Math 218D-1 Spring 2025
Welcome
Welcome to Math 218D! I will post almost all course materials on this page. Please check back for information about homework assignments, exams, and additional resources. You should probably read the syllabus.
Jump to Week: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Instructors
Joe Rabinoff
jdr@math.duke.eduMath & Physics 243
Office Hours:
| M | 11am | –1pm | Math & Physics 243 |
| W | 11am | –noon | Math & Physics 243 |
Soham Karwa
soham.karwa@duke.eduOffice Hours:
| M | 5 | –6:30pm | Math & Physics 259 |
| Tu | 6 | –7:30pm | Math & Physics 259 |
Eva Aggarwal
Undergraduate TAeva.aggarwal@duke.edu
Office Hours:
| Tu | 7 | –9pm | Math & Physics 274J |
| W | 7 | –9pm | Math & Physics 274J |
Classes
| Lecture | Instructor | Location | Time |
|---|---|---|---|
| 218D-1 (001) | Joe Rabinoff | Math & Physics 154 | TTh 10:05–11:20am |
| 218D-1 (002) | Soham Karwa | Math & Physics 130 | TTh 3:05–4:20pm |
| Problem Session | Instructor | Location | Time |
| 218D-1 (01D) | Conghan Dong | Gross Hall 105 | F 8:30–9:45am |
| 218D-1 (02D) | Conghan Dong | Gross Hall 105 | F 10:05–11:20am |
| 218D-1 (03D) | Arjun Nigam | Gross Hall 105 | F 11:45am–1:00pm |
| 218D-1 (04D) | Arjun Nigam | Gross Hall 105 | F 3:05–4:20pm |
Resources
Rabinoff's Reliable Row Reducer
For practicing row reduction when you're allergic to fractions.
Interactive Linear Algebra
A complement to Strang's book written by Joe.
Textbook
Strang, Introduction to Linear Algebra
The official text for the course is Introduction to Linear Algebra (6th Ed) by Gilbert Strang, Wellesley–Cambridge Press/SIAM (2016). However, we will follow Strang only loosely. Another good reference is the online text Interactive Linear Algebra, by Dan Margalit and Joe Rabinoff.Help
Office Hours
Your first stop for help should be office hours! Joe, Soham, Conghan, Arjun, and Eva will be holding plenty (times listed above), and you should not hesitate to show up with questions and chat with the other students there. If you can't make the scheduled office hours, send us an email and we'll set up a meeting.Ed Discussion
Ask questions on Ed Discussion. Your instructors and fellow students are happy to post answers for everyone to see, and usually do so much more quickly than someone can respond to an email message.
Help Rooms
The Math Department operates several Help Rooms to provide assistance to students in lower-level courses. Students are invited to drop by the Help Rooms whenever they are open. No appointment is necessary. Please check the MVLA + Probability HR Schedule to find the times dedicated to Math 218D-1.The Academic Resource Center
The Academic Resource Center (ARC) offers free services to all students during their undergraduate careers at Duke. Services include Learning Consultations, Peer Tutoring, Learning Communities, ADHD/LD Coaching, Outreach Workshops, GRE/MCAT Prep, Study Connect, and more. Because learning is a process unique to every individual, we work with each student to discover and develop their own academic strategy for success at Duke. Contact the ARC to schedule an appointment. Undergraduates in any year, studying any discipline can benefit!Schedule
| Topic and Section | Materials | Homework | |||||
|---|---|---|---|---|---|---|---|
| Week 1: 1/6 | Introduction | Joe's Notes Demo: Best fit ellipse Demo: Rabbit population Demo: A Plane Demo: Two Planes |
Homework #1 Due 1/15 11:59pm Note: HW#1 covers the recorded lecture. |
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| Vectors and Matrices (Ch. 1, 2.4, 2.7) | Joe's Notes Recorded: Watch on WarpWire (on Canvas) before the second lecture. |
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| Problem Session | |||||||
| Week 2: 1/13 | Gauss–Jordan Elimination (2.2) | Joe's Notes Gauss–Jordan Slideshow Rabinoff's Reliable Row Reducer |
Homework #2 Due 1/22 11:59pm |
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| Parametric Form, Inverses (2.5) | Joe's Notes Demo: Parameterized Line Demo: Parameterized Plane |
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| Problem Session | |||||||
| Week 3: 1/20 | LU Decomposition (2.3, 2.6) | Joe's Notes | Homework #3 Due 1/29 11:59pm |
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| Spans (2.2, 2.3) | Joe's Notes Demo: A vector Demo: Scalar multiplication Demo: Vector addition Demo: Vector subtraction Demo: Linear combinations of two vectors in R2 Demo: Linear combinations of two vectors in R3 Demo: Linear combinations of three vectors in R3 Demo: Span of 1 vector in R2 Demo: Span of 2 vectors in R2 Demo: Span of 1 vector in R3 Demo: Span of 2 vectors in R3 Demo: Span of 3 vectors in R3 Demo: Column Picture for Consistency: Consistent Demo: Column Picture for Consistency: Inconsistent Demo: Row/column picture: parallel lines |
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| Problem Session | |||||||
| Week 4: 1/27 | Subspaces (3.1, 3.2) | Joe's Notes | Homework #4 Due 2/5 11:59pm |
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| Linear Independence and Basis (3.4, 3.5) | Joe's Notes Demo: Coplanar vectors Demo: Redundant parameterization Demo: Linearly dependent vectors Demo: Linear Dependence of 3 vectors in R2 Demo: Linearly independent vectors Demo: Linear (In)dependence of 1 vector Demo: Linear (In)dependence of 2 vectors Demo: Linear (In)dependence of 3 vectors |
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| Problem Session | |||||||
| Week 5: 2/3 | Fundamental Subspaces (3.4, 3.5) | Joe's Notes Supplement: A Basis for Nul(AT) Demo: Rank-nullity: n=3, r=0 Demo: Rank-nullity: n=3, r=1 Demo: Rank-nullity: n=3, r=2 Demo: Rank-nullity: n=3, r=3 |
Homework #5 Due 2/12 11:59pm |
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| Orthogonal Complements (3.5, 4.1) | Joe's Notes Demo: Closest vector to b Demo: Orthogonal complement of one vector Demo: Orthogonal complement of two vectors Demo: Orthogonal complements in R2 |
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| Problem Session | |||||||
| Week 6: 2/10 | Orthogonal Projections (4.2) | Joe's Notes Demo: Decomposition relative a line in R2 Demo: Decomposition relative a line in R3 Demo: Decomposition relative a plane in R3 Demo: Projection onto a plane Demo: Projection onto a line Demo: Projection onto a line in R3 |
Homework #6 Due 2/19 11:59pm |
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| Projection Matrices (4.2) | Joe's Notes | ||||||
| Friday, 2/14: MIDTERM 1 | Practice Midterm and Solutions Midterm and Solutions |
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| Week 7: 2/17 | Least Squares (4.3) | Joe's Notes Demo: Least-Squares Demo: Best-fit line Demo: Best-fit parabola Demo: Best-fit trigonometric function Demo: Best-fit ellipse Demo: Best-fit ellipse: what is minimized? |
Homework #7 Due 2/26 11:59pm |
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| Gram–Schmidt and QR (4.4) | Joe's Notes | ||||||
| Problem Session | |||||||
| Week 8: 2/24 | Determinants I (5.1, 5.3) | Joe's Notes | Homework #8 Due 3/7 6:59pm |
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| Determinants II (5.2) | Joe's Notes Supplement: Cofactor Matrix |
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| Problem Session | |||||||
| Week 9: 3/3 | Eigenvalues and Eigenvectors (6.1) | Joe's Notes Demo: Rabbits Multiply Demo: Eigenspaces of a flip Demo: Eigenspaces of a shear Demo: Eigenspaces of a rotation Demo: Eigenspaces of the rabbit matrix Demo: Eigenspaces of a 3x3 matrix Demo: Eigenspaces of a projection matrix |
Homework #9 Due 3/19 11:59pm |
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| Diagonalization: Vector Form (6.2) | Joe's Notes | ||||||
| Problem Session | |||||||
| SPRING BREAK | |||||||
| Week 10: 3/17 | Diagonalization: Geometry (6.2) | Joe's Notes Dynamics demos: |
Homework #10 Due 3/26 11:59pm |
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| Complex Numbers; Euler's Formula (9.1) | Joe's Notes Recorded: Watch on WarpWire (on Canvas) before the next lecture. |
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| AM/GM and Diagonalizability (6.2) | Joe's Notes Supplement: Proof of the AM≥GM theorem Demo: Eigenspaces of a non-diagonalizable 3x3 matrix Demo: Eigenspaces of a diagonalizable 3x3 matrix |
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| Problem Session | |||||||
| Week 11: 3/24 | Stochastic Processes and PageRank | Joe's Notes Demo: Dynamics of a 2x2 Positive Stochastic Matrix Further Reading: Arnoldi Iteration Algorithm |
Homework #11 Due 4/2 11:59pm |
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| Spectral Theorem (6.4) | Joe's Notes Demo: Eigenspaces of a 3x3 symmetric matrix Demo: Eigenspaces of a 3x3 symmetric matrix (GM=2) Demo: Dynamics of a 2x2 symmetric matrix |
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| Friday, 3/28: MIDTERM 2 | Practice Midterm and Solutions Midterm and Solutions |
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| Week 12: 3/31 | LDLT; Quadratic Forms (2.7, 6.5) | Joe's Notes Supplement: Supplement: LDLT and Cholesky |
Homework #12 Due 4/9 11:59pm |
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| Quadratic Optimization (6.5) | Joe's Notes | ||||||
| Problem Session | |||||||
| Week 13: 4/7 | SVD: Outer Product Form (7.1, 7.2) | Joe's Notes Further Reading: Lanczos Algorithm for finding the largest eigenvalues and eigenvectors of a large symmetric matrix (e.g., for finding the most important singular values and singular vectors of ATA) |
Homework #13 Due 4/16 11:59pm |
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| SVD: Matrix Form (7.3) | Joe's Notes | ||||||
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| Week 14: 4/14 | PCA I (7.3, 7.4) | Joe's Notes | Homework #14 Due 4/23 11:59pm |
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| PCA II (7.3) | Joe's Notes | ||||||
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| Week 15: 4/21 | Final Exam Review | ||||||
Final exams:
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Practice Final and Solutions | ||||||