Math 200 - Linear Algebra

Math 200 - Linear Algebra
Spring 2026
Emily Proctor

Please read the syllabus to learn all the details about the course.

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Homework

Week beginning February 9

Due Wednesday, February 11:

Get yourself a copy of the textbook (let me know if you run into any trouble with this).

Solve the three systems of equations that I presented in class on Monday. You don't need to turn these in, but we will start class on Wednesday by talking about them.

Read Sections 1.1 and 1.2. When you are reading, please pay close attention to Example 1 in Section 1.1, and carefully read the steps of the row reduction algorithm in Section 1.2. These ideas/procedures will form the cornerstone of the work that we do in the first half of class, so it's good for you to get a solid foundation in them. We will go over them in class as well.

Please type (up to) a page about who you are as a mathematician (classes or experiences you've had, anything you've liked, anything you've disliked...) as well as your motivations for taking Linear Algebra this semester. Be honest! :) This is just a simple, informal assignment that will help me start to get to know you. If there is anything else you'd like me to know as we start the semester, please include that too. Print out two copies of your page and turn them in at the start of class on Wednesday.

Due Friday, February 13:

From Section 1.1 do problems: 1, 3, 7 (explain your answer), 13, 19 (explain your conclusion), 22, 23, 27, 29, 31, 33, 39, 41.

From Section 1.2 do problems: 1, 19, 24*, 25, 27, 29, 31, 33, 35, 40, 43.

*Problem 1.2.24 captures a lot of ideas. I'll pass out the solution for this one and it won't be graded, so just use it as a way to think things through. We'll use it in class on Friday to summarize and consolidate the ideas that you are exploring in Section 1.2.

Here is a copy of our notes from class on Wednesday. I apologize for the typo in the notes that I handed out in class! I have corrected it in the notes here, and highlighted where the typo was so that you can see it more easily. Thanks for your patience!

Read Section 1.3.

Week beginning February 16

Due Monday, February 16:

From Section 1.2 do problems: 10, 38, 41.

From Section 1.3 do problems: 5, 7, 9, 11, 13, 15 (do 3 instead of 5), 17, 22, 24, 26, 28, 30, 32, 33, 40, 41.

When doing your problem sets for this class, rather than simply doing the problems and moving on, try to ask yourself each time what skills and connections you are developing based on the problems you are doing. Often two (or more) different problems are asking essentially the same thing, but from two (or more) different points of view. Try to see if you can recognize this.

Here is a copy of our notes from class on Friday.

Read Section 1.4.

Due Wednesday, February 18:

From Section 1.4 do problems: 1, 3, 9, 13, 18, 21*, 24, 26, 28, 30, 32, 34, 36, 42*, 44, 45**.

*If you have geometric intuitions for Problems 1.4.21 and 1.42, make sure to support it with an algebraic argument.

**For Problem 1.4.45, see Theorem 5 on p.41. While you are there, go ahead and flip forward to p.217 and see the definition of a linear transformation. We will see that matrix multiplication is an example of a linear transformation!

Here is a copy of the handout from Monday about the relationship between matrix equations, vector equations, and systems of equations.

Read Section 1.5.

Due Friday, February 20:

From Section 1.5 do problems: 3, 7, 10, 18, 21, 25, 27, 29, 31, 33, 35, 38, 40, 43 (justify your response), 44 (justify your response), 45, 49, 52.

Read Section 1.7

Week beginning February 23

Due Monday, February 23:

From Section 1.7 do problems: 1, 4, 7, 11, 17, 18, 29, 31, 33, 34, 36, 37, 39, 43, 44, 46.

If you're curious to see how the things we have been looking at are going to come together, skip ahead and take a look at the invertible matrix theorem on p.121. (Note that this is a theorem about square matrices.) Not everything in the theorem will look familiar yet, but some of it should!

Read Section 1.8

Due Wednesday, February 25:

From Section 1.8 do problems: 1, 3, 7, 9, 11, 14, 15, 17, 19, 34, 38, 39*, 43.

*For Problem 1.8.39, try using the definition of linear dependence to answer the question.

Read Section 1.9.

Due Friday, February 27:

From Section 1.9 do problems: 1, 2, 5, 8*, 9*, 12, 13, 15, 19, 20, 21, 24, 25, 26, 29.

* For Problems 1.9.8 and 1.9.9, briefly show/explain how you arrived at your answer.

existence/uniqueness chart

Week beginning March 2

Due Monday, March 2:

From Section 1.9 do problems: 31, 32, 35 (justify your responses), 37, 38, 39, 40, 43, 44*.

*Here is a template that you can use for Problem 1.9.44. I hope it is helpful!

Read Section 2.1.

Due Wednesday, March 4:

From Section 2.1 do problems: 1, 7, 10, 11, 15, 17, 19, 20, 22, 25, 26, 27, 29*, 30*, 32, 39.

*For Problems 2.1.29 and 2.1.30, try using the definition of linear dependence (p.60). For Problem 2.1.30, consider your answer to Problem 2.1.27.

As you know, I did not spend much time in class today talking about the row-column method for computing a matrix product, or the definition of the transpose. Both ideas are fairly straightforward, and I think you'll pick them up well by reading about them. So, just to make sure that you do see them, go back and take a look at pages 102-105 in Section 2.1, making note of the warnings on the bottom of p.104. One thing of note is that the row-column method of computing a matrix product is much better than the definition of AB for finding the actual individual entries in AB (such as in computations). As we'll see, the definition of AB will be really helpful to us as we continue to investigate things from a theoretical standpoint.

Read Section 2.2.

Due Friday, March 6:

From Section 2.2 do problems: 1, 7, 9, 11, 13, 15, 17, 18, 23, 24, 25, 28, 31, 32, 33, 34, 36, 41.

Read Section 2.3.

Continue to prepare for our exam, which is Tuesday, March 10, 7-9pm, in Warner 105. We will have a(n optional) review session on Monday, March 9, 2:30-4pm, in Warner 105. Come with questions!

Week beginning March 9

Due Monday, March 9:

From Section 2.3 do problemss: 2, 3, 4, 11, 13, 15, 17, 23*, 24*, 26*, 27*, 30* (assume H is a square matrix), 34 (assume A is a square matrix. Here A^2 means AA), 38, 41, 45, 46.

*Although Problems 2.3.23, 24, 26, 27, and 30 can all be answered quickly using the IMT, a good way to review for the exam and solidify the ideas in your mind is to also argue directly using material we have learned earlier.

Here is a practice exam and solutions. Here, too, is a sheet that lists the main definitions that we have been working with. I hope these are a helpful tools for you for reviewing and consolidating the material!

As a reminder, we'll have class as usual on Monday, March 9. Our exam is Tuesday, March 10, 7-9pm, in Warner 105 (we will all be in Warner 105 after all). We have a review session Monday, March 9, 2:30-4pm, also in Warner 105. I won't have an agenda so please bring questions!

Read Section 3.1.

Due Wednesday, March 11:

No class! The homework from class on Monday is due on Friday. I hope you enjoy a little time off from linear algebra!

Due Friday, March 13:

From Section 3.1 do problems: 1, 5, 9, 19, 20, 21, 22, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 43.

Note: Problems 19-21, 25-30, and 33-36 are meant to prepare you for what we are going to talk about in class on Monday. Read p.114-115 from Section 2.2 for a discussion about elementary matrices. The upshot is that multiplying on the left by an elementary matrix does the same thing as performing one of our row reduction operations. Problems 25-30 have you compute the determinants of some elementary matrices, and Problems 33-36 have you see the effect of multiplying by one of these matrices (i.e. performing a row operation) on the the determinant.

Read Section 3.2.

Week beginning March 16

Due Monday, March 16:

From Section 3.2 do problems: 2, 3, 15, 17, 20, 23, 25, 27, 29, 32, 33, 36, 37, 38, 39, 40, 42, 46.

Read Section 4.1.

Due Wednesday, March 18:

On Monday, we introduced a major new concept, the definition of a vector space. We haven't gotten far enough to do many book problems, so to prepare for class on Monday, finish working through our worksheet of examples. You don't have to turn this in but do spend some time with it so that the ideas can start to sink in. We'll go over it at the start of class on Wednesday.

Read through your exam and the solutions (everyone: put a little focus on Problem 7). If there is anything you missed on the exam, take some time to review it and/or ask for help. I'd be happy to talk!

No new reading! We'll continue with Section 4.1 on Wednesday. See you then!

Due Friday, March 20:

From Section 4.1 do problems: 1*, 3*, 5**, 8**, 11, 13***, 15, 17, 20a, 21, 23, 27, 29, 30, 34****, 35, 39.

*For Problems 4.1.1 and 4.1.3, please give an answer that is different from the back of the book.

**For Problems 4.1.5 and 4.1.8, see Examples 4 and 7 about polynomials. The upshot is that the set of all polynomials of degree at most n is a subspace of F(R;R), and hence is a vector space itself! We'll talk more about this space as we go along.

***For Problem 4.1.13, have a look back at Problem 1.3.33!

****Note that Problem 4.1.34 follows a very typical format for a uniqueness proof: suppose that w acts like -u then prove that w must actually equal -u. If you look at Problem 4.1.33, you'll see the same pattern.

Read Section 4.2, making notes of Remarks 1 and 2 on p.214.

Week beginning March 23

Happy Spring Break!

Week beginning March 30

Due Monday, March 30:

From Section 4.2 do problems: 2, 5, 7*, 10, 15, 20, 21*, 25, 27, 29, 31, 33, 35, 39, 40, 41**, 43a, 47.

*For Problems 4.1.7 and 4.1.21, please give reasons/answers that are different from the back of the book.

**Notice that Problem 4.1.41 is a direct proof that Col A is a subspace of R^m. We proved this in class using the span theorem!

Here is the (somewhat complex!) example of a linear transformation that I passed out in class on Friday.

Read Section 4.3.

In order to get more out of class on Monday, before class review Theorem 1.4.4 (p.39), Section 1.7, and the Invertible Matrix Theorem.

Due Wednesday, April 1:

From Section 4.3 do problems: 3, 5, 6, 9, 20*, 21, 23, 25, 27, 33, 34, 35, 39, 40, 41.

*Problem 4.3.20 is a warm-up to the Spanning Set theorem, which we will cover in class on Friday. Try it out and think about why what you are doing works. We will talk more on Friday.

Read Section 4.4.

Due Friday, April 3:

From Section 4.3 do problems: 13*, 14*, 22, 24, 28, 30, 36**, 43, 44***.

From Section 4.4 do problems: 21****, 24.

*For Problems 4.3.13 and 4.3.14, please do Nul A and Col A only (you do not need to find a basis for Row A).

**For Problem 4.3.36, you may find a double angle trig identity helpful!

***For Problem 4.4.44, "by inspection" means that the linear dependence relation is straightforward and you can probably figure it out by just taking a good look at (inspecting) the given polynomials.

****For Problem 4.4.21, please make sure to show your work and justify your thinking.

Reread Section 4.4.

Week beginning April 6

Due Monday, April 6:

From Section 4.4 do problems: 3, 5, 9, 11, 15, 16, 17, 18, 22, 23, 25, 26, 27*, 28.

*Note the structure of a uniqueness proof in Problem 4.4.27! (Recall that one-to-one means that there is a unique input mapping to a given output.) Yay!!

Read Section 4.5.

Due Wednesday, April 8:

From Section 4.4 do problems: 13*, 19, 20, 35a, 36.

From Section 4.5 do problems: 1, 4, 7, 9, 12**, 13**, 14**, 17, 20, 23, 26, 29.

*For Problem 4.4.13, consider translating the whole question into R^3, using the standard basis for P_2.

**For Problems 4.5.12-14, just determine the dimensions of Col A and Nul A (you don't have to do Row A).

Read Section 4.6.

Due Friday, April 10:

From Section 4.5 do problems: 27, 31, 35, 38*, 39, 41**, 42**, 43, 44, 45, 46, 51.

*For Problem 4.5.38, just find the dimension of the column space (you don't need to do the row space).

**For Problems 4.5.41 and 42, we didn't talk much in class about infinite-dimensional vector spaces, but give these problems a shot. I will pass out solutions and they won't be graded, but see if you can imagine how you could show that a vector space is infinite-dimensional. Theorem 12 on p.243 is a helpful tool here.

Read Section 4.6 again.

Continue to prepare for our exam, which is Tuesday, April 14, 7-9pm, in Warner 105. The exam will focus on Sections 3.1 through 4.5. Here is a topic list; I hope it will be useful to you as you prepare! We will have a(n optional) review session on Monday, April 13, 2:30-4pm, in Warner 105. Come with questions!

In order to get more out of class on Friday (and simultaneously prepare for the test!), it'd be a good idea to go back and review the material in Section 4.4 on the coordinate mapping before class.

Week beginning April 13

Due Monday, April 13:

From Section 4.6 do problems: 1, 3, 6, 7, 11, 12, 13, 14, 15, 17, 18*, 22a**.

*For Problem 4.6.18, note the structure of a uniqueness proof!

**For Problem 4.6.22a, it might be helpful to review the discussion of matrix multiplication on p.100-101.

Here is a practice exam and solutions. Here, again, is a list of topics that we have been working on. I hope these are a helpful!

As a reminder, we'll have class as usual on Monday, April 13. Our exam is Tuesday, April 14, 7-9pm, in Warner 105. We have a review session Monday, April 13, 2:30-4pm, also in Warner 105. I won't have an agenda so please bring questions!

Read Section 5.1.

Due Wednesday, April 15:

No class! The homework from Monday's class will be due on Monday of next week.

Due Friday, April 17:

No class! I hope you enjoy the Spring Symposium!

Week beginning April 20

Due Monday, April 20:

From Section 5.1 do problems: 1, 3, 6, 11, 15, 17, 19, 21, 23, 24, 25, 31, 34*, 37, 39, 43.

*For Problem 5.1.34, suppose Ax=cx for some nonzero c. How would this lead to a contradiction?

Read Section 5.2.

Due Wednesday, April 22:

From Section 5.2 do problems: 1, 7*, 9, 11, 15, 16, 18, 19, 21, 25, 27, 28, 32**.

*For Problem 5.2.7, you will need the quadratic formula.

**For Problem 5.2.32, see the definition of similar matrices at the bottom of p.285. Problem 5.2.32 is meant to get you famililar with the term "similar" but the actual work of the problem is based on things you've done before (see Problem 3.2.40!).

Read Section 5.3.

Due Friday, April 24:

From Section 5.3 do problems: 2, 3, 5, 9, 10, 17*, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 35, 36.

*Note that the matrix for Problem 5.3.17 is triangular.

Read Section 5.4.

In order to get more out of class on Friday, before class review Section 1.9 (the standard matrix of a linear transformation R^n to R^m) and Section 4.4 (the coodinate map).

I apologize that I think I had a typo on the board for the last example that I gave in class today, so I am attaching my notes if you'd like to update your notes and clear up any confusion. Conveniently, the next page of notes has a statement of the diagonalization theorem, which I had wanted to pass out to you anyway. I hope this all helps!

Take some time to look through your exam and read the comments. Focus on the good things and, for the places where you had challenges, take some time to review or ask questions. I'm here to help!

Week beginning April 27

Due Monday, April 27:

From Section 5.4 do problems: 1, 2, 3, 4, 5, 6, 7, 13, 15, 25.

To prepare for Monday, review what we did in class Friday, along with Section 1.9 (the standard matrix of a linear transformation....this is a special case of what we did in class Friday!), Section 4.4 (the coordinate mapping), Section 4.6 (the change of basis matrix), and Section 5.3 (diagonalization). We'll put them all together on Monday!

Reread Section 5.4.

If you didn't have a chance to look at your exam and read the comments, set aside a little time this weekend to do that and review if you need to. Stay in touch with questions if you have them!

Due Wednesday, April 29:

From Section 5.4 do problems: 9, 12, 14, 23, 24, *27, *28, *29.

*Problems 5.4.27, 28, and 29 in this assignment come from an older edition of the book. Please do these problems from the linked handout, rather than from our book!

Several ideas came together in class today. In addition to doing these problems, spend some time rereading your notes and internalizing how all the pieces fit together. In case it's helpful, here is a copy of the portion of my notes that I handed out in class today.

Read Section 6.1.

Due Friday, May 1:

From Section 6.1 do problems: 1, 3, 5, 7, 10, 13, 16, 17, 19, 20, 25, 26, 30, 33, 35*, 36*, 37*.

*Make note of the results from Problems 6.1.35, 6.1.36, and 6.1.37. We'll use them in class on Friday!

Here is a copy of the vector space summary handout from class today.

Read Section 6.2.

Week beginning May 4

Due Monday, May 4:

From Section 6.1 do problems: 23, 38, 39.

From Section 6.2 do problems: 2, 9, 17, 23, 25, 26, 27, 34, 40.

Read Section 6.3.

Due Wednesday, May 6:

From Section 6.2 do problems: 11, 13, 16, 39, 41.

From Section 6.3 do problems: 1, 3, 6, 10, 16, 19*, 21, 22, 23, 26.

We did not have an example problem of the Orthogonal Decomposition Theorem in class today, so please see Example 2 on p.369 and Practice Problem 1 on p.373.

*Pay particular attention to Problem 6.3.19. We will use this idea quite a bit in class on Wednesday!

Read Section 6.4 through p.378, paying careful attention to Example 2 on p.376-7.

Due Friday, May 8:

From Section 6.4 do problems: 3, 7, 11, 17, 18, 19, 20, 26.

Read Section 6.5.

We'll be making use of the transpose of a matrix in class on Friday. We haven't done much with the transpose this semester, so to make class feel a little smoother go to Example 8 on p.105 to see/remind yourself how we form the transpose of a matrix.

Week beginning May 11

Due Monday, May 11:

From Section 6.3 do problems: 11*, 13*.

From Section 6.5 do problems: 3, 5, 9*, 12*, 13, 17, 18, 19, 20, 21, 22.

*Note that in Problems 6.3.11, 6.3.13, 6.5.9, and 6.5.12, the given vectors/columns are orthogonal (but they are not in Problems 6.5.3, 6.5.5, and 6.5.13).

Read Section 6.6 through Example 1, p.392.

There will be time in class on Monday for you to fill out course response forms, so please bring your favorite device for doing this. Thank you in advance for your feedback!

If you have used the Q-center this semester, please consider filling out this anonymous survey.

Consider beginning your review of the material from our course in preparation for our final on May 15. Here is a list of topics from our course. And here is a practice exam and solutions.

We will have an optional review session on Thursday, May 14, 12:30-2:00pm in Warner 105. See you there!