Math 2240-102 (Summer 2023 - Session 1) Homepage

AsULearn [eText access] Syllabus Schedule Zoom Office/Classroom

Videos Suggested Homework [4th ed.] Old Exams SAGE Linear Algebra Sandbox

News & Announcements

6/30

Final grades have been sent out.

If you ever have any questions about this class, other math classes, or whatever, please feel free to drop by my office or send me an email.

Have a great 4^th and rest of the summer!

6/26

The Final Exam is Friday at 8:00am. I plan to log into Zoom (our usual office/classroom link) on Thursday morning at 8:00am to answer questions / review for the final.

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, 4.1-4.6 (& "old" 4.6 too), 5.1-5.3, 6.1-6.7.

I would encourage you to prepare 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.
• 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.

6/23

Don't forget Homework #3 [Source: (.tex)] is due today Sunday!

Our last quiz is on Wednesday. What do I want you to know for Quiz #4?

• Know how to run the Gram-Schmidt orthogonalization process.
• Know how to find the basis for an orthogonal complement.
  ($W^\perp$ is the orthogonal complement of $W$)
• Know what it means to be an orthogonal set and an orthonormal set.
• Given an orthogonal basis, know how to compute coordinates with inner products.

As always, I'm happy to meet via Zoom or in person. Just email to let me know.

Schedule for our last week:

• Monday: Watch "Orthog. Sets and Proj." and "Gram-Schmidt"
• Tuesday: Watch "Orthog. Comp." and "Proj. Mat. and Least Sq."
• Wednesday: Quiz #4 [No Zoom - I will email out the quiz at 7:30am. It is to be turned in by 9:45am.]
  Watch "Survey of..." (last video)
• Thursday: [Zoom at 8am for final exam questions and review.]
• Friday: Final Exam [Log into Zoom at 8am.]
  Review materials to be posted early next week.

6/19

Test #3 is Thursday at 8:00am. I plan to log into Zoom (our usual office/classroom link) on Wednesday morning at 8:00am to answer questions / review for the test.

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 and in Homework #3). Be aware that there aren't many (?any?) old questions covering complex eigenvalues/vectors. Relevant handout:

• 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 perfectly to our current text. You can find old tests here. 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.
• 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.

6/16

Don't forget Homework #2 [Source: (.tex)] is due today!

We have a quiz on Monday. What do I want you to know for Quiz #3?

• Know how to compute 2x2 and 3x3 determinants. In particular, know the 3x3 "trick".
• Know how to compute determinants using row and column expansions.
• Know how elementary operations affect the determinant
• Know determinant properties.
  Example: If det(A)=5, det(B)=2, and det(C)=3, then det(C^T A^(-1) B^3) = det(C) det(A)^(-1) det(B)^3 = 3(2^3)/5 = 24/5.

As always, I'm happy to meet via Zoom or in person. Just email to let me know.

Schedule for next week:

• Monday: Quiz #3 [Zoom at 8am for questions then quiz.]
  Watch "Eigen...and Char. Eqns." and "Eigen. Theory and Diag." Referenced Handout:
  An Eigenhandout [Source: (.tex)]
• Tuesday: Watch "Diagonalization Exs. and Why..."
• Wednesday: [Zoom at 8am for test questions and review.]
  Watch "Complex Eigen.", "Diag. App.: Lin. ODEs", and "Diag. App.: Markov..."
• Thursday: Test #3 [Log into Zoom at 8am.]
  Review materials to be posted early next week.
• Friday: Homework #3 due [Submit via email - due by 5pm.]
  Watch "Inner Prod. Sp." and "Inner Prod. Sp.: Orth. and C.-S."

Note: Homework #3 [Source: (.tex)] is due Friday, June 23^th.

6/12

Test #2 is Thursday at 8:00am. I plan to log into Zoom (our usual office/classroom link) on Wednesday morning at 8:00am to answer questions / review for the test.

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 4.1-4.6 & "Old 4.6" [you may skip 4.7-4.8]. 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:

• Coordinate Matrix Example (.pdf) [Source: (.tex)]
• Computing Bases (.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 perfectly to our current text. You can find old tests here. 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.
• 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.

6/09

Don't forget Homework #1 [Source: (.tex)] is due today!

We have a quiz on Tuesday. What do I want you to know for Quiz #2?

• Know how to find bases and dimensions of the Row Space, Column Space, and Null Space of a matrix.
• Given a basis for a vector space and a vector, know how to find that vector's coordinates. Conversely, given a coordinate vector (and specified basis), know how to recover the vector that has those coordinates.
• Know what a subspace is. Know the "subspace test". Be able to show something is a subspace by either running the subspace test or by recognizing it as a kernel/null space or range/span/column space.

As always, I'm happy to meet via Zoom or in person. Just email to let me know.

Schedule for next week:

• Monday: Watch "Vector Spaces: Bases and Dim." and "Vector Spaces: Coords. and More.."
• Tuesday: Quiz #2 [Zoom at 8am for questions then quiz.]
  Watch "Vector Spaces: Theory Rev...", "Vector Spaces: Comp. with Bases", and "Coord. Mats. and Change..."
  Referenced Handouts:
  Coordinate Matrix Example (.pdf) [Source: (.tex)]
  Computing Bases (.pdf) [Source: (.tex)]
• Wednesday: [Zoom at 8am for test questions and review.]
  Watch "Coord. Matrices: Comp...." and "Simple Determinants"
  Referenced Handout:
  Kernel, Range, & Composition Example (.pdf) [Source: (.tex)]
• Thursday: Test #2 [Log into Zoom at 8am.]
  Review materials to be posted early next week.
• Friday: Homework #2 due [Submit via email - due by 5pm.]
  Watch "Determinants: Gen. and Props.", "Determinants: Geom.", and "Cramer's Rule"

Note: Homework #2 [Source: (.tex)] is due Friday, June 16^th.

6/05

Test #1 is Thursday at 8:00am. I plan to log into Zoom (our usual office/classroom link) on Wednesday morning at 8:00am to answer questions / review for the test.

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 perfectly to our current text. You can find old tests here. The following problems are relevant:

• 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.
• 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.

6/02

Schedule for next week:

• Monday: Watch "Linear Transformations: Intro." and "Linear Transformations: Standard Mat..."
• Tuesday: Watch "Matrix Algebra: Add,..." and "Matrix Algebra: Inverses..."
• Wednesday: [I will log in 8am on Zoom to answer test questions and review.]
  Watch "Matrix Algebra: Partitioned Matrices", "Matrix Algebra: LU-Decomp...", and "Graphics: Hom. Coords."
• Thursday: Test #1 [Log into Zoom at 8am.]
  Review materials to be posted early next week.
• Friday: Homework #1 due [Submit via email - due by 5pm.]
  Watch "Vector Spaces: Axioms and Subspaces" and "Vector Spaces: Linear Transformations"

Note: Homework #1 [Source: (.tex)] is due Friday, June 9^th.

5/30

Schedule for this week:

• Tuesday: Meet to discuss syllabus, logistics, etc.
  Watch "What is Linear Algebra?", "Linear Systems: Row Operations", & "Linear Systems: RREF"
• Wednesday: Watch "Linear Systems: Parameterizing Solutions and Multiple Systems" & "Vectors, Linear Combinations, and Spans"
• Thursday: Watch "Linear Independence and The Linear Correspondence", "Matrix Vector Multiplication", & "Homogeneous Systems"
• Friday: Meet for Quiz #1 which covers 1.1-1.5 plus the linear correspondence between columns.
  Also, watch the application videos & "Span and Linear Independence: Recap"
  
  What I want you to be able to do on Friday's quiz:
  • 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 reconstruct 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.
  
  

5/29

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.

5/28

From Genie Griffin (griffine@appstate.edu) our tutoring coordinator:

Hi All,

This summer, University Tutoring Services will only be offering appointment based tutoring. There will be no drop in tutoring. Students can make these appointments through TracCloud.


Info for your students on how to make an appointment:

• The expectations of tutoring follow Appalachian State University Conduct, Academic Integrity, and Title IX Compliance office's standards of student behavior. Additionally, all tutors are allowed to discontinue service to students if they feel threatened, harassed, uncomfortable, unsafe, etc. It is up to the discretion of the Tutoring Coordinator as to whether or not the privilege of tutoring will continue to be offered to each student, based on behavior.
• You can schedule an appointment as early as two weeks in advance and up to 24 hours before the scheduled appointment via TracCloud.
• You are able to schedule two appointments per week per course.
• To cancel or reschedule an appointment without a penalty, contact the Student Learning Center at least twenty four hours before the scheduled appointment by calling 828-262-2291, email student-learning-center@appstate.edu, or visit us in 208 DD Dougherty.
• Three missed appointments will result in the loss of tutoring privileges.
• Missed appointments will be received for the following:
  • Canceling less than 24 hours prior to the appointment
  • Failure to arrive to the appointment
  • Consistent tardiness or leaving the appointment early

Please email me with any questions!
Genie

5/23

For the first day of class (5/30), we will meet at 8am 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 have provided (on our AsULearn page) 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.

5/18

Homework sets:
Homework #1 [Source: (.tex)] is due Friday, June 9^th.
Homework #2 [Source: (.tex)] is due Friday, June 16^th.
Homework #3 [Source: (.tex)] is due Friday, June 23^rd.

5/16

Questions about this upcoming course? Email me: cookwj@appstate.edu

Syllabus, schedule, & suggested homework have been updated.
To do -- Update: AsULearn and Pearson [eText].