Oh Snow! So our final exam got derailed yesterday due to snow.
Most of the classes opted to take their test average as a final exam score.
A few brave souls actually took the exam. I have just sent out a final grade report.
Also, all tests including the final exam and their keys are posted now.
Grades posted. Done!
Finally, just because this class is over, don't feel like you can no longer
ask me questions. If you ever have a question (mathematical or otherwise), please
stop by my office or send me an email. I'm happy to help if I can.
Merry Christmas!
12/01
Upcoming:
• Wednesday - last day of class - Review for the Final Exam
• Thursday - reading day - Review Session at 9am (until whenever)
in our regular classroom: Walker 304.
• Monday - December 8th - Final Exams 8am-10:30am & 11am-1:30pm.
Grading & Final Notes:
• I dropped the lowest quiz and lowest homework score.
• I will replace your lowest test score with the final exam score (if it helps).
• You may bring one page of notes to the final exam.
(one sided, standard sized paper, printed and handwritten are ok)
The Final Exam is Monday 8:00am-10:30am (for the earlier section) and
11:00am-1:30pm (for the later section). This exam is cumulative, but you may skip some material.
The sections of our text covered are: 1.1-1.5, 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 carefully prepare your page of notes. Please copy down formulas, theorems,
examples, etc. that might be of use during the exam.
General notes:
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
(onto vectors and onto subspaces - including projection matrices), orthogonally
diagonalize a matrix, find a basis for an orthogonal complement, find least
squares solutions, and find a 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 - except possibly a "standard matrix",
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 our tests and quizzes. Then notes and our homeworks and
suggested homework. In addition, old exams and finals can provide some helpful extra review
problems.
Here's a list of relevant old exams to consider...
[Note: Overall, 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 2025:
Test #1 -- all of it.
Test #2 -- all of it.
Test #3 -- all of it. [I don't plan on asking anything like 2b.]
Fall 2012:
Test #1 -- all of it.
Test #2 -- all of it.
Test #3 -- problems 2-3,5.
Test #4 -- problems 1,2a,3-6.
Final Exam -- problems 1,2,3a,4,5c,6,7,8bc,9a,10.
Spring 2011:
Test #1 -- all of it.
Test #2 -- problems 1,2a,3-6.
Test #3 -- problems 2-5.
Test #4 -- problems 1,2ac,3-6.
Final Exam -- problems 1,2,3a,4,5bc,6,7,8bc,9a,10.
Today's Examples:
• Gram-Schmidt Examples
[Source: (.tex)]
[Note to future self: This should be expanded and fold in Su'23 Q4.]
Reminder/Upcoming:
• Monday: Tie up loose ends & review for Test #3.
• Tuesday: Sam & Eddie's review session is at 11am in Walker 308.
[We will try to get this recorded.]
• Wednesday: Test #3
Test #3 is Wednesday (11/19).
Test #3 will cover
Sections 3.1-3.3 (determinants, properties of determinants, & what determinants mean
geometrically - you may skip Cramer's rule),
Sections 5.1-5.3 (eigenstuff),
Sections 6.1, 6.2, & 6.4 (inner product spaces, orthogonal sets, & Gram-Schmidt).
Once again, I will design this test to be done without technology, but calculators are allowed.
I recommend looking at your homework, quizzes, notes, and suggested homework when studying.
In addition, parts of old tests could be helpful.
You can find old tests here.
Note: My old exams don't have many inner product space (i.e., Chapter 6) problems.
There are some dot product related geometric "basics" problems on my old Test #1's from Calculus 3.
You can find Calculus 3 tests here. While not all parts of their
problem #1's are relevant, you might find problems:
Spring 2016 1abc, Summer 2016 1bc, Fall 2016 1bc, Summer 2019 1ac, & Spring 2021 1c to be useful
as practice. In addition, Summer 2023 Quiz #4
[Source: (.tex)]
and its Answer Key (skip 2 part I for now)
have some practice questions about Chapter 6 stuff.
Reminder:
• Quiz #6 is Wednesday. We have been studying eigenstuff!
You should know stuff from our An Eigenhandout
[Source: (.tex)].
This is Section 5.1-5.3 in our textbook.
In particular, be ready to find a characteristic polynomial then
eigenvalues and eigenvectors of a matrix and thus determine
algebraic and geometric multiplicities. Be able to determine if
a matrix is diagonalizable and if so be able to find a matrix
P that diagonalizes it. In addition, know that a matrix is
singular exactly when zero is an eigenvalue and that the
determinant (resp. trace) is the product (resp. sum) of the
eigenvalues (counting multiplicity).
For some good practice problems check out old tests/exams.
In particular, check out:
We are about to look at defining and computing determinants. Determinants are deeply connected
with geometric ideas like area and volume. While the following links are mostly beyond what I intend
to discuss in class, I will leave them here for anyone who is interested.
Reminder/Upcoming:
• Friday: Homework #2
[Source: (.tex)]
is due today.
We will discuss kernels and ranges of linear transformations today.
• Monday: No class (Fall Break)
• Wednesday: We will tie up loose ends and review for Test #2.
Sam & Eddie are offering a review session at 3pm in Walker 308.
• Friday: Test #2
Test #2 is Friday (10/17).
Test #2 will cover
Sections 1.3-1.9 (geometric stuff, spanning, independence, linear transformations, & standard matrices),
Sections 2.8-2.9 (special subspaces, bases, dimension, rank, & coordinates),
Sections 4.1-4.6 (subspaces, special subspaces, linear transformations, independence, bases, coordinates,
dimension, change of basis).
Again, I will design this test to be done without technology, but calculators are allowed.
I recommend looking at your homework, quizzes, notes, and suggested homework when studying.
In addition, parts of old tests could be helpful.
You can find old tests here.
Note: For problems that ask you to use the subspace test, I will ask you to show this is a subspace by
recognizing it as a span or null space instead. Also, showing a set is linearly independent (like Fall 2012 #3b)
could/should be done using coordinates instead. Instead of showing a map is a linear transformation, I plan to
ask for a standard matrix.
The following problems are relevant:
Fall 2012:
Test #3 -- all of it. [keeping the above note in mind]
Test #4 -- problems 1-3.
Final Exam -- problems 5-9.
Spring 2011:
Test #3 -- all of it. [keeping the above note in mind]
Note to future self: Maybe add in a template for such a homework
...assignment like I did for Homework #1.
09/30
Reminder:
• Quiz #4 is Wednesday. We have been studying spans, linear
independence, bases, dimension, & column/row/null spaces.
You should know definitions and how to tell if a set of vectors
is linearly independent or not and if it spans or not. Know how
to find bases for column, row, & null spaces of a matrix
along with their dimensions. If you look at Test #3's from
Fall 2012 and Spring 2011 problems 2, 3, and 4ab are good
ones to consider.
Where are we in the textbook?
Sorry we've been jumping around. Here is an attempt to reconcile the text's order with what we've been discussing in class:
4.1 introduced abstract vector spaces and subspaces
1.8, 1.9, & 4.2 discuss linear transformations
(the rank nullity theorem is in 4.5)
4.6 looks at changing bases
09/19
Reminder:
• Quiz #3 is Monday. We have been looking at section 4.1 plus some
things from 1.3-1.5. In particular, be ready to sketch the results of
adding and scaling (= taking linear combinations of) vectors. Expect a
question asking you to explain why a particular set is not a subspace.
Also, be ready to show particular sets are subspaces by recognizing
them as the span of a collection of vectors (as done in class today).
Also, know how to tell if a vector belongs to a span or not.
Test #1 was passed back today. The test and its answer key are posted on our
page of old exams.
Upcoming:
• Quiz #2 is Monday. It will cover sections 2.1-2.3. You should know how
to add, scale, multiply, and transpose matrices. Also, you should
know how to find out if a matrix is invertible and if so, find its inverse.
Finally, you should know the 2x2 inverse formula.
• I plan to review for Test #1 during Wednesday's class. In addition,
Sam and Eddie will be offering a review session on Wednesday (9/10)
at 3pm in Walker 210. Feel free to come for however long you can.
This test is to be done by hand (no technology). I recommend looking at your quizzes, notes, and
suggested homework when studying. In addition, I have several old tests posted.
Fall 2012 Test #1 plus Fall 2012 Test #2 problem 1 make up a decent guide for what to expect
on our first test.
You can find old tests here. The following problems are relevant:
Homework #1 [Source: (.tex)]
is due Friday, September 5th.
→ Homework #1 Template (.mw) for working in Maple.
[Note: I fixed a typo in the template file.
Initially problem #2's parts were mislabeled.]
08/25
Quiz #1 is Wednesday! It will cover our first three handouts. In particular, you
should...
• Know how to translate from a linear system to a matrix.
• Know how to perform Gauss-Jordan elimination.
• Know what is meant by...
coefficient/augmented matrix, REF/RREF, & basic/free variables.
• Know how to translate back a matrix to a system.
• Know how to write a general solution in (vector) parametric form.
• Know how the linear correspondence between columns works.
Where are we in the textbook?
• Sections 1.1 & 1.2 correspond to our RREF handout.
• Section 1.3 - We haven't fully discussed this yet.
• Sections 1.4 & 1.5 correspond to our Vector Form handout.
• Section 2.1 - We have just started discussing matrix operations.
08/18
Today we went over syllabus nonsense and "What is Linear Algebra?" Where are we in the textbook?
More-or-less section 1.1.
Mistakes were Made:
I realized during the noon section that I had made a mistake when rushing through my example
at the end of the 9am section. The system's translation, RREF, and solution were fine...but when I added
-7 times row 1 to row 3, I messed that up. Basically I ended up skipping a step or two. I plan to go
over this example again at the beginning of both classes on Wednesday.
No (in person) class Friday!
As I mentioned in class, I will be helping my son move in on Friday. In lieu of in person class, I will
send out a link to a video. Also, as I mentioned, if you would like to come when I record Friday's
video, you are welcome to. I will be recording Friday's class on Thursday at 8am in Walker 304
(our usual classroom).
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 provided on our coursepage in AsULearn).
I have a video introduction, but it's taylored for Calculus 3: Maple Intro. (38mins).
My Maple examples (mostly for calculus) can be found here.
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
?was? 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.
08/11
From Genie Griffin (griffine@appstate.edu) our tutoring coordinator:
Hi All,
Hope everyone had a great summer!
I wanted to let you all know about the free tutoring offered by the University to our students for this fall semester. Please share with your students!!
Drop-in Tutoring: In person drop in tutoring for general math, business math, and stats will be offered Monday-Thursday from 4:30-7:30 pm in DD Dougherty room 208.
*Also note--Drop-in tutoring will continue to be in-person in the evenings. However, this fall semester, tutoring services are piloting a new online 24/7 asynchronous tutoring option with a small committee of tutors. More information will be sent out about this at a later date!
Appointment Based Tutoring: Many of our math/stats and business math tutors are opening their availability for tutoring appointments between the hours of 9am-4pm Monday through Thursday and 9am-2pm on Fridays. These appointments can be for online or in-person tutoring. Because of drop-in hours in the evenings Monday through Thursday, any appointments that start at 4pm or later will be online only. They can find more information here: https://studentlearningcenter.appstate.edu/tutoring.
Weeknight football:We have a home football game on Thursday, Nov. 7. Tutoring is usually limited on these days. I will update you all as we get closer to that date if there are changes in operations.
Dates Tutoring will not be offered: September 1 (Labor Day), October 13-14 (Fall Break), and November 26-28 (Thanksgiving Break).
New, Starting this Semester! Students are now able to use the Student Learning Center (located in DD Dougherty Room 208) as a study space between the hours of 9am-4pm Monday through Thursday and 9am-2pm on Fridays. They do not need an appointment for this. However, if they find that they need assistance while working in that space, a Front Desk Assistant will be available to help them schedule tutoring appointments or inform them about drop-in hours and other resources.
Tutoring will start on Monday, Aug. 25 and end on Wednesday, Dec. 3.