We will periodically use Maple throughout the term. I have some demos that can be
found on my Maple Examples page.
I will provide a link & activation code so you can download Maple for free.
Alternatively, you can access Maple and other software by logging on to a campus
lab computer using udesk.appstate.edu
You should be aware of the Math Lab – free tutoring for Math 2240 (and other math classes).
More information about the Math Lab can be found at https://mathsci.appstate.edu/programs-courses/resources/math-tutor-lab.
This summer's math lab meets via Zoom. Its hours of operation are Monday through Thursday from 5 to 8pm.
The lab's Zoom link is: https://appstate.zoom.us/my/genmatlearninglab.
06/22 Course Data
MAT 2240 Section 101
INTROD TO LINEAR ALG
Meeting Times MTWRF 10:20am-12:00pm ONLINE
[Dates: 07/01-08/04]
Course Title & Description:
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MAT 2240. Introduction to Linear Algebra (3 credits)
A study of vectors, matrices and linear transformations, principally
in two and three dimensions, including treatments of systems of linear
equations, determinants, and eigenvalues.
Prerequisite: MAT 1120 or permission of the instructor.
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Any questions about this class?
Send me an email at cookwj@appstate.edu