MAT-345, *Applied Linear Algebra 1*, <br>Fall '22, Dr. Richard Ketchersid</br>
==
###### tags: `345`

# Resources useful throughout the course
* The Math Center (16-202) offers help with MAT 345. The following hours have people available to help with MAT345.

|Person |Days |Time |
|-|-|-|
|Ayak|M|11:00 - 3:00|
|Clayton|TR|1:00 - 3:00|
|Jeff|TR|3:00 - 5:00|
|Segla|MW|1:00 - 3:00|

* In addition, I can meet with you in person after class or after my MAT264 class at 12:45, 16-402 (next door). I am also available on ZOOM, you just need to schedule with me by [email](mailto:[e-mail]) or [Remind](https://www.remind.com/join/m345090622).
* Here are some [fantastic 3Blue1Brown videos](https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab) on linear algebra.
* Dr. Gil Strang: (Linear Algebra guru :male-teacher:)
* [Here is a sequence of videos](https://ocw.mit.edu/courses/res-18-010-a-2020-vision-of-linear-algebra-spring-2020/video_galleries/videos/) that overviews linear algebra in 70 minutes. This is intended for teachers of linear algebra, but is great as an overview.
* [18.06 MIT from 2020 on OCW]((https://web.mit.edu/18.06/www/videos.shtml)) is a very popular full set of lectures.
* [Here is a podcast](https://youtu.be/lEZPfmGCEk0) interview with Dr. Strang discussing the relevance of linear algebra.

# Class Administration

## Time and location
Our class meets Wednesday and Friday 3:00 PM - 4:45 PM in building 16, room 404.

**Communication**: As an adjunct there are no fixed office hours. I am happy to talk after and if arranged before class. [Email is good](mailto:[e-mail]) and [Remind](https://www.remind.com/join/m345090622) is a great option for quick messages and questions.

## Participation

I will use [PollEverywhere](https://pollev.com/rketcher) to ask questions during class and use the responses for attendance, so please always respond to the polls. Participation is 10 points per week for a total of 150/1000 points, is 15% of the class grade.

## Homework

Homework, i.e., practice, is a critical component to learning math. The homework in this class is all done by hand, there is no machine graded work in MAT-345. Due to this, and the finiteness of time, here is what we will happen:
* Each homework ends on a Friday, often there is a quiz on that same material on that same Friday.
* I will post solutions to the homework by Wednesday. You should have attempted to complete the work by then.
* You will **compare your homework to my solutions** and write down any questions that you have to ask in class.
* **Make corrections** to your work and **submit** it to Halo ([as described below](#Homework-submission-requirements)) a copy of your corrected work to Halo by the end of the day on the Friday that it is due.

The ten homework assignments count for 100/1000 points, which is 10% of the class grade. I will, more-or-less, assume that you have done as asked, completed the work, compared the work with the solutions, made corrections, and submitted the work as indicated. If this is all true, then you should receive full credit for this work. This is not an "effort" grade. I expect that you have learned the material and have done a good job at these assignments.

:::warning
:warning: I will not accept homework that is not submitted in the format described below.
:::

### Homework submission requirements
I want each homework submitted via Halo as a single multi-page file. Here are three options:
1. Create a multi-page PDF which is the standard for most scanning apps.
2. You may format your work electronically. This requires some time and effort in Word, or better yet, in LaTeX.
3. Word Docs with embedded images. If you embed images in Word, then read [this](https://ketchers.github.io/Teaching/345/345Example.pdf) concerning how I want this done.

If you need to resize your pages to a standard size you can use [this](https://www.pdf2go.com/resize-pdf) site. The default is "A4" size, which is fine. If you need to compress the pdf use [this](https://www.pdf2go.com/compress-pdf) site and choose "small file size medium quality."

## Quizzes

Quizzes will reflect what you have done in the homework. You will have 30 - 45 minutes at the end of class to complete the quiz. The quiz is taken on the same day that the homework is due, so as mentioned above, I expect that by this time you have completed, reviewed, and corrected your homework. The four quizzes count for 160/1000 points in this class, which is 16% of the grade.

## Exams
Exams are in class. The entire class period will be provided for the exam. Exams count for 580/1000 points in this class, which is 58% of the class grade. There will not be quizzes on the weeks that we have exams. To do well in the class, it is important that you take these exams seriously. They will generally have at least three sections:
1. True/False - with explanations.
2. Short answer - Statements of theorems and/or definitions
3. Computational problems.
4. Theory - proofs or explanations.

## Benchmark Assessment

There is one *benchmark assessment*, this will actually be absorbed as a couple of problems on the second exam so you will not notice it, but it does count as 10/1000 points, which is 1% of the class grade.

## Summary of class grade breakdown

|Participation|Homework|Quizzes|Exams|Benchmark|Total|
|:-:|:-:|:-:|:-:|:-:|---|
15%|10%|16%|58%|1%|100%|

## [Extra Credit](https://hackmd.io/LIwcMrL2S2yZaxNzrHtjzg?view)

[Here is a list](https://hackmd.io/LIwcMrL2S2yZaxNzrHtjzg?view) of bonus questions, correct responses to these are worth five points each and all bonus points will be added at the end of class.

# Notes on Topics
:::spoiler Expand to see the list of topics

In order to make the page load faster, I will split off the topic specific notes into separate pages.

## [Topic 1: Systems of Equations, Gaussian Elimination](/LiT8wjLfTvOmk7IHbMkKjA)

## [Topic 2: Determinants and Elementary Matrices](/XQrf-fmJQS2g77_jqKs34Q)

## [Topic 3: Vector Spaces, Independence, and Basis](/Vtgu8UEESUmPyhuDUwBhDw)

## [Topic 4: Linear Transformations](/EopmNhKhSEWhx6tULXF-CQ)

## [Topic 5: Inner Product and Orthogonality](/d_w9BrsjSO2aQKzi826BSg)

## [Topic 6: Eigenvectors, Eigenvalues, and Diagonalization](/cNMbONbtSYOHOIMjs0C9ug)

## [Topic 7: Hermitian Matrices, Spectral Decomposition, and SVD](/W070QXZ9T6KGhTCrXyxZMA)
:::

[TBA]:
/DCvTLDbxRBKXK4R5QWsEtQ?view "TBA"

# Exam information and results

## Exam 1

:::spoiler **Exam 1 Information and Results**

Your first exam falling on Wednesday 9/28. I am posting here a couple of old quizzes covering the material from Topic 2 for practice.
* [Quiz on topic 2 (version 1)](https://ketchers.github.io/Teaching/345/Topic%202/Quiz_2_121921_soln.pdf) from a past class.
* [Quiz on topic 2 (version 2)](https://ketchers.github.io/Teaching/345/Topic%202/Quiz_2_051021_soln.pdf) from a past class.

Of course, study my solutions, compare them to your own. Make sure you are doing what I expect to see on these. It is not the "answer" I am most interested in. I am most interested in you showing me the process and your understanding of the material.

**The Exam will have:**
* Seven T/F questions (50 points; 7 points each). You will not be asked to supply justification.
* Four computational problems (80 points; 20 points each). Example of such might include, but are not restricted to:
* Using row operations to solve a system along with all that follows from this technique. Quiz 1 had a good example of this.
* Algebraic manipulation of vectors and matrices, including viewing matrix multiplication row-wise, column-wise, and element-wise. Rules for inverse and transpose for products.
* Calculating determinants, the definition of $A^{-1}$ interms of the adjoint, and Cramer's Rule.
* Elementary matrices. Their use in row operations and computing determinants as well as results derived from their use.
* LU decomposition. We also briefly discussed something called CR decomposition, but I won't hold you to recall this. You should know the interpretation of L and the very simple way to compute it by keeping track of the row ops you use in an elimination.
* Three theoretical problems or proofs. (60 points; 20 points each). I will provide four options and you choose the three you want to do.
* Look at the T/F questions on the quizzes. Justifying a T/F question is a good example of what might be asked here.
* Understanding the theoretical application s of the Gauss-Jordan elimination. For example, show that an $n\times n$ matrix is invertible iff $A\mathbf{x}=\mathbf{0}$ has exactly one solution.
* When you prove an "if and only if" ("iff", "$\iff$") statement, you must prove both directions. (See example just below.)
* When you prove a thing like $(AB)^T=B^TA^T$ you must prove this generally for arbitrary matrices $A$ and $B$ there the products make sense.
* When you say that something like $AB=BA$ is not always true, then you must provide an example.

**Example "iff" proof:** Prove that for any square matrix $A$:
$$
\text{ There is }\mathbf x\neq \mathbf 0\text{ such that }A\mathbf x=\mathbf 0 \iff \det(A)=0.
$$

**Proof** **($\implies$)** ("only-if" direction) Suppose $A\mathbf x=\mathbf 0$ for some $\mathbf x \neq \mathbf 0$. Let $R=\text{rref}(A)$, since $\mathbf x\neq \mathbf 0$ we know $R\neq I$ and hence $\det(R)=0$, but we know $\det(A)=c\cdot \det(R)$ for some $c\neq 0$, so $\det(A)=0$.

**($\Longleftarrow$)** ("if" direction) Suppose $\det(A)=0$ and $R=\text{rref}(A)$. As above $\det(R)=0$ and so there is at least on free variable in the solution of $A\mathbf x=\mathbf 0$ and hence there is $\mathbf x\neq\mathbf 0$ so that $A\mathbf x\neq\mathbf 0$.

### Solutions and corrections for Exam 1
Here are the [solutions for Exam 1](https://ketchers.github.io/Teaching/345/Exam_1_092822_soln.pdf), minus Part III. I would like those who wish to earn back 50% of the points missed on Part III.

The **theory** in this course is important and you should know how to reason, express your reasoning, provide proofs, and explanations. This is a hard skill and one you need to practice. I would like to encourage you to do so here. You may complete all five problems for part III for 75% of your missed points, or 20 points, whichever is largest. Part III was worth 60 points, so those of you availing yourselves of this opportunity are guaranteed between 20-45 additional points for Exam 1 - assuming the work is correct.

The best way to learn how to **do** proofs is to read and **understand** proofs. I provide many proofs in my lectures and notes and, of course, the book is filled with them. I expect that you are reading and understanding these along the way. If not, you should be asking questions. Refer to the notes and to the book for examples of how to argue while making these corrections.

[Here is a blank copy of Exam 1](https://ketchers.github.io/Teaching/345/Exam_1_092822.pdf) for you to use for this purpose. Corrections are due by Oct 9 and I will accept these in class (on paper) or submitted via Halo messages. Same rules as the homework, i.e., single multipage PDF, correctly scaled, etc.

**Exam 1 Distribution**:

:::

## Exam 2

:::spoiler **Exam 2 Information and Results**

This exam is primarily over Topic 3 and Topic 4, of course, these involve material from Topic 1 and Topic 2. The structure will be:
* (In Class 11/2) 60 points on T/F with part of the value based on explanations (This will be the "in-class" theory)
* Look at all T/F in the text at the end of the chapters.
* Look at all T/F (and explanations) in the quizzes and make-up quizzes as well as old exams.
* Know definitions and theorems: (Vector space, subspace, spanning set, linearly independent, basis, similar matrices, linear transformation, row-space, null-space, column-space, kernel, image, etc.)
* (In Class 11/2) 100 points (5 x 20 points) for computational problems
* Find basis, spanning sets, linearly independent subsets.
* Find representations of vectors and linear transformations from given basis.
* Find and use change of basis matrices. (similarity)
* (Take-Home due 11/4 in class if handing in in-person, or submitted PDF by email by midnight Friday 11/4) 30 points (3 x 10 points) Theory/Proof problems (I will entertain clarifications in class Friday, but not provide answers. We will be moning into Topic 5 Friday.)

I am making the "Theory" part worth just 15% (approximately). This way the theory part might determine the difference between a B or an A, but it will harpoon your grade if you are struggling with this.

You may use:
* One (regular sized) piece of paper worth of notes, front and back.
* You may lisrten to music using a headset as long as this is not disturbing anyone around you.
* You may use MATLAB, Python, other online matrix calculators (just get my approval), any handheld calculator is fine. Here are a couple students have asked for that are fine:
* [Matrix reshish](https://matrix.reshish.com/multiplication.php)
* [Desmos Matrix Calculator](https://www.desmos.com/matrix)

It is still your job to let me know exactly what you are doing. don't just answer a question like "What is the matrix for $L$ wrt the standard basis?" with a single matrix, show the in-between steps, like $[L]=B[L]_{\cal B}B^{-1}$ and give all the intermediate matrices. Use the matrix calculator, just to actually do the multiplication and find the inverse.

[**Take Home Portion** (Due 11/5@Midnight)](https://ketchers.github.io/Teaching/345/Exam_2_Ground_100222_(take_home).pdf) Use [this form](https://forms.gle/6PqooBj9Q4v21NxB6) to submit your work, or send an <a href="mailto:[e-mail]?subject=Exam 2 Take Home Portion">email</a> with subject "Exam 2 Take Home Portion"

[Exam 2 Take Home Portion solutions](https://ketchers.github.io/Teaching/345/Exam_2_Ground_100222_take_home_solutions.pdf)

**Second Chance** You may ***retake*** Exam 2 in the comfort of your own home (dorm?) for the opportunity of earning back ~~50%~~ **70%** of the point lost (per problem). I will not set a due date. This is an all or nothing on each problem. Either you provide a **perfect** solution and get the full ~~50%~~ **70%** or an imperfect solution and get nothing. I want to see the you understand the material!

**Important** When you correct the T/F you must tell me what your score was on the original. Give me a table like:
|a|b|c|d|e|f|g|h|i|j|
|-|-|-|-|-|-|-|-|-|-|
|3|4|2|2|0|3|6|6|4|2|

[**Copy of in-class portion of the exam**](https://ketchers.github.io/Teaching/345/Exam_2_Ground_100222_(take_home)(solutions).pdf)

**Perfect** = correct + complete with all details + clearly written

**Warning** Scratchwork = imperfect = 0% credit

**Grade distribution for Exam 2**

:::info
[**Exam 2 Solutions**](https://ketchers.github.io/Teaching/345/Exam_2_Ground__100222.pdf)
:::

## Exam 3

:::spoiler **Exam 3 Information and Results**

As discussed in class. Exam 3 will be sort of like Quiz 4. There is no take-home portion. There will be T/F section in which you do **not** need to justify your answers and a computational section. I will post the solutions to the computational problems after the exam so that you have some info on how you did and I will post the T/F answers. For ***make-up*** on the T/F part you may provide full reasons for each T/F question. This must be done by 12/20.

The material is based on Ch 5 and 6, but this is by nature a cumulative course, so all previous material is fair game. That said, the particular computations will be from Ch 5 and 6.

**Here is the distribution of Exam 3 grades**:

:::info
[**Exam 3 Solutions**](https://ketchers.github.io/Teaching/345/Exam_3_Ground__121322.pdf)
:::

# Midterm Distribution
In case you are interested, here is the distribution of midterm grades:

# Final Grade distribution
In case you are interested, here is the distribution of midterm grades.