Math 110: Linear Algebra (Summer 2019)

In the summer of 2019, I am teaching Math 110: Linear Algebra. This is the course website and it will also serve as the syllabus. The math on this page is rendered by MathJax.

Announcements

Wednesday, June 26 @ 3:29PM	The due date for HW1 has been changed to Tuesday night. All future HWs will also be due on Tuesday nights. This is so that you have ample time after the Monday office hour. A .tex template has also been uploaded to the Materials section.
Tuesday, June 25 @ 9:03PM	My Friday office hours have been changed to 9-11AM. The table in Meeting times on this page has been updated to reflect this.

Contents of this page

Course description
Grading
Meeting times
Lecture synopses
Materials

Course description

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The official math department listing for Math 110 reads as follows:

Prerequisites: 54 or a course with equivalent linear algebra content

Description: Matrices, vector spaces, linear transformations, inner products, determinants. Eigenvectors. QR factorization. Quadratic forms and Rayleigh's principle. Jordan canonical form, applications. Linear functionals.

The official text for this class is Linear Algebra Done Right by Sheldon Axler.

As you may gather from the catalogue description of the course, the topics in Math 110 overlap significantly with those in Math 54. However, Math 54 is designed for a much broader audience than Math 110 and aims to give students working knowledge of basic linear algebra by exposing them to matrices for a whole term. Unfortunately, students are likely to leave Math 54 thinking linear algebra is only about matrix bashing. Moreover, even though students spend the whole semester dealing with matrices, they might leave the course without answers to important questions such as:

Why is matrix multiplication done the way it is?
Why is the determinant computed the way it is?
Why is the notion of a "symmetric" matrix meaningful? (i.e. why do we care about symmetric matrices, other than that they look pretty?)
What does the transpose of a matrix represent?
What's the geometric significance of being "upper triangular"?

My two central goals of the term are as follows. The first is to impart a deeper understanding of the fundamental concepts of linear algebra and answer questions such as these (and many more!). Then we will explore relevant applications as time permits. Priority will be given to topics which are conceptually profound/illuminating because those are the ones which benefit the greatest from being explored in a lecture setting. The second is to train students in mathematical exposition. For many students, this may be their first proof-based math course. I will strive to be a good role model by writing out proofs fully in complete sentences on the blackboard in lectures.

Pacing

Dear students: please do not underestimate this class, even if you think the grading scheme (see below) appears somewhat lenient! There is a LOT to cover and we move with twice the intensity of fall/spring offerings (8 hours a week instead of 4).

Grading

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Here is the breakdown:

40% Homework
30% Midterm
30% Final
0% Starred Problems

Now let's discuss each of these items in more detail.

Homework

Homework is the most important part of this class. Indeed, it is the most important part of any math class: you must get your hands dirty to get comfortable with the material. There will be two kinds of homework assigned: "Exercises" from Axler and "Problems" composed by yours truly (or taken from other sources). The latter (the "problems") will be hard, and they are the only ones you should turn in. But you are also expected to do the "exercises"---if you need some incentive for it, refer to the "Exams" paragraph below.

Collaboration is allowed and also encouraged, but you must write up and submit your own solutions, and you absolutely must not copy other people's writeups. Copying is not collaboration! Consultation of other sources is allowed, however not really encouraged. In any case, you must credit anyone (other than myself) and anything (other than Axler's text) you get help from on the homework.

Your homework must be typeset in LaTeX. You should view this requirement as part of the "mathematical exposition" half of this class. Of course, this is in part out of consideration for the grader, but there is also benefit to yourself as well. I've noticed from experience that students' handwritten arguments tend to be overall less coherent than their typed ones. In any case, I assure you that the time cost of learning the basics of LaTeX is not big. The amount of time spent typesetting and polishing your solutions will certainly be incomparable to the time spent thinking and trying things out by hand! Here are some resources for LaTeX:

Detexify can help you find the code for a math symbol.
Overleaf/ShareLaTeX is a convenient online editor, and they also have a tutorial series for getting started with LaTeX.
TeXStudio is the (offline) editor I personally use, but there are plenty of others.

When writing up your homework, you should think of your peers as the intended audience. This is also a good guideline for how much detail to include in your solutions as well. You do NOT need to typeset purely arithmetical computations. To avoid the possibility of copying, you should NOT share your writeup with other students for proofreading.
Homeworks will be assigned weekly and will be due on Tuesday. They should be submitted in .PDF format to Gradescope.

Exams

Exams will be comprised of some combination of multiple-choice, short-answer, and free-response (proof) problems. At least half of the proof problems will be taken directly from the assigned homework "exercises." The rest will be of comparable difficulty (and likely also from Axler). While I expect your very best on the homework, it would be unreasonable to have the same standards for exams, so the exams will be generally much easier than the homework. The exams will be open-book but NOT open-note, to avoid the possibility of students just copying exercise solutions into a cheat-sheet. Your exam grade will be determined by the higher of the following two schemes:

Both exams weighted equally at 30%.
The final weighted at 50% and the midterm at 0%. (If you miss the midterm because of an emergency outside your control, the value of 60% is used instead of 50%.)

Starred Problems

Occasionally you may encounter a problem that is marked with an asterisk (*). These problems are completely optional and also of a higher difficulty level. At the end of the course, I will assign a letter grade to each student, capped at A. Then, for the students with the A letter grade, I will examine their performance on the starred problems to decide whether to instead give the grade of A+. For GPA calculations there is no difference whatsoever between A and A+, so this is purely for bragging rights.

Meeting Times

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If you would like to meet but cannot make it to my listed office hours, please send me an email so we can arrange a time.

Section	Meeting Times	Location
14124 Lecture 003	Mon-Thurs 09:00-10:00	Wheeler 222
14125 Discussion 301	Mon-Thurs 10:00-11:00	Wheeler 222
My Office Hours	Mon&Wed 11:00-12:00, Fri 09:00-11:00	Evans 834

Lecture Synopses

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Part I: The Fundamentals

Week 1: Introduction

		Lecture Content	Homework Exercises	Relevant Material
#1	6/24	Board pictures and audio recording (50MB, I will try to find ways to make it smaller in the future). Some of the board pictures are blurry, probably because I took them at a bad angle. Content: Logistics, definition of a vector space. Homework 1 has been posted to Gradescope and it is also available on this page if you scroll down to the Materials section. Note that it covers material from the rest of the week, so don't expect to be able to solve it yet!	None (went over some basic proofs in discussion).	Please peruse Prof. Hutching's primer on proof writing at your leisure. Also, here is the article that was distributed during class.
#2	6/25	Board pictures (the subspace conditions are partially erased, sorry---please see 1.34 on p.18 in Axler) and audio recording (large!). Examples of fields. \( \mathbb{F}^n \). Subspaces, and how to check a subset is a subspace. Sums of subspaces. Quotient spaces (very informally and briefly).	Exercises	p.5-10, 12-17, 18-21 (up to but not including direct sums). Please see 1.39 in particular (my proof in class was kind of a mess, sorry).
#3	6/26	Board pictures (better quality this time!) and audio recording (large!). Digression on sets and functions (injective, surjective, bijective). Will use tomorrow. Cartesian product (of sets, and then of vector spaces). Graphs, briefly. Direct sums. Condition for \( W_1 + \cdots + W_n \) to be a direct sum. Simpler condition for \( W_1 + W_2 \) to be a direct sum. Definition of linear maps (linear transformations).	Exercises	p.91 and Example 3.72 on p.92 (we will talk about isomorphism tomorrow). p.21-24 for direct sums. Definition 3.2 for linear maps on p.52 (we will return to this section in more detail later; right now I just want to use linear maps to motivate tomorrow's topics).
#4	6/27	Board pictures and audio recordings: first hour and second hour. Linear maps \( \mathbb{F}^n \to V \) and the "standard basis" of \( \mathbb{F}^n \). From this: independence, span, basis. Standard theorems about these. I didn't get to state the theorem that every vector space has a basis, but we wouldn't have proved it anyway (and it's only briefly mentioned in the optional problem on the HW). The crazy proof on the last board was completed after lecture and serves more as an illustration of a more abstract proof technique; please read the proof in Axler as that one is likely much more digestible.	None. Don't worry, we will definitely review the topics from today next week and cement them with examples and exercises.	p.27-36, 39-42, 51-54 (3.5 on p.54 and 2.21 on p.34 may be useful for certain homework problems...)

Homework 1 (due the night of Tuesday, July 2) can be found under Materials.

Week 2: Linear transformations

		Lecture Content	Homework Exercises	Relevant Material
#5	7/01	Board pictures and audio recording. Finish up some results about bases, spanning, and independence. Definition of dimension. Dimension of a (proper) subspace. Dimension of Cartesian product. Theorem on complementary subspaces.	Exercises	The stuff from last Friday, also p.44-47. I plan to prove 2.43 later this week in a different way, though you are certainly encouraged to read Axler's proof of it too.
#6	7/02	Board pictures and audio recording. Vector spaces are classified up to isomorphism by their dimension. Dimension of direct sum via Cartesian product. Null space, range. Quotient spaces. The first isomorphism theorem (and the only of the three that we will consider). Rank-nullity.	Exercises	Mostly section 3.E. My treatment of quotient spaces and the first isomorphism theorem is more self-contained (because my goal was to prove rank nullity as a consequence). So the "logical order" and proofs from class will differ from Axler somewhat.
#7	7/03	Board pictures and audio recording. Quotient spaces (picture again). \( \mathscr{L}(V,W) \) as a vector space. Bases and matrices. Matrix multiplication. Row operations, elementary matrices.	Exercises	2.43 on p.27. p.54-57. Section 3.C. Wikipedia article for elementary matrices.
	7/04	Independence day, no class!

Homework 2 (due the night of Wednesday, July 10) can be found under Materials.

Week 3: The dual space

		Lecture Content	Homework Exercises	Relevant Material
#8	7/08	Board pictures and audio recording. Writing down the matrix of a linear map. Definition of invertibility and inverse for matrices. The effects of row reduction, and why we care. Solving problems using row reduced matrices. Column space \( \cong \) Range. Definition on last board is wrong; column space is a subspace of \( \mathbb{F}^{n,1} \) and as such is not literally equal to the range of the linear map that the matrix encodes---that is a subspace of \( W \). But they are isomorphic; see end of §3.F.	Exercises	3.C, 3.D. Again, it's worthwhile to review row reduction if you don't remember much of it.
#9	7/09	Board pictures and audio recordings: part 1 and part 2. See my writeup (linked in the rightmost column). Duality, the beginning.	Next time!	Writeup §1-3.
#10	7/10	Board pictures and audio recording. Talk about misconceptions from HW1. Proof of the theorem I stated in the writeup. Duality between surjections and injections. The annihilator of a subspace and its dimension.	Exercises	p.101-106 of §3.F. The writeup I posted above: remark at the end of §3, and §4.
#11	7/11	Board pictures and audio recording. Given \( T \colon V \to W \), the null space and range of the dual map \( T' \). Matrices and duality: transpose matrix, row vectors. How a relationship between bases of \( V \) gives a relationship between the dual bases of \( V' \). Extracting an entry of a matrix: \( \mathcal{M}(T)_{i,j} = \psi_i T(v_j) \).	Exercises	Read the rest of §3.F. I forgot to state the "dimension" half of 3.107,109.

Homework 3 (due the night of Tuesday, July 16) can be found under Materials.
Duality is the last topic for the Midterm (being at the end of Chapter 3) and is also the peak of conceptual difficulty in this course. If you have found this topic overwhelming, this is intended to assure you that it is not "setting the tone" for the rest of the term. That is certainly not to say that the course will be easier in the second half---indeed the overall difficulty in any course tends to go up with progress---it is only to say that no other topics will be quite as "mind-bending" as this one.

Week 4: Miscellany

The aim of this week is to explore a few topics without introducing new major concepts, while reviewing material in the process. Relationship between real and complex vector spaces: restriction and extension of scalars. Preservation of flags and "upper-triangular" matrices. The evaluation pairing and "bilinearity."

		Lecture Content	Homework Exercises	Relevant Material
#12	7/15	Board pictures and audio recording. "Restriction of scalars": a study of the process by which a \( \mathbb{C} \)-vector space can be viewed as a \( \mathbb{R} \)-vector space (of twice the original dimension).	Exercises	N/A
#13	7/16	Board pictures and audio recording. Reminder on checking well-definedness. Upper (and lower) triangular matrices. Relationship to flags.	Exercises	N/A
#14	7/17	Board pictures and audio recording. Revisit theorem on lists of functionals from a matrix rank perspective. The evaluation pairing. Bilinear forms. A glimpse of the relationship between duality (namely bases and dual bases) and inner products, to be explored later on.	Exercises	N/A
	7/18	Midterm in Evans 740/736 (7th floor)		Chapters 1 to 3 of Axler

Homework 4 (due the night of Tuesday, July 23) can be found under Materials.

Part II: Algebra of Operators

The structure of vector spaces is entirely characterized by dimension. The structure of linear operators is much richer, and over the complex numbers \( \mathbb{C} \) is classified by Jordan form. A central goal of this second half is to discuss Jordan form (although we might omit the proof) and how it ties together various other concepts.

Week 5: The determinant

The actual content may shift or change in other ways as we actually progress through this week.

		Lecture Content	Homework Exercises	Relevant Material
#15	7/22	Board pictures and audio recordings: part 1 and part 2. Polynomial expressions of an operator. The minimal polynomial (and why it's useful). Reduction of high-degree expressions in an operator via polynomial division with remainder (and how to compute the remainder by plugging in roots). Example used throughout: the Fibonacci sequence. Polynomials over \( \mathbb{C} \), the Fundamental Theorem of Algebra.	Exercises	Writeup, §1 and Appendix A
#16	7/23	Board pictures and audio recordings: part 1 and part 2. Eigenstuff in relation to the minimal polynomial. Guaranteed existence of an eigenvector over \( \mathbb{C} \). Axiomatic characterization of the determinant, geometrically motivated. Proof of uniqueness. Multiplicative property of the determinant. There is a mistake: right-multiplying by \( L_{i,j}(1) \) adds the jth column to the ith column rather than vice versa! This is reflective of the fact that its transpose matrix is \( L_{j,i}(1) \). Also I updated the last proof to have more details.	Exercises	§2 through §3.3 of the writeup above
#17	7/24	Board pictures and audio recordings: part 1 and part 2. Existence of the determinant via cofactor expansion recursive formula. I forgot to take a picture of the board where I talked about block matrices. The determinant doesn't depend on choice of basis (so the "determinant of an operator" makes sense).	Exercises	§3.4 through §5 of the writeup above. I didn't do §5.1 (block matrix from restriction and quotient operators) in class, but it's a worthwhile result and it helps streamline a later argument.
#18	7/25	Board pictures and audio recording. Properties of similar matrices. The characteristic polynomial. Relationship between characteristic and minimal polynomials: the Cayley-Hamilton theorem. Over \( \mathbb{C} \), every matrix is upper-triangularizable.	Exercises	§6 through §7 of the writeup above. Also see Axler §5.B, and §5.A up through Example 5.8. I'll do 5.10 as part of something next week.

Homework 5 (due the night of Tuesday, July 30) can be found under Materials.

Week 6: Inner product spaces

		Lecture Content	Homework Exercises	Relevant Material
#19	7/29	Board pictures and audio recordings: part 1 and part 2. Explicit example of Cayley-Hamilton. Eigenspaces and generalized eigenspaces. I gave a correct but bad definition of generalized eigenspaces; see the writeup for a better one. If \( T \in \mathscr{L}(V) \) then \( V \) is the direct sum of the generalized eigenspaces of \( T \). Diagaonlizability and relation with the minimal polynomial. (Generalized) eigenvectors of different eigenvalues are independent.	Exercises	Example at the end of §7 from the writeup posted in Week 5, then §8 and §9 of that same writeup.
#20	7/30	Board pictures and audio recordings: part 1 + part 2. A brief discussion of the structure of operators, and then the content of Axler §6.A.	Exercises	Axler §6.A
#21	7/31	Board pictures and audio recording. The content of Axler §6.B.	Exercises	Axler §6.B
#22	8/01	Board pictures and audio recordings: main part and brief discussion of least squares. I realize now that my notation for orthogonal projection conflicts with my notation for restriction operator! All instances of \( P_U \) in this lecture refer to "orthogonal projection onto \( U \)." From here on out, I will write \( \operatorname{proj}_U \) instead for clarity. The content of Axler §6.C.	Exercises	Axler §6.C

Homework 6 (due the night of Tuesday, August 6) can be found under Materials.

Week 7: Structure of Operators

First we'll examine the structure of special types of operators on inner product spaces. And then we'll return to the general setting and investigate Jordan decomposition / Jordan form.

		Lecture Content	Homework Exercises	Relevant Material
#23	8/05	Board pictures and audio recording. Adjoint and properties. Operators with special interactions with their adjoint.	Exercises	Axler §7.A
#24	8/06	Board pictures and audio recording. Spectral theorem.	Exercises	Axler §7.A, §7.B
#25	8/07	Board pictures and audio recordings: main part and examples. Rayleigh's principle. Isometries, examples.	Exercises	Axler §7.C but not the stuff on positive operators
#26	8/08	Board pictures and audio recordings: part 1 and part 2. Jordan form. Computations. Given a matrix in JCF, how to read off: eigenvalues, characteristic and minimal polynomials, trace, determinant, etc.	Exercises	Writeup from Week 5, §10 and Proposition 7.7

Homework 7 (due the night of Tuesday, August 13) can be found under Materials.

Week 8: Review and Miscellany

Any new concepts introduced this week are not in the scope of the final exam. One of these days will also be dedicated to a review.

		Lecture Content	Homework Exercises	Relevant Material
#27	8/12	Board pictures and audio recording. "Conic sections" and linear algebra.		N/A
#28	8/13	Board pictures and audio recordings: part 1 and part 2. Calculus and linear algebra: \( f(T) \) for general \( f \). Matrix exponential and logarithm.	Exercises	N/A
#29	8/14	Board pictures and audio recordings: part 1 and part 2. Review of requested topics		N/A
	8/15	Final exam, location TBA	Practice exam Solutions	Second half of course

Materials

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Link	Comments
Final Exam Solutions	Solutions to the final exam administered on 8/15.
Homework 7 Solutions 7	Posted 8/4, and due the night of 8/13. LaTeX templates: matrix version for Problems 3, 4 and operator version for Problems 3, 4. You can choose whether to do 3 and 4 (matrix version) or 3' and 4' (operator version).
Homework 6 Solutions 6	Posted 7/31, and due the night of 8/6. Here is a .tex template.
Homework 5 Solutions 5	Posted 7/24, and due the night of 7/30. Here is a .tex template. Edit (July 27): A correction has been made to Problem 3.
Midterm Solutions	Solutions to the midterm administered on 7/18.
Homework 4 Solutions 4	Posted 7/18, and due the night of 7/23. There aren't many problems, and each individual problem is not long either. Here is a .tex template.
Homework 3 Solutions 3	Posted 7/9, and due the night of 7/16. Here is a .tex template.
Homework 2 Solutions 2	Posted 7/2, and due the night of 7/10. Here is a .tex template. Edit (July 7 at 6:07PM): The formatting of Problem 2 has been adjusted slightly to make it more clear that the arbitrary vector spaces and maps from the first part do not "persist/carry over" into the second part. Part of 2b is coming up with what they ought to be. Edit (July 9): Due date postponed. Also the \( W' \) has nothing to do with dual spaces, sorry for the bad notation.
Homework 1 Solutions 1	Posted 6/24, and due the night of 7/2. Here is a .tex template that you can use. You can copy the contents into a blank Overleaf document, for instance. If you do use this template, pay attention to the comments scattered throughout. Of course you do not need to use this template if you do not want to. Edit: A \( \subset \) sign has been replaced with a \( \subseteq \) sign. In general I will avoid the use of \( \subset \) because it is ambiguous.
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