Linear Algebra

Professor: Erika L.C. King

Email: eking@hws.edu

Office: Lansing 304

Phone: (315) 781-3355

Home Page

Office Hours: M: 10:30am-Noon, W: 4:00-5:15pm, Th: 11:15am-12:45pm, F: 11:00am-Noon, and by appointment

Class Schedule: held MWF 1:55-2:50pm in Stern 303

Course Syllabus

Proof Writing and Presentation Tips

Course Grade Scale

WeBWorK Home Page for Our Class

WeBWorK Instructions and FAQs

WeBWorK Syntax and List of Functions

**BONUS WeBWorK due Sunday, December 10 at 8:00pm:**

- This is optional! Gain points toward your WeBWorK grade!
- Practice material on Eigenvalues and Eigenvectors with the WeBWorK assignment here. This is good practice for the final exam. Be sure to read the directions and hints to help you understand what is being asked. Ask me if you have questions. This is due SUNDAY at 8:00pm!

**Reading and Practice Problems:**

- You should have a copy of the Final Exam Preparation sheet. If you have lost yours here is another copy.
- Reread Section 5.3 (pages 283-288). Do you feel confident with this material now?
- Complete the Practice Problem for Section 5.3 on page 288 of the text. This will not be collected, and you should check your answers on page 290.
- Complete these practice problems from Section 5.3 (pages 288-289): 3, 5, 11, 17, 21, 23, 25, 27 and 29. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.

**Review Session:** We will have a review session on Monday, December 11th from 11:00am until Noon in Napier 102. Attendance is optional and you are welcome to come for
any portion of the review session. Bring questions.

**Office Hours:** I will hold office hours Monday, December 11th from 4:00pm until 5:00pm. If you have conflicts at that time and still wish to see me, please
make an appointment.

**Final Exam:** The final exam is on Tuesday, December 12th from 8:30am until 11:30am in Stern 303.

**THANK YOU FOR A GREAT CLASS!!! HAVE A GREAT WINTER BREAK!!! KEEP IN TOUCH!!!**

**Quiz 9 will take place on Monday at the beginning of class. It will cover Sections 4.5 and 4.6. Be prepared to state definitions and theorems,
give examples with explanations, and complete short exercises.**

**There will be time in class on Wednesday to complete course evaluations. If you have a laptop or other device on which it is easy to type,
it would be useful to bring it on Wednesday for this purpose. I will also have a few devices.**

**Looking Ahead: I have reserved a room (Napier 102) so that we can have a review session for the final on Monday, December 11 at 11:00am. This is not a
required meeting, and you are welcome to come and go as you need. I hope you all find it a good opportunity to review. I will also have office hours in
the afternoon that day.**

**Homework for class Monday, December 4:**

- Review your class notes from Friday's class. Please let me know if you have any questions on the material we covered from Section 4.6 and from Section 5.1.
- Complete the Practice Problems for Section 4.6 on page 238 of the text. This will not be collected, and you should check your answers on page 240.
- Complete the group worksheet from Friday. Be prepared to discuss this in class. Use Section 5.1 in your text to guide you for question 2.
- Reread Section 5.1 (pages 268-273) in your text. Keep track of what you understand and make of list of any questions you have.
- Read Section 5.2 (pages 276-281) in your text. As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions, theorems and facts. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet on the second half of Section 5.1 and Section 5.2. Bring your work to class to discuss and present!

**BONUS ASSIGNMENT due Wednesday, December 6 at 1:55pm:**

- No collaboration allowed. You may not discuss these bonus submissions with anyone but Professor King. You are highly encouraged to discuss them with Professor King!
- No outside resources allowed. You may only use your notes and the textbook.
- You may NOT submit a solution for a question for which you received full credit on the exam.
- If you submit the BONUS assignment (which I hope you do), one of the problems you submit must be the one from this list for which you missed the most points: Problem 4(b), Problem 7(a), Problem 8.
- You may submit solutions for up to THREE of the following problems (the total possible points you can earn for complete correct solutions of these questions are listed in brackets after each question): Problem 4(b) [3], Problem 7(a) [4], Problem 8 [5], Bonus 2 [2].
- You must staple your resubmission to your original exam before turning it in.
- Since this is for bonus points, I will be looking for much more detail than I did on the exam where you had little time for reflection. Don't assume that if you missed only one point on a question that it will be easy to get full credit on the bonus assignment for it. In fact, you should expect it to be harder than getting points on other questions where you missed more points! You are highly encouraged to work on those things which you found the most challenging while taking the exam.
- Be sure to read my original comments and respond to them, follow directions, and include thorough proofs!

**Homework for class Wednesday, December 6:**

- Remember to bring a laptop or other device on which it is easy to type to class on Wednesday to do course evaluations.
- Review your class notes from Monday's class. In particular, review the proof of Theorem 5.2 that we outlined in class and make sure you believe it and can follow all the details. Gather any questions you have about Section 5.1, and be prepared to ask them at the beginning of Wednesday's class!
- Reread Section 5.1 (pages 268-273) in your text. Do you feel confident with this material now?
- Complete the Practice Problems for Section 5.1 on page 273 of the text. This will not be collected, and you should check your answers on pages 275-76.
- Complete these practice problems from Section 5.1 (page 273): 5, 9, 11, 13, 15, 19, 21 and 23. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Reread Section 5.2 (pages 276-281) in your text. Keep track of what you understand and make of list of any questions you have.
- Make sure you have completed this worksheet on the second half of Section 5.1 and Section 5.2. Bring your work to class to discuss and present!
- Read Section 5.3 (pages 283-288) in your text. As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions, theorems and facts. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet on Sections 5.2 and 5.3. Bring your work to class to discuss and present!
- OPTIONAL (Also listed below): Practice material on Eigenvalues and Eigenvectors with the WeBWorK assignment here. This is good practice for the final exam. Be sure to read the directions and hints to help you understand what is being asked. Ask me if you have questions. This is due SUNDAY at 8:00pm!

**Collected Homework (Due Friday, December 8 at 1:55pm):**

**Note that these assignments are due right at the beginning of class! You must be on time and be ready to turn in your work - already stapled, etc. If you turn it in after 1:55 it is late and you will have to use your free late or lose points.**- Remember that although you may discuss this assignment with others, your write up should be your own.
**Do not share your write-up, look at other's write-ups, discuss word for word how something should be proved, etc.**Be sure to note with whom you collaborate if you do. - Complete the following problems:
- Complete the problems on this worksheet.

- This assignment has been finalized!

**Homework for class Friday, December 8:**

- Review your class notes from Wednesday's class! Come to office hours!
- Make sure you have completed this worksheet on the second half of Section 5.2 and Section 5.3. In particular, make sure you have worked through Section 5.3 Exercise 7 (question 7 on the handout). This will be a good warm-up for the one on your collected homework! Bring your work to class to discuss and present!
- Reread Section 5.2 (pages 276-281). Do you feel confident with this material now?
- Complete the Practice Problem for Section 5.2 on page 281 of the text. This will not be collected, and you should check your answers on page 283.
- Complete these practice problems from Section 5.2 (page 282): 9, 11, 13, 15, 17 and 21. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Reread Section 5.3 (pages 283-288) in your text. Keep track of what you understand and make of list of any questions you have.
- While rereading Section 5.3, make a note of Theorem 7 and try Exercise 19 on page 289, comparing it to Example 6. Doing this is essentially your reading-prep worksheet, but there are only two things! Bring your work to class to discuss and present!
- Idea for reviewing for the final: Choose a partner. Each of you make up a practice exam using problems from the "Practice Problems" at the end of each section or odd numbered exercises in the text. Include 6-10 problems. Mix up the problems so that they don't necessarily appear in the same order as they are in the text. Exchange exams. Write up the solutions to each other's exams in a pseudo-test-taking atmosphere. Trade papers again and "grade" your partner's work! This might help you figure out which sections you need to focus on for studying!
- OPTIONAL (Also listed below): Practice material on Eigenvalues and Eigenvectors with the WeBWorK assignment here. This is good practice for the final exam. Be sure to read the directions and hints to help you understand what is being asked. Ask me if you have questions. This is due SUNDAY at 8:00pm!

**Have a great Thanksgiving break!**

**Since we just had an exam and then break, we will NOT have a quiz this week!**

**Homework for class Monday, November 27:**

- Read Section 4.5 (pages 227-230) in your text (this is really just two pages!). The ideas of ${\mathbb R}^n$ as a vector space give us some intuition into this idea of dimension! As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions, theorems and facts. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- If you haven't already, complete this worksheet on the Section 4.5 reading that was due Friday, November 17th. Bring your work to class to discuss and present!
- Then complete this worksheet on the second half of Section 4.5. Bring your work to class to discuss and present!
- Complete the Practice Problems for Section 4.5 on page 230 of the text. This will not be collected, and you should check your answers on page 232.

**BONUS ASSIGNMENT due Wednesday, November 29 at 1:55pm:**

- No collaboration allowed. You may not discuss these rewrites with anyone but Professor King. You are highly encouraged to discuss them with Professor King!
- No outside resources allowed. You may only use your notes and the textbook.
- You must staple your resubmission to your original assignment before turning it in.
- Since this is for bonus points, I will be looking for even more detail than on a regular assignment.
- You can earn as many points as you lost on your original submission. Be sure to read my original comments and address them all!

**Homework for class Wednesday, November 29:**

- Review your class notes from Monday's class. Gather any questions you have about Section 4.5, and be prepared to ask them at the beginning of Wednesday's class!
- Complete these practice problems from Section 4.5 (page 231): 5, 9, 11, 15, 19, 21, 23 and 25. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Read Section 4.6 (pages 232-238) in your text. As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions, theorems and facts. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet on Section 4.6. Bring your work to class to discuss and present!

**Collected Homework (Due Friday, December 1 at 1:55pm):**

**Note that these assignments are due right at the beginning of class! You must be on time and be ready to turn in your work - already stapled, etc. If you turn it in after 1:55 it is late and you will have to use your free late or lose points.**- Remember that although you may discuss this assignment with others, your write up should be your own.
**Do not share your write-up, look at other's write-ups, discuss word for word how something should be proved, etc.**Be sure to note with whom you collaborate if you do. - Complete the following problems:
- Complete the problems on this worksheet.

- This assignment has been finalized!

**Homework for class Friday, December 1:**

- Review your class notes from Wednesday's class! Make a list of your questions, especially on Section 4.6. Come to office hours!
- Finish working on the group worksheets we were working on at the beginning and end of Wednesday's class. We will begin class by discussing any questions there are on the latter one especially. Be prepared to participate in discussion. The class did a great job in Wednesday's class with the reading prep for Wednesday!
- Reread Section 4.6 (pages 232-238). Do you feel confident with this material now?
- Complete these practice problems from Section 4.6 (page 231): 3, 5, 7, 9, 11, 13, 15 and 17. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Read the motivation for Chapter 5 on pages 267-268 -- a real world owl application!
- Read Section 5.1 (pages 268-273) in your text. As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions, theorems and facts. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet on Section 4.6 and 5.1. Bring your work to class to discuss and present!

**WeBWorK due Saturday, November 18 at 1:00pm:**

- I think this is a good check as you study for your exam, but instead of making it due Wednesday or Friday, I extended it to Saturday so that you can choose when you want to focus on this material.
- Practice material on Bases, Null Spaces and Column Spaces with the WeBWorK assignment here. This is good practice for the exam on Monday. Be sure to read the directions and hints to help you understand what is being asked. Ask me if you have questions. This is due SATURDAY at 1:00pm!

**Homework for class Monday, November 20:**

- Prepare for Exam 3!
- You should have a copy of the Exam 3 Preparation sheet. If you have lost yours here is another copy.
- Be sure to do extra practice problems with any material for which you feel less comfortable. Make sure you are confident when you come to class on Monday!
- Feel free to email me with questions any time over the weekend!!! Remember that I also have office hours on Monday morning.
- Here are some more practice problems from Section 4.4 (pages 224-225): 3, 7, 9, 13, 15, 19, 21, 23, 27 and 29. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- REMEMBER TO BE AT THE EXAM IN STERN 303 AT 1:25!!!

**Quiz 8 will take place on Monday at the beginning of class. It will cover Sections 4.2 and 4.3. Be prepared to state definitions and theorems,
give examples with explanations, and complete short exercises.**

**Homework for class Monday, November 13:**

- Review your class notes from Friday's class. Please let me know if you have any questions on the material we covered from Section 4.3.
- Each group was assigned a problem from the group worksheet on Friday. Work through the details of your group's problem. You are encouraged to discuss your work with me in office hours or via email before class. Be ready to put the finishing touches on it to present directly following the quiz. Do not assume you will have time to figure it out in class! Ideally, write your solution neatly on a piece of paper that can be projected with the document camera. Putting your solution on the board after the quiz is also a possibility.
- Complete the other parts of the group worksheet from Friday.
- Is the linear transformation on the group worksheet from Friday one-to-one? Why or why not? Be ready to discuss or ask questions related to this!
- Reread Section 4.3 in your text (pages 210-215). Where do you still have questions and where have questions been cleared up for you?
- Complete the Practice Problems for Section 4.3 on page 215 of the text. This will not be collected, and you should check your answers on pages 209-210.
- Read pages 218-221 of Section 4.4 in your text. As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet. Bring your work to class to discuss and present!

**Remember there is a colloquium on Tuesday, November 14th at 4:45 in Napier 101 about mathematical phylogenetics and the Mathematical
Phylogenetics Scholars Program! You can earn up to five bonus points towards your homework grade for attending!**

**Homework for class Wednesday, November 15:**

- Practice material on Bases, Null Spaces and Column Spaces with the WeBWorK assignment here. Be VERY careful! The first two only allow TWO submissions! Be sure to read the directions and hints to help you understand what is being asked. Ask me if you have questions. This is due Wednesday at 1:00pm.
- Review your class notes from Monday's class. Gather any questions you have about the groupwork we finished up and Section 4.3, and be prepared to ask them at the beginning of Wednesday's class!
- Read Section 4.4 in your text (pages 218-224). As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- If you haven't finished the reading prep sheet that was assigned for Monday, do it now! Bring your work to class to discuss and present!
- Complete this worksheet to cover the last few pages of Section 4.4. Bring your work to class to discuss and present!
- Complete these practice problems from Section 4.3 (pages 215-217): 5, 7, 9, 11, 15, 19, 21, 23 and 29. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Practice material on Bases, Null Spaces and Column Spaces with the WeBWorK assignment here. This is good practice for the exam on Monday. Be sure to read the directions and hints to help you understand what is being asked. Ask me if you have questions. This is due SATURDAY at 1:00pm, but I highly recommend that you complete it sooner!

**Collected Homework (Due Friday, November 17 at 1:55pm):**

**Note that these assignments are due right at the beginning of class! You must be on time and be ready to turn in your work - already stapled, etc. If you turn it in after 1:55 it is late and you will have to use your free late or lose points.**- Remember that although you may discuss this assignment with others, your write up should be your own.
**Do not share your write-up, look at other's write-ups, discuss word for word how something should be proved, etc.**Be sure to note with whom you collaborate if you do. - Complete the following problems:
- Complete the problems on this worksheet.

- This assignment has been finalized!

**Homework for class Friday, November 17:**

- Review your class notes from Wednesday's class! Make a list of your questions. Come to office hours!
- Email me any questions you have on any of the material in Sections 3.2 and 4.1-4.4 by noon on Friday! This includes any questions you have on past due WeBWorKs. (I am happy to address questions on the WeBWorK due Saturday individually.) Even if you don't have time to email me, bring the questions to class. We will be able to address more questions if I know what they are ahead of time. You are also highly encouraged to ask questions in office hours instead of waiting until Friday's class!!!
- Reread Section 4.4 in your text (pages 218-224). Neat stuff!
- Complete the Practice Problems for Section 4.4 on page 224 of the text. This will not be collected, and you should check your answers on pages 226-227.
- If you haven't completed the reading preparation for Wednesday, do so now (this worksheet). Bring any questions you have on this to class.
- Read Section 4.5 from the beginning on page 227 through the top of page 229 in your text (this is really just two pages!). The ideas of ${\mathbb R}^n$ as a vector space give us some intuition into this idea of dimension! As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions, theorems and facts. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet. Bring your work to class to discuss and present!
- This is listed below as well since it is not actually due on Friday: Practice material on Bases, Null Spaces and Column Spaces with the WeBWorK assignment here. This is good practice for the exam on Monday. Be sure to read the directions and hints to help you understand what is being asked. Ask me if you have questions. This is due SATURDAY at 1:00pm!

**WeBWorK due Saturday, November 18 at 1:00pm:**

- I think this is a good check as you study for your exam, but instead of making it due Wednesday or Friday, I extended it to Saturday so that you can choose when you want to focus on this material.
- Practice material on Bases, Null Spaces and Column Spaces with the WeBWorK assignment here. This is good practice for the exam on Monday. Be sure to read the directions and hints to help you understand what is being asked. Ask me if you have questions. This is due SATURDAY at 1:00pm!

**Quiz 7 will take place on Monday at the beginning of class. It will cover Sections 4.1 and 4.2. Be prepared to state definitions and theorems,
give examples with explanations, and complete short exercises.**

**Homework for class Monday, November 6:**

- Practice material on Subspaces with the WeBWorK assignment here. Be VERY careful! The first three only allow ONE submission! Be sure to read the directions and hints to help you understand what is being asked. Ask me if you have questions. This is due Monday at 1:00pm.
- Work through the details of your group's problem you put on the board before we left class on Friday.
**Have one member of your group write it up very neatly on a piece of paper that we can project for all to see instead of rewriting it on the board.**Be ready to discuss the results with the class. - Complete the third part of the proof in the first problem on the group worksheet. That is, prove that the range of $T$ is close under scalar multiplication. Then do question 2 of the group worksheet. Be ready to discuss these on Monday!
- Reread Section 4.2 in your text (pages 200-207). Where do you still have questions and where have questions been cleared up for you?
- Complete the Practice Problems for Section 4.2 on page 207 of the text. This will not be collected, and you should check your answers on pages 209-210.
- Read pages 210-212 of Section 4.3 in your text. As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. For example, you should know the definitions of basis and standard basis. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!

**My meeting was moved, so my Wednesday office hours are the same as usual!!! Due to another commitment, however, I will likely need to change
part of my Thursday office hours.**

**Homework for class Wednesday, November 8:**

- Review your class notes from Monday's class. Please let me know if you have any questions on the material we covered from Section 4.2.
- Be sure you have completed question 2 from the worksheet for last Friday's groupwork. I will be calling on people to answer each part of the question - there are five vectors and two questions for each vector, so this means just about everyone will be called on! Be sure to have justification for your answers.
- Complete these practice problems from Section 4.2 (pages 208-209): 3, 5, 9, 11, 15, 17, 21, 23 and 25. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Read Section 4.3 in your text (pages 210-215). As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions and theorems. For example, you should know the definitions of basis and standard basis, and the statements of Theorems 4.4-4.6. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet. Bring your work to class to discuss and present!

**Due to a meeting, my Thursday office hours are slightly morphed. They will be 10:30-10:50am and 12:15pm-1:15pm.**

**BONUS ASSIGNMENT due Wednesday, November 8 at 1:55pm:**

- No collaboration allowed. You may not discuss these bonus submissions with anyone but Professor King. You are highly encouraged to discuss them with Professor King!
- No outside resources allowed. You may only use your notes and the textbook.
- You may NOT submit a solution for a question for which you received full credit on the exam.
- If you submit the BONUS assignment (which I hope you do), one of the problems you submit must be the one from this list for which you missed the most points: Problem 7(a), Problem 8, Problem 9, Problem 10.
- You may submit solutions for up to THREE of the following problems (the total possible points you can earn for complete correct solutions of these questions are listed in brackets after each question): Problem 7(a) [3], Problem 8 [3], Problem 9 [4], Problem 10 [4], Bonus 1 [2], Bonus 2 [2].
- You must staple your resubmission to your original exam before turning it in.
- Since this is for bonus points, I will be looking for much more detail than I did on the exam where you had little time for reflection. Don't assume that if you missed only one point on a question that it will be easy to get full credit on the bonus assignment for it. In fact, you should expect it to be harder than getting points on other questions where you missed more points! You are highly encouraged to work on those things which you found the most challenging while taking the exam.
- Be sure to read my original comments, follow directions, and include thorough proofs!

**Collected Homework (Due Friday, November 10 at 1:55pm):**

- Remember that although you may discuss this assignment with others, your write up should be your own.
- Complete the following problems:
- Complete the problems on this worksheet.

- This assignment has been finalized!

**Homework for class Friday, November 10:**

- Review your class notes from Wednesday's class. Please let me know if you have any questions on the material we covered from Section 4.2.
- Complete these practice problems from Section 4.2 (page 209): 31, 32 and 35. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- If you haven't completed the reading preparation for Wednesday, do so now (this worksheet). Bring any questions you have on this to class.
- At the end of class we claimed that in order to have a basis of ${\mathbb R}^3$ we needed to have three vectors. WHY? What would happen if we only had two? What if we had four? Use Theorems, Facts and/or definitions to help you precisely justify your answers. Remember that a basis has two properties. Be ready to share your results on Friday in class!
- Read/reread Section 4.3 in your text (pages 210-215). Add to your collection of definitions and theorems you created for Wednesday.
- Complete this worksheet. Bring your work to class to discuss and present!

**Quiz 6 will take place on Monday at the beginning of class. It will cover Section 4.1. Be prepared to state definitions and theorems,
give examples with explanations, and complete short exercises.**

**Homework for class Monday, October 30:**

- If you didn't finish working through the proof of why the set of all polynomials of degree less than or equal to $n$ is a vector space, do so here and collect any questions you have to bring to office hours or class.
- Review your class notes and group work solutions from Friday's class. If you did not finish the group worksheet in class, do so. Be ready to discuss problem 4 in class!
- Reread Section 4.1 in your text (pages 192-197). Where do you still have questions and where have questions been cleared up for you?
- Complete these practice problems from Section 4.1 (pages 198-199): 3, 5, 7, 9, 11 and 27. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Read pages 200-203 of Section 4.2 in your text. As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. For example, you should know the definitions of null space and column space. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet. Bring your work to class to discuss and present!

**BONUS ASSIGNMENT due Wednesday, November 1 at 1:55pm:**

- Choose from: Problem 1, Problem 4 part a, Problem 4 part b, Problem 4 part c, Problem 4 part d
- No collaboration allowed. You may not discuss these rewrites with anyone but Professor King. You are highly encouraged to discuss them with Professor King!
- No outside resources allowed. You may only use your notes and the textbook.
- You must staple your resubmission to your original assignment before turning it in.
- Since this is for bonus points, I will be looking for even more detail than on a regular assignment.
- You can earn as many points as you lost on your original submission. Be sure to read my original comments!

**Homework for class Wednesday, November 1:**

- Review your class notes from Monday's class. Please let me know if you have any questions on the material we covered from Section 4.1.
- Prepare a detailed proof of your group's axiom from exercise 1 of the group worksheet handed out at the end of Monday's class. As soon as you get to class on Wednesday, meet with your group to compare your proofs and finalize what you will put on the board soon after class begins.
- Give a proof for exercise 1(e) on the group worksheet handed out in Monday's class. Be prepared to share your results with your group and the class as a whole. I will be looking for a volunteer to share this!
- Problems 2-5 on the group worksheet are about subspaces. Outline solutions for these exercises.
- If you did not complete the worksheet due on Monday, do it now. We discussed exercises 1 and 2 in class. Be sure you have 3-6 finished. Bring your work to class to discuss and present! Note you should write down any questions that you have even though I didn't provide a specific space on the handout for this! This worksheet is mostly on the first 3-4 pages of Section 4.2.

**Collected Homework (Due Friday, November 3 at 1:55pm):**

- Remember that although you may discuss this assignment with others, your write up should be your own.
- Complete the following problems:
- Complete the problems on this worksheet.

- This assignment has been finalized!

**Homework for class Friday, November 3:**

- Carefully review all returned work. Read my comments and make sure you understand them and how that helps solve the exercise or explain a concept with which you were struggling.
- Review your class notes from Wednesday's class. Please let me know if you have any questions on the material we covered from Sections 4.1 and 4.2.
- If you haven't worked through problems 2-5 on the group worksheet handed out on Monday, do so now. These are great examples for understanding subspaces and preparing for future exams.
- Read/reread Section 4.2 in your text (pages 200-207). As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. For example, you should know the definitions of null space and column space. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet. Bring your work to class to discuss and present!

**WeBWorK due Sunday, October 22 at 7:30pm:**

- I think this is a good check as you study for your exam, but instead of making it due Friday, I extended it to Sunday so that you can choose when you want to focus on this material.
- Practice material on Determinants with the WeBWorK assignment here. This should be a good check of your understanding as you study for the exam! Most of these should be a quick! Be sure to read the directions and hints to help you understand what is being asked. This is due SUNDAY at 7:30pm.

**Homework for class Monday, October 23:**

- Prepare for Exam 2!
- You should have a copy of the Exam 2 Preparation sheet. If you have lost yours here is another copy.
- Feel free to email me with questions any time over the weekend!!! Remember that I also have office hours on Monday morning.
- REMEMBER TO BE AT THE EXAM IN STERN 303 AT 1:25!!!

**BONUS ASSIGNMENT due Wednesday, October 25 at 1:55pm:**

- Choose from: Problem 1, Problem 4 part 1, Problem 4 part 2, Problem 4 part 3, Problem 5
- No collaboration allowed. You may not discuss these rewrites with anyone but Professor King. You are highly encouraged to discuss them with Professor King!
- No outside resources allowed. You may only use your notes and the textbook.
- You must staple your resubmission to your original assignment before turning it in.
- Since this is for bonus points, I will be looking for even more detail than on a regular assignment.
- You can earn as many points as you lost on your original submission. Be sure to read my original comments!

**Homework for class Wednesday, October 25:**

- REVIEW: Complete these practice problems from Section 3.2 (pages 177-178): 21, 25, 27, 28, 35 and 39. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Read the motivation for Chapter 4 on pages 191-192.
- Read pages 192-193 of Section 4.1 in your text (JUST TWO PAGES!). As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. For example, you should know the definition of vector space. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this SHORT worksheet. Bring your work to class to discuss and present! Note there is a place on the worksheet to write down your questions!

**Collected Homework (Due Friday, October 27 at 1:55pm):**

- Remember that although you may discuss this assignment with others, your write up should be your own.
- Complete the following problems:
- Complete the problems on this worksheet.

- This assignment has been finalized!

**Homework for class Friday, October 27:**

- Review your class notes and group work solutions from Wednesday's class. If you did not finish the group worksheet in class, do so. Be ready to discuss problem 3
in class.
**I will be looking for a volunteer to present this solution...or calling on someone to present!** - Read/reread Section 4.1 in your text (pages 192-197). As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. For example, you should know the definition of vector space, subspace, etc. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet. Bring your work to class to discuss and present! Note there is a place on the worksheet to write down your questions!
- Complete the Practice Problems for Section 4.1 on page 197 of the text. This will not be collected, and you should check your answers on page 200.

**Quiz 5 will take place on Monday at the beginning of class. It will cover Sections 2.1, 2.2 and 2.3. Be prepared to state definitions and theorems,
give examples with explanations, and complete short exercises.**

**Optional Chance to Rework Homework Question: If you would like to present me with a new version of Problem 1 from your homework due Friday, October 13, I am inviting you to turn
in a new version by the beginning of class on Monday. Keep in mind the reminders that were mentioned in question 4. It would be great if everyone was able to really nail this proof!**

**Homework for class Monday, October 16:**

- Practice material on Elementary Matrices with the WeBWorK assignment here. Many of these should be a quick check of your understanding. Be sure to read the directions and hints to help you understand what is being asked. This is due Monday at 1:00pm.
- Reread Section 2.3 in your text (pages 113-116), and review your class notes from Friday's class. Particularly work through the details of the first part of the proof of Theorem 9. Then work through the second part of the proof using the text as a guide, but making sure to fill in details the text left out. Do you have any questions?
- Complete the Practice Problems for Section 2.3 on page 116 of the text. This will not be collected, and you should check your answers on pages 118-119.
- Complete these practice problems from Section 2.2 (pages 111-113): 9, 11 ($X$ is a matrix. This is good practice in proving uniqueness: first prove that it is a solution and then assume two solutions exist and show they must be the same. See our work in class or the proof of Theorem 5 in the text for examples.), 13, 15 (this is generalizing the Shoes and Socks Theorem!), 17 and 21. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- We know how to find the determinate of a $2\times 2$ matrix. Now we will learn how to find determinants of larger matrices. Read pages 166-167 of Section 3.1 in your text. Yes, we are skipping ahead! This is technical so I am only asking you to read the first two pages so that you can work on understanding the notation.
- Complete this short(!) worksheet. Bring your work to class to discuss and present! Note there is a place on the worksheet to write down your questions!

**Homework for class Wednesday, October 18:**

- Make sure you have worked through the worksheet that was due on Monday and reviewed your class notes. Bring any questions you have to class.
- Look back at the proof of Theorem 7 on page 109. Carefully study how Theorem 6 was vital to the result.
- Read/reread Section 3.1 in your text (pages 166-169).
- Complete the ONE Practice Problem for Section 3.1 on page 169 of the text. This will not be collected, and you should check your answers on page 171.
- Complete these practice problems from Section 2.3 (pages 117-118): 11, 15, 17, 21, 27 and 31. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems (or even - these are GREAT QUESTIONS!!!) if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Read Section 3.2 in your text (pages 171-176).
- Complete this worksheet. Bring your work to class to discuss and present! Note there is a place on the worksheet to write down your questions!

**Collected Homework (Due Friday, October 20 at 1:55pm):**

- Remember that although you may discuss this assignment with others, your write up should be your own.
- Complete the following problems:
- Number 16 from Section 2.2, page 112. Note that you CANNOT say, "since $AB$ is invertible, then $(AB)^{-1}=B^{-1}A^{-1}$," because you do not know that $A^{-1}$ exists -- you are trying to PROVE that $A$ is invertible. (You can only apply the Shoes and Socks Theorem when you already know that the two matrices in question are both invertible.) After you use the Hint in the text, you will find that $A$ equals some other matrix. Explain why this other matrix is invertible. So what can you conclude about $A$?
- Find the inverse of the matrix $D$ in Exercise 11 of Section 2.1 (page 102) and the inverse of the matrix in Section 2.2, Exercise 32 (page 112). Show all work. Ignore the original directions for those two questions!
- Complete problem 3 on this worksheet.
- Complete problem 4 on this worksheet.
- Complete problem 5 on this worksheet.
- Number 4 from Section 3.1, page 169. Read the instructions. Do it two ways.
- Number 14 from Section 3.2, page 177. ALSO decide whether the matrix is invertible.

- This assignment has been finalized!

**Homework for class Friday, October 20:**

- The text gives a very brief summary of induction in the paragraph before Theorem 5 on page 174. This is insufficient for understanding the
technique. If you are new to induction or it is just feeling a bit rusty, check out the first three pages of
this chapter of Richard Hammack's book,
__Book of Proof__. We will be using induction to prove Theorem 2 on page 169. Hammack's book is quite nice, and if you would like to review other types of proof or concepts, feel free to check out the other chapters here. - Review your group work and class notes from Wednesday. Make sure you have a firm grasp of questions 1-7 on the group worksheet or identify specific questions you have about these exercises.
- Complete these practice problems from Section 3.1 (pages 170-171): 11, 17, 25, 27, 29, 37, 39 and 40. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Reread Section 3.2 in your text (pages 171-176).
- Complete the Practice Problems for Section 3.2 on pages 176-177 of the text. This will not be collected, and you should check your answers on page 179.
- Complete this worksheet. Bring your work to class to discuss and present! Note there is a place on the worksheet to write down your questions!
- This is listed below as well since it is not actually due on Friday: Practice material on Determinants with the WeBWorK assignment here. This should be a good check of your understanding as you study for the exam! Most of these should be a quick! Be sure to read the directions and hints to help you understand what is being asked. This is due SUNDAY at 7:30pm.

**Have a great fall break!**

**Since we have fall break this week, we will NOT have a quiz!**

**Homework for class Wednesday, October 11:**

- Review your notes from class. Carefully work through each step of our proof of Theorem 4 and see if you have any questions. Note that the text guides you through an alternate proof of Theorem 4 in Exercise 25 of Section 2.2.
- Make sure you understand the approach to completing the first problem on the group worksheet from class.
- Reread Section 2.1 in your text (pages 94-102).
- Complete these practice problems from Section 2.1 (pages 102-103): 5, 7, 8, 9, 10, 23 and 25 (this is a great problem!). These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Complete this worksheet. It focuses on the second half of Section 2.2. Bring your work to class to discuss and present! Note there is a place on the worksheet to write down your questions!
- Read Section 2.3 in your text (pages 113-116). This is really only three pages!

**Collected Homework (Due Friday, October 13 at 1:55pm):**

- Remember that although you may discuss this assignment with others, your write up should be your own.
- Complete the following problems:
- Suppose that $A$ is an invertible matrix. Prove that its inverse is unique. That is, show that if $B$ and $C$ are both inverses of $A$, then $B=C$.
- Consider the matrices $A$ defined in Exercises 4, 6 and 10 of Section 1.8 and do the
following for each one; this means you have three questions with two parts each. (Ignore
the text instructions, and ignore the given vector ${\bf b}$.)
- (a) Decide whether the matrix transformation $T({\bf x})=A{\bf x}$ in question is onto its co-domain. Justify your answers with a short proof and appropriate theorems/facts.
- (b) Decide whether the matrix transformation $T({\bf x})=A{\bf x}$ in question is one-to-one. Justify your answers with a short proof and appropriate theorems/facts.

- Number 7 from Section 2.2, page 111. For 7(a), see Example 4 in the text. For 7(b) reduce the entire "super-augmented" matrix they give in the question all at once. Check that the solutions you found in part (a) appear in the last four columns.
- Suppose $(B-C)D=0$, where $B$ and $C$ are $m\times n$ matrices and $D$ is an $n\times n$ invertible matrix. Prove that $B=C$. [Notes: Remember that one of the goals of the course is to work on our proof-writing. Proofs like these are asking you to identify and utilize basic properties of matrix multiplication. Remember that matrix multiplication is more complicated than scalar multiplication, and we must do one step at a time and justify each step with an appropriate theorem. As we discussed, when you multiply we must mention which side we are using. If we use an inverse, we must mention/justify that we know it exists, since not all matrices have inverses!]

- This problem set is now complete!

**Homework for class Friday, October 13 (Happy Friday the 13th!):**

- Carefully read through your Exam 1 and rework any questions on which you lost points. Come to office hours to go over any questions you have. Everyone should do this no matter how few points you missed!
- Review our group work from class. Carefully work through each step of the proof for number 3. Also be sure to complete number 2. Bring any questions you have on this or any other part of Section 2.2 to class.
- Reread Section 2.2 in your text (pages 104-111).
- Complete the Practice Problems for Section 2.2 on page 111 of the text. This will not be collected, and you should check your answers on page 113.
- Read Section 2.3 in your text (pages 113-116). This is really only three pages!

**Quiz 4 will take place on Monday at the beginning of class. It will cover Sections 1.8 and 1.9. Be prepared to state definitions and theorems,
give examples with explanations, and complete short exercises.**

**Homework for class Monday, October 2:**

- Practice material on Linear Transformations with the WeBWorK assignment here. Many of these should be a quick check of your understanding. For the last question, think of what the matrix equation would look like with each of the vectors whose images are given. This is due Monday at 1:00pm.
- Reread Section 1.9 in your text (pages 71-78), and review your class notes from Friday's class. Do you have any questions?
- You should have three standard matrices in your notes that match up with three linear transformations. These were on the board at the end of class. Determine which of these transformations are onto and which are one-to-one. Develop clear arguments as to why your conclusions are correct. Be ready to discuss these directly after the quiz on Monday.
- Read pages 94-98 of Section 2.1. Take notes and write down any questions you have. This section is about Matrix Operations. Now we will look at operations with not only a matrix and a vector, but with two matrices (where the second is more than just an $n \times 1$ matrix)! See how these operations work similarly to real numbers, and how they are different.

**Homework for class Wednesday, October 4:**

- Complete the Practice Problems for Section 1.9 on page 78 of the text. This will not be collected, and you should check your answers on pages 80-81.
- Complete these practice problems from Section 1.9 (pages 69-70): 3, 9, 17, 19, 21, 23, 25, 27, 29, 31 and 35. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Read the motivation for Chapter 2 on pages 93-94. Aircraft Design!!!
- Reread/read Section 2.1 in your text (pages 94-102). As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. For example, you should know the definitions of diagonal matrix, zero matrix, transpose, etc. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet on Section 2.1 and bring your work to class to discuss and present! This is meant to help to guide and direct you through the reading. Note questions you have as you are working through these!

**Collected Homework (Due Friday, October 6 at 1:55pm):**

- Remember that although you may discuss this assignment with others, your write up should be your own.
- Complete the following problems:
- As one of you brought up in class, at the bottom of page 66 the text claims, "Every matrix transformation is a linear transformation." The pieces to prove this claim are within the text, mostly in Section 1.8. Bring those pieces together in a nice short proof that shows the statement is true. Be sure to use the definitions of matrix transformation and linear transformation, as well as to quote specific theorems and/or facts. Proofs should use full sentences!
- Number 6 from Section 1.8, page 69. Read the directions carefully! They ask you to answer two questions!
- Number 10 from Section 1.8, page 69.
- Carefully prove the following theorems. Be sure to refer to any theorems or facts very specifically.
- Theorem 0.1: Assume that $T:{\mathbb R}^n\to{\mathbb R}^m$ is a linear transformation with standard $m\times n$ matrix $A$. If $T$ is onto, then $m\leq n$.
- Theorem 0.2: Assume that $T:{\mathbb R}^n\to{\mathbb R}^m$ is a linear transformation with standard $m\times n$ matrix $A$. If $T$ is one-to-one, then $m\geq n$.
- Theorem 0.3: Assume that $T:{\mathbb R}^n\to{\mathbb R}^m$ is a linear transformation with standard $m\times n$ matrix $A$. If $T$ is both onto and one-to-one, then $m=n$.

- Notice that none of the theorems in the previous question were if and only if theorems. Choose one of the theorems and justify carefully why we cannot say that if the inequality (or equality) holds then the transformation has that property. (Hint: A good way to justify this is to find a counterexample!)
- Real World Applications: Carefully read pages 81-82 of Section 1.10. Then do the following:
- Number 2 from Section 1.10, page 87. Do part (a) and then SET UP part (b) and EXPLAIN briefly how you would solve the question, but do NOT solve it!
- Number 4 from Section 1.10, page 87. These numbers will be messy! This is more like the real world!

- Number 6 from Section 2.1, page 102. Read the directions carefully! They ask you to show your work in two ways!
- Please return Quiz 3 with your Week 6 Homework.
- This assignment has been updated and is complete!

**Homework for class Friday, October 6:**

- Review your notes from class. If you did not complete the group worksheet in class, do so now. In particular, prove the new rule for transposition (2g of the worksheet). [Hint: Compare the $(i,j)$-entry of each matrix!] Also, what is the warning to take away from 3c? Bring questions to class about this worksheet and be ready to ask them at the beginning of class.
- Reread Section 2.1 in your text (pages 94-102).
- Complete the Practice Problems for Section 2.1 on page 102 of the text. This will not be collected, and you should check your answers on page 104.
- Read pages 104-107 of Section 2.2 in your text. As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. For example, you should know the definitions of a invertible, inverse, determinant, etc. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet. It begins with a few questions from Section 2.1 and then follows your reading for Section 2.2. Bring your work to class to discuss and present! Note there is a place on the worksheet to write down your questions!

**Homework for class Monday, September 25:**

- Prepare for Exam 1!
- You should have a copy of the Exam 1 Preparation sheet. If you have lost yours here is another copy.
- Feel free to email me with questions any time over the weekend!!! Remember that I also have office hours on Monday morning.
- REMEMBER TO BE AT THE EXAM IN STERN 303 AT 1:25!!!

**Office hours are moved from Wednesday to Tuesday this week! I will have office hours 11:30am-12:30pm on Tuesday and no hours on Wednesday.**

**Homework for class Wednesday, September 27:**

- Reread/read Section 1.8 in your text (pages 63-69). As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. For example, you should know the definitions of a transformation (and its domain, codomain, image and range), and what it means for a transformation to be linear, etc. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete the Practice Problems on page 69 of the text. This will not be collected, and you should check your answers on page 71.
- Complete this worksheet on Section 1.8 and bring your work to class to discuss and present!

**Collected Homework (Due Friday, September 29 at 1:55pm):**

- Remember that although you may discuss this assignment with others, your write up should be your own.
- Complete the following problems:
- Complete the problems on this worksheet.

**Homework for class Friday, September 29:**

- Review your notes from class and the worksheets we discussed. Work through the last problem on the group worksheet. Make sure you feel confident about the material in Section 1.8. Bring your questions to office hours and class.
- Complete these practice problems from Section 1.8 (pages 69-70): 1, 5, 9, 15, 17, 21 and 31. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Read Section 1.9 in your text (pages 71-78). As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. For example, you should know the definitions of the standard matrix of a transformation, one-to-one, onto, etc. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet on Section 1.9 and bring your work to class to discuss and present!

**Quiz 3 will take place on Monday at the beginning of class. It will cover Sections 1.4 and 1.5. Be prepared to state definitions and theorems,
give examples with explanations, and complete short exercises.**

**Homework for class Monday, September 18:**

- Choose an office hour to attend this week! Bring questions! Our first exam is next Monday so be sure you are asking all the questions you have now!!!
- If you are still having questions about row reduction, check out this video!
- Review your notes from class and the worksheet we discussed. Bring your questions to office hours and class.
- Make sure you have completed the Practice Problems for Section 1.5 (page 47).
- Complete these practice problems from Section 1.5 (pages 48-49): 17, 19, 31, 33 and 23.
**When you come to class, put your solution to one of the first four of these questions on the board with your group.**If you weren't one of the ones writing on the board on Friday, you should be one on Monday! - Read Section 1.7 in your text (pages 56-61). As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. For example, you should know the definitions of homogeneous system, trivial solution, nontrivial solution, parametric vector form, etc. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete this worksheet on Section 1.7 and bring your work to class to discuss.

**Office hours are moved from Wednesday to Tuesday this week! I will have office hours 11:30am-12:30pm on Tuesday and no hours on Wednesday.**

**Homework for class Wednesday, September 20:**

- Practice material in Section 1.7 with the WeBWorK assignment here. Note that this is over all parts of Section 1.7, even the Theorems we have not yet discussed in class. Thus, I am going to allow you until Friday to finish it. You should be able to finish it by Wednesday using your reading of the section together with our discussions, but I am giving you some flexibility here. Nevertheless, I would get started now, since you have a collected assignment and will also have classwork due for Friday! This is due Friday at 1:00pm.
- Remember to come to office hours with any questions you have! At this point you should have at least four problems (or more partial problems) completed for the collected assignment due Friday!
- Reread Section 1.7 in your text (pages 56-61). Review your flashcards and other notes! Write down questions to bring to class.
- Complete the Practice Problems for Section 1.7 on page 61 of the text. This will not be collected, and you should check your answers on page 63.
- Complete this worksheet on Section 1.7 and bring your work to class to discuss. I will be looking for a volunteer to present the proof of Theorem 8. Please email me or come to my Tuesday office hours if you want to volunteer. If I don't hear from anyone by the end of office hours, I will roll a die to select someone randomly from the class to present!
- Complete these practice problems from Section 1.7 (page 62): 9, 11. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.

**Collected Homework (Due Friday, September 22 at 1:55pm):**

- Remember that although you may discuss this assignment with others, your write up should be your own.
- Complete the following problems:
- Complete the problems on this worksheet.
- The above worksheet has been finalized! Note that one more problem was added on Wednesday.
- BONUS (5 points): Prove Theorem 6 from page 47 of the text! Remember that one of the hypotheses of Theorem 6 is that the system is consistent, that is, this theorem is ONLY talking about consistent systems!

**Homework for class Friday, September 22:**

- REMEMBER that you have a WeBWorK assignment due on Friday! It is on Section 1.7 and is here. This is due Friday at 1:00pm.
- Remember to come to office hours with any questions you have! Review your class notes from our discussions and problem sets. Make a list of specific questions that you can ask in office hours. You can also ask some questions in class as well, but you should not save all of your questions for class as others may have some as well.
- Reread Section 1.7 in your text (pages 56-61). Review your flashcards and other notes!
- Be sure you complete the practice problems we started in class on Wednesday from Section 1.7 (page 62): 11, 15, 17, 19, 21, 23, 27 and 31. Also try exercises 33 and 35. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Read pages 63-66 of Section 1.8 in your text. As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. For example, you should know the definitions of a transformation (and its domain, codomain, image and range), and what it means for a transformation to be linear, etc. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!

**Quiz 2 will take place on Monday at the beginning of class. It will cover Sections 1.2 and 1.3. Be prepared to state definitions and theorems,
give examples with explanations, and complete short exercises.**

**Homework for class Monday, September 11:**

- Choose an office hour to attend this week!
- Review your notes from class and the worksheet we discussed. Bring your questions to office hours and class.
- Reread Section 1.4 in your text (pages 35-40). Review your flashcards and other notes! Write down questions to bring to class.
- Complete the Practice Problems on page 40 of the text. This will not be collected, and you should check your answers on pages 42-43.
- Complete this worksheet on Section 1.4 and bring your work to class to discuss.
- If you didn't complete the Section 1.3 practice problems assigned for Friday, do them now. Then try these practice problems from Section 1.3 (pages 32-33): 23 and 24. Be sure you can explain why the statements are true or false.
- Start your collected problem set due Friday this Sunday night!

**Homework for class Wednesday, September 13:**

- Practice material in Sections 1.3 and 1.4 with the WeBWorK assignment here. It isn't as long as it looks! But be very careful about how many attempts you have for each question!!! This is due Wednesday at 1:00pm.
- Remember to come to office hours with any questions you have! If you haven't started the assignment due Friday, do so NOW!
- Work on a proof for Theorem 5(a). Fill in the blanks the text leaves in their proof! Remember that every step should be justified with a theorem or a definition.
- Reread Section 1.4 in your text (pages 35-40). Review your flashcards and other notes! Write down questions to bring to class.
- As part of your reading of Section 1.4, work carefully through the proof of part of Theorem 4 at the end of the section (page 40). What methods of proof are they using?
- Read Section 1.5 in your text (pages 43-47). As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. For example, you should know the definitions of homogeneous system, trivial solution, nontrivial solution, parametric vector form, etc. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete these practice problems from Section 1.4 (pages 40-42): 1, 5, 7, 9, 11, 13, 15, 17, 21, 23 and 25. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.

**Collected Homework (Due Friday, September 15 at 1:55pm):**

- Remember that although you may discuss this assignment with others, your write up should be your own.
- Complete the following problems:
- Complete the problems on this worksheet.
- The worksheet above has been added to and finalized! Make sure you see the new problems!

**Homework for class Friday, September 15:**

- Reread Section 1.4 in your text (pages 35-40). As part of your reading of Section 1.4, work carefully through the proof of part of Theorem 4 at the end of the section (page 40). Review the first half we did in class. Work out the details of the second half. In particular, what method of proof are they using in the second half?
- Proof presentation! I would like to have a volunteer to present the second half of the Theorem 4 proof at the beginning of Friday's class. If you are interested in doing so, please contact me ahead of time.
- Reread Section 1.5 in your text (pages 43-47). Review your flashcards and other notes! Write down questions to bring to class.
- Complete this worksheet on Section 1.5 and bring your work to class to discuss.
- The worksheet above has been updated! Make sure you use the updated version!
- Complete these practice problems from Section 1.5 (page 48): 1, 3, 9, 11. Then do problems 5 and 15 noting their similarities and differences! These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.

**Quiz 1 will take place on Monday at the beginning of class. It will cover Sections 1.1 and 1.2. Be prepared to state definitions and theorems,
give examples with explanations, and complete short exercises.**

**Homework for class Monday, September 4:**

- Practice using WeBWorK with the DemoSet assignment which can be accessed on the WeBWorK Home Page for Our Class. This is due Monday at 1:00pm.
- Complete the Row Reduction Algorithm worksheet I handed out in class on Friday. Fill in the blanks (the steps), and then apply the row reduction algorithm to the matrix on the sheet.
- Reread Section 1.2 in your text (pages 12-21).
- Read pages 24-29 of Section 1.3 in your text. As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete these practice problems from Section 1.2 (pages 21-22): 1, 5, 11, 13, 15, 21, 23, 25. These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.

**Homework for class Wednesday, September 6:**

- Last chance to bring in your picture if you forgot to bring it to your appointment!
- Practice material in Sections 1.1 and 1.2 with the first real WeBWorK assignment here. It isn't as long as it looks! But be very careful about how many attempts you have for each question!!! This is due Wednesday at 1:00pm.
- Reread Section 1.2 in your text (pages 12-21). Make special note of the Theorem on Existence and Uniqueness of Solutions, Theorem 2 on page 21 in the text. This brings together many of the ideas we have been discussing.
- Solve this question and be ready to discuss it at the beginning of class: How many solutions does a consistent linear system of 3 equations and 4 unknowns have? Why?
- Read/reread Section 1.3 in your text (pages 24-31). As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete the Practice Problems on page 32 of the text. This will not be collected, and you should check your answers on pages 34-35.
- Complete these practice problems from Section 1.2 (pages 21-22): 27, 29 and 31 (Draw what is happening geometrically here!). These will not be collected, but you should do ALL of these at least in your head if not on paper. In fact, you should do MORE odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.
- Answer the following and be ready to discuss it in class: True or false: If there is a free variable in the RREF of an augmented matrix, then the system must have an infinite number of solutions. Justify your answer.

**Collected Homework (Due Friday, September 8 at 1:55pm):**

- Remember that although you may discuss this assignment with others, your write up should be your own.
- Complete the following problems:
- Number 33 from Section 1.1, page 11. Show your work as was done for $T_1$ in the problem statement above the question.
- Number 34 from Section 1.1, page 11. Use the hint!!!
- Number 4 from Section 1.2, page 22. Read the directions carefully and do all the parts of the question!
- Number 12 from Section 1.2, page 22. Be sure to show your work and explain your reasoning where appropriate!
- Number 14 from Section 1.2, page 22. Be sure to show your work and explain your reasoning where appropriate!
- Number 20 from Section 1.2, page 22. Be sure to show your work and explain your reasoning where appropriate! Don't forget to use Theorems in explanations!
- For each of the following, decide whether or not it is possible for a system to satisfy the given description. If it is possible, give an
augmented matrix (in row-echelon or reduced row-echelon form) that corresponds to such a system and
**prove**that the corresponding system does in fact fulfill the requirements; if it is not possible,**prove**that it is not possible. (Hint: Theorems are helpful in proving!) (a) A system of 3 equations in 5 unknowns that has infinitely many solutions. (b) A system of 3 equations in 5 unknowns that has no solutions. (c) A system of 3 equations in 5 unknowns that has exactly 1 solution. - Repeat question 7 for the following statements. (a) A system of 5 equations in 3 unknowns that has exactly 1 solution. (b) A system of 5 equations in 3 unknowns that has infinitely many solutions. (c) A system of 5 equations in 3 unknowns that has exactly 2 solutions.
- Prove part (vii) of the Algebraic Properties of Vectors in $R^n$ Theorem (p. 27). See the solution to Practice Problem 1 of Section 1.3 for an example of how such a proof should go. Note that the sample proof on page 34 is really an outline for a proof. Be sure to include sentences and write your proof in paragraph form. You will quote definitions and properties of real numbers in your proof.

**Homework for class Friday, September 8:**

- Reread Section 1.3 in your text! What is the span of a set of vectors? How is it connected to what we discussed Wednesday in class?
- Complete this example and bring your work to class to discuss.
- Read Section 1.4 in your text (pages 35-40). As you read, take notes, record questions and start a list (or a set of flashcards!) of definitions. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Try these practice problems from Section 1.3 (pages 32-33): 1, 5, 9, 11, 15, 21 and 25. These will not be collected, but you should do as many as you need to in order to feel comfortable with the material. In fact, you should do more odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.

**Welcome to Linear Algebra!!!**

**Collected Homework (Due Tuesday, August 29 by 2:30pm):**

- Write an essay as assigned on the syllabus.

**Homework for class Wednesday, August 30:**

- Read the syllabus! In fact, read it at least two times. You should be sure you have read all of it and understand what is expected. Please ask if you have questions. Note the paper copy I gave you is green so that you can easily find it. Refer to it often.
- Put the exam dates from the syllabus on your calendar. Note that the midterm exams begin at 1:25, a half hour before our usual class time. Let me know ASAP if you have any issues with this.
- I put together a website for my MATH 135 class to help with proof writing and presentations. Read the Proof Writing and Presentation Tips website. Use this as a reference when you are preparing your homework and presentations for class. Note there is a link to this site and the syllabus at the top of this web page.
- Read the Introduction to Chapter 1 and Section 1.1 in your text. This covers pages 1-9. As you read, take notes, record questions and start a list of definitions. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Following the techniques in Example 1 on pages 5-6, solve the system of linear equations I put on the board near the end of class on Monday (the one for which we wrote the coefficient and augmented matrices). I would like student volunteers to take us through these steps in class on Wednesday! Be prepared!!!
- Complete the Practice Problems on page 9 of the text. This will not be collected, but you should check your answers on page 11.
- Read pages 12-17 of Section 1.2 in your text. Again, as you read, take notes, record questions and start a list of definitions. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!

**Collected Homework (Due Friday, September 1 at 1:55pm):**

- Remember that although you may discuss this assignment with others, your write up should be your own.
- Complete the following problems:
- Consider the system of linear equations consisting of $2x+y=1$ and $6x+3y=b$. For what value(s) of $b$ is the system inconsistent? For what value(s) of $b$ does the system have infinitely many solutions? For what value(s) of $b$ does the system have a unique solution? Show your work and justify your answers.
- Number 10 from Section 1.1, page 10. Continue to reduced row echelon form and determine the solution. Use matrix notation and label each step. Note that each step should consist of a SINGLE elementary row operation.
- Number 12 from Section 1.1, page 10. Use matrix notation and label each step. Note that each step should consist of a SINGLE elementary row operation.
- Number 16 from Section 1.1, page 10. Use matrix notation and label your steps (you may carefully do two steps at once as long as they are carefully labeled). You can answer the question from the echelon form (RREF not required).
- Number 18 from Section 1.1, page 10. How does this question translate into a question about systems of equations? (Be sure to answer this question as part of your solution!) Use matrix notation to solve the relevant system, and show your work/reasoning.
- Number 20 from Section 1.1, page 10. Show your work/explain your reasoning.

**Homework for class Friday, September 1:**

- Complete the question we discussed at the end of class about which values of $h$ made the system consistent.
- Reread Section 1.1 in your text!
- Read/reread Section 1.2 in your text (pages 12-21). As you read, take notes, record questions and start a list of definitions. You should also feel free to fill in the blanks for any skipped steps in the reading by working through calculations yourself!
- Complete the Practice Problems on page 21 of the text. This will not be collected, and you should check your answers on pages 23-24.
- Be sure that you know definitions. How many do we have now? You may find it helpful to make flashcards to practice. Remember that if you really understand a definition, you should be able to produce and example of something that satisfies the definition and something that does not.
- Try these practice problems from Section 1.1 (pages 10-11): 1, 5, 9, 13, 15, 23, 25 and 27. These will not be collected, but you should do as many as you need to in order to feel comfortable with the material. In fact, you should do more odd problems if you are not confident after completing these! Note that AFTER you have solved these questions you can check your answers in the back of the textbook.