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

**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 Sections 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):**

**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.
- The above worksheet has been updated including fixing exercise 1. More may be added to this assignment 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!

**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.
**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.
- 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.
**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:
- 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.