Linear Algebra Done Right (3rd Edition) by Sheldon Axler

• The textbook has its own webpage, which contains a lot of useful resources, including links to further video lectures on the various sections, as well as lists of errata

We aim to cover (the majority of) Chapters 1-3 and 5-8 of of Axler's textbook, as well as possibly Chapter 10. In contrast with your first linear algebra class (MAT 211 or AMS 210), we will focus on abstract vector spaces and linear maps as opposed to Euclidean spaces and matrices, which are only meant to serve as examples. Furthermore, in following Axler's textbook, we will be notably develop the theory of linear algebra from a determinant-free perspective. (Compare with the math department's list of MAT 310 topics, which also includes the official description from the undergraduate bulletin.)

Homework is perhaps the most pivotal part of the course (hence why it is worth 50% of the grade). The only way to learn linear algebra is by working through examples to thoroughly develop a mathematical toolkit and to become familiar with the subject. Your homework, with exceptions (in orange below), are to be submitted in envelopes posted outside of the TA's offices by Wednesday night, unless otherwise noted, and will typically cover material through the previous Wednesday. Late homework will not be accepted except in the event of some extenuating circumstance. See the syllabus for more information.

• Homework 1 (due W, Sep 1, 11:59 PM, BY E-MAIL)

• Homework 2 (due W, Sep 8)

• Homework 3 (due W, Sep 15)

• Homework 4 (due W, Sep 22)

• Homework 5 (due W, Sep 29)

• Homework 6 (due W, Oct 6)

• Homework 7 (due W, Oct 13)

• Homework 8 (due W, Oct 27)

• Homework 9 (due W, Nov 3)

• Homework 10 (due W, Nov 10)

• Homework 11 (due W, Nov 17)

• Homework 12 (due W, Dec 1)

FOR PRACTICE: "Homework 13" (not to be submitted)

Midterm Exam: W, Oct 20, in class

Practice Midterm and its Solutions

Midterm and its Solutions

• Statistics:

  • Grade Distribution (Including 5 out of 84 students who did not take the exam.)

  • Average: 35.2%; Median: 24.5% (Including 5 out of 84 students who did not take the exam.)

  • Average: 37.4%; Median: 32.0% (Excluding 5 out of 84 students who did not take the exam.)

• Approximate Letter Grades

  • 70 <= A

  • 40 <= B < 70

  • 15 <= C < 40

  • 10 <= D < 15

  • F < 10

Final Exam: W, Dec 15, **8:00-10:45 AM** (Note the different time)

Recording from Review Session (link will expire mid-2022)

Practice Final and its Solutions

Final and its Solutions

• Statistics (only including students who took the exam)

  • Grade Distribution

  • Average: 88.7/160 (~55%); Median: 84/160 (52.5%)

• Approximate Letter Grades

  • 120 <= A

  • 75 <= B < 120

  • 50 <= C < 75

  • 40 <= D < 50

  • F < 40

Any subsections listed after the date of the most recent lecture are tentative, and based upon previous semesters, though we may end up going a little faster or a little slower. The course should be at a rate of about 10 minutes per page of the textbook (8 pages per lecture), though you will of course spend more time with the material outside of class. It will be helpful for you to read ahead, as it is most difficult to learn mathematics during a first pass with the material.

Lecture 1 (M, Aug 23): 1.A R^n and C^n

Lecture 2 (W, Aug 25): 1.B Definition of Vector Space, 1.C Subspaces

Lecture 3 (M, Aug 30): 1.C Subspaces, 2.A Span and Linear Independence

Lecture 4 (W, Sep 1): 2.A Span and Linear Independence, 2.B Bases

Lecture X (M, Sep 6): LABOR DAY - NO CLASSES

Lecture 5 (W, Sep 8): 2.B Bases, 2.C Dimension

Lecture 6 (M, Sep 13): 3.A The Vector Space of Linear Maps

Lecture 7 (W, Sep 15): 3.B Null Spaces and Ranges

Lecture 8 (M, Sep 20): 3.C Matrices

Lecture 9 (W, Sep 22): 3.D Invertibility and Isomorphic Vector Spaces

Lecture 10 (M, Sep 27): 3.E Products and Quotients of Vector Spaces

Lecture 11 (W, Sep 29): 3.E Products and Quotients of Vector Spaces, 3.F Duality

Lecture 12 (M, Oct 4): 3.F Duality

Lecture 13 (W, Oct 6): 3.F Duality, 4. Polynomials, 5.A Invariant Subspaces

Lecture X (M, Oct 11): FALL BREAK - NO CLASSES

Lecture 14 (W, Oct 13): 5.A Invariant Subspaces, 5.B Eigenvectors and Upper-Triangular Matrices

Lecture 15 (M, Oct 18): 5.A Invariant Subspaces, 5.B Eigenvectors and Upper-Triangular Matrices, Review

Lecture X (W, Oct 20): MIDTERM EXAM (Chapters 1-4)

Lecture 16 (M, Oct 25): 5.B Eigenvectors and Upper-Triangular Matrices, 5.C Eigenspaces and Diagonal Matrices

Lecture 17 (W, Oct 27): 5.C Eigenspaces and Diagonal Matrices; Review of how to write proofs

Lecture 18 (M, Nov 1): 6.A Inner Products and Norm

Lecture 19 (W, Nov 3): 6.A Inner Products and Norms, 6.B Orthonormal Bases

Lecture 20 (M, Nov 8): 6.B Orthonormal Bases, 6.C Orthogonal Complements and Minimization Problems

Lecture 21 (W, Nov 10): 6.C Orthogonal Complements and Minimization Problems

Lecture 22 (M, Nov 15): 6.C Orthogonal Complements and Minimization problems, 7.A Self-Adjoint and Normal Operators

Lecture 23 (W, Nov 17): 7.A Self-Adjoint and Normal Operators

Lecture 24 (M, Nov 22): 7.A Self-Adjoint and Normal Operators, 7.B The Spectral Theorem; See also further remarks

Lecture X (W, Nov 24): THANKSGIVING BREAK - NO CLASSES

Lecture 25 (M, Nov 29): 8.A Generalized Eigenvectors and Nilpotent Operators, 8.B Decomposition of an Operator

Lecture 26 (W, Dec 1): 8.A Generalized Eigenvectors and Nilpotent Operators, 8.B Decomposition of an Operator, 8.D Jordan Form

Lecture 27 (M, Dec 6): Review

FINAL (W, Dec 15): FINAL EXAM **8:00-10:45 AM** (Note the different time)

Your grade is computed as 50% HW, 20% Midterm, 30% Exam.

The HW grade itself is computed as the average percentage of all of the homework for the class, with the lowest such grade dropped.

There are a lot of very good linear algebra materials out there, so I've listed a few below. It is worth noting that Axler's book comes from a deliberately determinant-free perspective (just read the first three paragraphs of his Preface to the Instructor). I've tried to choose somewhat complementary resources which give a different perspective.

• The YouTube series "Essence of linear algebra" by 3blue1brown (Grant Sanderson), which you will watch for HW 1. Here you will see that much of the insight into linear algebra comes from considering the Euclidean spaces R^2 and R^3. You should keep this geometric insight in mind, and think of linear algebra as extracting from this intuitive viewpoint a powerful algebraic machinery that can describe R^n (or C^n or abstract vector spaces in general) and related constructions.

• Sergei Treil's Linear Algebra Done Wrong, which obviously offers an entirely different perspective.

• Jim Hefferon's free textbook, which is how the instructor first learned linear algebra.

• Paul Halmos' Linear Algebra Problem Book, which is quite idiosyncratic, but very rewarding for the patient learner willing to learn by doing.

• The following website of Joseph Khoury which is technologically outdated, but which details a number of applications of linear algebra.

Linear algebra is in many ways the bread and butter of mathematics. It plays a central role in almost every mathematical subject you will encounter from here on in, and even if this is your last true brush with abstract mathematics, there are so many applications that a solid understanding may just change the way you look at the world.

Let us put aside, for a second, what exactly linear algebra is about, except to say that it studies objects which are "linear." The real world, on the other hand, is fundamentally non-linear. So why, then, is linear algebra still so ubiquitous?

From my viewpoint (I'm a geometer mainly), the most salient is that all smooth phenomena are approximately linear. Let us think about calculus. When you take the derivative of a function f, you obtain a new function f', the value of which is giving the slope of a tangent line. In other words, even though the function is non-linear, when x is close to p, we have the approximation f(x) - f(p) ≈ f'(p)(x-p). The field of analysis (in which one rigorously defines the concepts in (multivariable) calculus) is in fact founded on this concept: show that properties of linear functions also hold for non-linear functions by using these approximations. The main technical detail in analysis is understanding what the sign "≈" means in a rigorous fashion. The underlying linear algebra (which we may conflate with understanding linear functions), on the other hand, is taken for granted because it is so well understood!

This idea is more powerful than you may think. Most physical phenomena are modelled using differential equations, so suppose I give you some differential equation modelling the real world, and I give you some input parameters, and ask you to simulate what will happen in the future. We often input such systems into computers to see the result, and any computer ever is using ideas from numerical analysis in order to approximate the solutions to these differential equations. For example, you may be familiar with the Euler method for numerically solving a differential equation. Notice that the figure on the Wikipedia page is specifically using straight lines. Nearly every physical simulation which has been performed on a computer essentially works via a souped-up version of this idea.

So no matter what you end up doing in the future, with just a little effort, it's not too hard to find linear algebra around you.

Before taking this class, you should have familiarity with proofs. A letter grade of at least a C in MAT 200 or MAT 250 is a prerequisite, though if you have familiarity with proofs, you may obtain my permission by sending me an e-mail.

What makes proofs somewhat difficult is that it is somewhat easy to convince oneself that a proof has been understood. For this reason, it is worth pointing out that there are (at least) five levels of understanding of a given proof (in my opinion). Being aware of the level at which you are operating is key to achieving your desired letter grade in this course. The first three levels should be your main concern; the fourth and fifth levels are relevant if you are looking to continue on to graduate school, and we include them here for the sake of completeness.

1. You understand the reasoning of each line of the proof, and that each such line is completely rigorous, and hence the entire proof is rigorous. (Understanding the course material at this first level corresponds approximately to a letter grade of C.)

2. You understand the reason why the proof goes the direction it does. (Understanding the course material at this level corresponds approximately to a letter grade of B.)

3. Given the theorem or proposition statement, you could write the proof yourself. Note that I do not mean that you have simply memorized the proof, but that you really understand why the proof works, and you could write it even a few days after studying the proof. (Understanding the course material at this level corresponds approximately to a letter grade of A.)

4. You understand the theorem or proposition statement well enough that you can readily translate between a "slogan" for it and a rigorous statement. Again, this means without simply memorizing the statement. Take, for example, Proposition 1.44 in the textbook. It comes with the slogan Condition for a direct sum. From that slogan alone, can you reproduce the proposition statement in a rigorous way? Can you do it a few days after looking at the statement of Proposition 1.44? (Understanding the course material at this level corresponds to just starting to be able to lecture on the material.)

5. You understand how the theorem or proposition relates to the subject at large. To keep Proposition 1.44 in mind, you may ask, "Why is this proposition useful?" That's a great question, and it's not intrinsically baked into the statement nor the proof of the theorem and proposition itself. Instead, you have to look at the context. In this way, you begin to understand that the subject at hand has a narrative, with the propositions and theorems acting as plot points. Achieving this level can only be achieved by seeing many examples and working through many exercises. (Understanding the course material at this level corresponds to being able to lecture well on the material, and to rewrite the textbook yourself, perhaps with your own mathematical biases.)