Please note that all times on this website and the course Canvas site refer to Stanford local time.
Linear Algebra and Multivariable Calculus are two of the
widely used mathematical tools in quantitative work across all fields of study. This
course develops conceptual understanding and problem-solving skills in both, highlighting how
multivariable calculus is most naturally understood in terms of
linear algebra, and the course text addresses a variety of real-world applications.
Our focus is on teaching you skills that underlie a wide array of applications and preparing
you for all courses involving advanced quantitative work (across all sciences, engineering, economics, computer science, statistics, and so on).
By the end of this course, you should be able to:
- relate the algebra of systems of linear equations to the geometry of vectors, acquiring the ability to "see" in dimensions far beyond 3 (thinking is seeing!);
- analyze the behavior of multivariable functions via partial derivatives, and combine that information with tools of linear algebra (such as matrices and eigenvalues) to solve optimization problems;
- apply your newly acquired visual skills in high dimensions to gain insight into a variety of real-world applications across data science, natural sciences, and economics.
For a detailed syllabus see
the Syllabus page.
First Day Checklist
- Enrollment in lectures and sections:
Math 51 students attend lectures on MWF, starting on the first Monday of the quarter, and discussion sections on TuTh, starting on the
first Tuesday of the quarter.
Enroll in lectures and discussion sections on Axess.
Please see the enrollment tab for more information.
The links to zoom lectures and discussion sessions are provided in the course canvas site.
- Recommendations on how to optimize your Zoom learning experiences:
During each lecture and discussion section Zoom session, we invite you to
- Turn on video (or display a photo yourself) if you can, but it's fine to stop at any time.
- Choose "Gallery View" (top right) so you can see everyone in the meeting.
- Open "Participants" panel and "Chat" if joining on your computer.
- "Mute" your mic when not speaking
- Contribute in "Chat", note that the facilitators will acknowledge comments in discussion as appropriate and to benefit everyone.
- Use the "Raise Hand" feature in Participant Panel to ask a question or contribute to discussion.
- If you can, remember to start your comments with "Hi, I am/this is..."
- Required materials:
- The textbook has been specially created by the Stanford Math department in consultation with colleagues in many other departments; it is free and electronic-only. To get the book using your SUNet ID, visit the textbook page by clicking here or selecting the Textbook menu item at the top of this page.
The book contains much more than is covered in the course. It also includes many fully worked examples, helpful for studying. We hope it will be a useful resource for topics that you may encounter in later coursework. On the second page of the introduction, you will find the e-mail address for reporting any corrections, typos, etc. The authors of the text are very eager to hear from you.
- Calculators are neither required nor recommended for any exams in Math 51 (we keep the numbers simple on exams). There are a small number of homework problems for which any basic scientific calculator (even a free online one) is useful to convert some expressions into decimal approximations (and such calculations never arise on exams). There is no programming anywhere in the course.
- Check for exam conflicts right away and contact us: Except in case of emergency, you must inform us of exam conflicts at least one week prior to the exam, together with a valid reason for the conflict. The allowable reasons are course-related or competition-related schedule.
There will be 5 bi-weekly short exams. The exams administered on Canvas will consist of mostly numerical answer, True/False and Multiple Choice questions. There will be some randomization element in exams: different students will not have exactly the same exams, however the length and difficulty levels of different exams are designed to be consistent.
The dates of exams are: January 21st, February 4th, February 18th, March 4th, and March 18th. Each exam is administered on Canvas in a 24-hour window starting on Thursday 5PM and ending on following Friday 5PM. Within the 24-hour window, once you start the exam, you will have 1 hour to complete and submit the exam on Canvas. No downloading, scanning or uploading is needed.
See all exam details and policies here.
- Students with documented disabilities:
Students who may need an academic accommodation based on the impact of a disability must initiate the request with the Office of Accessible Education (OAE). See the Registrar's page on academic accommodations. Professional staff will evaluate the request with required documentation, recommend reasonable accommodations, and prepare an Accommodation Letter for faculty dated in the current quarter in which the request is made. Students should contact the OAE as soon as possible since timely notice is needed to coordinate accommodations. The "OAE" is located at 563 Salvatierra Walk (phone: 1-650-723-1066).
You must provide
an accommodation letter, dated in the current quarter, at least one week prior to
an exam, for us to have adequate time to arrange the accommodations. Renewing OAE accommodation is NOT equivalent to alerting the staff of Math 51!
Please email your accommodation letter.
- Technology Support:
Stanford is expanding its current technology support for students by providing additional financial support for remote students to improve or establish home internet and technology capabilities and increasing the amount of technology equipment, including laptops, that students can check out from The Hub. The initial wave of support since spring quarter has focused on low-income students and those who have the greatest demonstrated need. This latest program expansion will provide even more students with access to equipment and home internet improvements.
Math 51 has an
"active learning" structure; research
has shown that pre-class reading, combined with daily participation in class activities targeted to specific learning goals, improves student
learning outcomes in math and science courses. Furthermore, active learning increases student performances and narrows achievement gaps for underrepresented students. Here's what this means for us:
Both MWF class sessions and a majority of TuTh discussion sections are more interactive than traditional math classes:
- Twice each week (exception on weeks with exams) there will be a modest amount of reading in the course text to introduce
some of the motivation behind the topic(s) to be discussed in class, along with an associated questionnaire
on Canvas to be completed before class. We strongly urge you to watch the instructional videos on How to read the course textbook before the first Pre-Class Reading Questionnaire. In addition, there will be one check-in question on material from the previous chapter and lecture to re-enforce your learning; this check-in question will be graded for accuracy, you can think of it as a practice exam question. The responses to the rest of the questionnaire are not
graded for correctness, just for a good-faith effort, to inform
how the instructor organizes the classroom time around the learning goals for that day.
- Most of the TuTh discussion sections focus on small-group collaboration with worksheets consisting of
problems designed around the learning goals and themes in the homework and exam questions. These worksheets are posted on Canvas 2-3 days before each section. Please try to work on the worksheet problems before going to sections, and bring your questions about the worksheet to sections.
The goal is to engage with the new skills and concepts, and to learn from your peers as well as from the guidance of a graduate student
who answers questions. The work in discussion sections is aimed at giving practice with the material
recently learned in the course; it is not graded, and complete solutions are provided later in the day for each TuTh worksheet.
Canvas questionnaire assignments on the twice-weekly pre-class reading: a typical questionnaire consists of 1 check-in question (always the first question) and 3 to 5 "low-stress" questions. Except for the check-in question, you needn't answer more than one or two sentences per question, and you get full marks for ANY good-faith answer. These assignments are intended to give the instructor feedback on how the reading went and how the course is going; think of them as surveys in which students are voting for which topics need more motivation in class (and which need less or none). Because we will have to review your feedback in a limited time period, the firm deadlines are:
The course grade is based on the following components:
- Mondays at 8am (about 5 or 6 questions).
- Wednesdays at 8am (typically about 4 questions), these are ungraded and optional, intended only for those who wanted some guidance on pre-class reading for the Wednesday class.
- Fridays at 8am (typically about 4 questions) except on exam weeks (where there is only ungraded optional pre-class reading for the Friday class); and
- Exception: As a "warm-up," in the first week we'll have an ADDITIONAL graded pre-class reading questionnaire, due on the first Wednesday at 8am. (It will be posted in Canvas by the first Monday morning.)
- 60% for bi-weekly exams.
See exams page for dates, policies, and previous exams;
- 32% for weekly written homework assignments (best 8 out of 9 assignments);
- 8% for pre-class reading questionnaires on Canvas (total points earned divided by 85% of total possible points, not to exceed 100%);
By Math Department policy, any student found to be in violation of the Honor Code on any assignment or exam in this course will receive a final course letter grade of NP.
You are fully responsible to adhering to the requirements of the Honor Code document. In particular, it is forbidden to
The university is well-aware of "academic educational sites," such as Chegg, Slader, CourseHero, etc. Their use in connection with the exam is an Honor Code violation that is taken very seriously at Stanford. To reinforce this, here is cautionary true story. Last Spring, some students set up accounts with fake names and fake e-mail addresses on some such sites and cheated on exams in that way. University staff enforcing academic integrity found uploaded exam material on such sites (some behind paywalls) and then secured cooperation of those websites to determine the real identities of students how posted exam material or viewed solutions there. This led to some students being punished by Stanford for violating the Honor Code.
More information about the Stanford Honor Code can be found here.
- Collaborating with another student or any other person.
- Copying from another's exam or allowing another student to copy your work.
- Communicating with a person other than the teaching staff via email, text messaging, Google, any form of social media, messenger, chat rooms, message boards, etc., about anything related to the exam.
- Plagiarism of material that you did not create, such as copying parts of posted solutions or text wholesale from anywhere, including the internet. The work that you submit must be your own. This also includes representing another's work as your own.
- Sharing the exam questions or anything in your solutions with any other person for any reason. The restrictions on sharing exam content applies until Friday 8:30AM PDT, which is the designated time for the last exam to end. For most of you, this embargo may go well beyond the finish time of YOUR exam.
You are encouraged to attend the office
hours provided by the
instructors and teaching assistants. You
may attend the office hours of any teaching staff member. No appointment is ever necessary, just drop in at the scheduled office hours with your questions! The office hours page also lists some other help resources.
The scheduled office hours for any given week can be found on the office
hours page. Note that they might change slightly from week to week so it's always a good idea to check both the time and zoom link on Canvas.
The Stanford Math Department does not use Piazza or similar platforms in its courses. This decision is based on a careful review of a variety of issues. We strongly encourage students working with and assisting one another, as well as with TA's and instructors. But we believe that (despite FERPA compliance) Piazza does not sufficiently protect student privacy, and there are other potentially adverse effects that give us additional concern.