Math 218D-1 Spring 2023


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.

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Joe Rabinoff
Joe Rabinoff

Joe Rabinoff
Math & Physics 243
Office Hours:
Tunoon–1pmGross Hall 104
W1:30–3:30pmGross Hall 111
Eylem Yildiz
Eylem Yildiz

Eylem Yildiz
Math & Physics 114
Office Hours:
Tu5:30–6:45pmMath & Physics 113
Th1–2:15pmZoom (see Sakai)
Edna Jones
Edna Jones

Edna Jones
Office Hours:
W9:30–11:30amMath & Physics 205
Bowen Li
Bowen Li

Bowen Li
Office Hours:
M7:30–10pmGross Hall 330


Lecture Instructor Location Time
218D-1 (001) Joe Rabinoff Math & Physics 130 TTh 10:15–11:30am
218D-1 (002) Eylem Yildiz Math & Physics 130 TTh 3:30–4:45pm
Problem Session Instructor Location Time
218D-1 (01D) Bowen Li Social Sciences 119 F 8:30–9:45am
218D-1 (02D) Bowen Li Perkins 071 F 10:15–11:30am
218D-1 (03D) Edna Jones LSRC A247 F noon–1:15pm
218D-1 (04D) Edna Jones Social Sciences 119 F 3:30–4:45pm


Access Gradescope by clicking on the tab in Sakai.


Strang, Introduction to Linear Algebra
The official text for the course is Introduction to Linear Algebra (5th 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.


Office Hours
Your first stop for help should be office hours! Joe, Eylem, Edna, and Bowen will be holding plenty (times listed above), and you should not hesitate to show up with questions and chat with the other students there.
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
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. If you can't make the scheduled office hours, send us an email and we'll set up a meeting.
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!


  Topic and Section Materials Homework
Week 1: 1/9 Introduction Joe's Notes
Demo: Best fit ellipse
Demo: Rabbit population
Demo: A Plane
Demo: Two Planes
Homework #1
Due 1/18

Note: HW#1 covers the recorded lecture.
Vectors and Matrices (Ch. 1, 2.4, 2.7) Joe's Notes

Recorded: Watch on WarpWire (on Sakai) before the second lecture.
Problem Session
Week 2: 1/16 Solving Equations: Elimination (2.2) Joe's Notes Homework #2
Due 1/25

Gauss–Jordan Algorithm (2.2, 2.5) Joe's Notes
Gauss–Jordan Slideshow
Rabinoff's Reliable Row Reducer
Problem Session
Week 3: 1/23 LU Decomposition (2.3, 2.6) Joe's Notes Homework #3
Due 2/1

Parametric Form Joe's Notes
Demo: Parameterized Line
Demo: Parameterized Plane
Problem Session
Week 4: 1/30 Spans (3.2, 3.3) 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
Demo: Row/column picture: sometimes inconsistent
Homework #4
Due 2/8

Subspaces (3.1, 3.2) Joe's Notes
Demo: Column and Null Space
Problem Session
Week 5: 2/6 Linear Independence and Basis (3.4, 3.5) Joe's Notes
Demo: Coplanar vectors
Demo: Redundant parameterization
Demo: Linearly dependent vectors
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
Demo: Linear Dependence of 3 vectors in R2
Homework #5
Due 2/15

Fundamental Subspaces (3.4, 3.5) Joe's Notes
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
Problem Session
Week 6: 2/13 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
Homework #6
Due 2/22

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 plane
Friday, 2/17: MIDTERM 1 Practice Midterm and Solutions
Midterm and Solutions
Week 7: 2/20 Projection Matrices (4.2) Joe's Notes Homework #7
Due 3/1

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?
Problem Session
Week 8: 2/27 Gram–Schmidt and QR (4.4) Joe's Notes
Demo: Projection formula
Homework #8
Due 3/8

Determinants I (5.1, 5.3) Joe's Notes
Problem Session
Week 9: 3/6 Determinants II (5.2) Joe's Notes
Supplement: cofactor matrix
Homework #9
Due 3/22

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
Problem Session
Week 10: 3/20 Diagonalization: Vector Form (6.2) Joe's Notes Homework #10
Due 3/29

Diagonalization: Geometry (6.2) Joe's Notes
Dynamics demos:
Problem Session
Week 11: 3/27 Complex Numbers; Euler's Formula (9.1) Joe's Notes

Recorded: Watch on WarpWire (on Sakai) before Tuesday's lecture.
Homework #11
Due 4/5

AM/GM and Diagonalizability (6.2) Joe's Notes
Supplement: Proof of the AM≥GM theorem

AM/GM demos:
Dynamics with a Complex Eigenvalue
Systems of ODEs (6.3) Joe's Notes
Friday, 3/31: MIDTERM 2 Practice Midterm and Solutions
Midterm and Solutions
Week 12: 4/3 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
Homework #12
Due 4/12

LDLT; Quadratic Optimization (2.7, 6.5) Joe's Notes
Supplement: LDLT and Cholesky
Problem Session
Week 13: 4/10 Quadratic Optimization II (6.5) Joe's Notes Homework #13
Due 4/19

SVD: Outer Product Form (7.1, 7.2) Joe's Notes
Problem Session
Week 14: 4/17 SVD: Matrix Form (7.3) Joe's Notes Homework #14
Due 4/26

PCA I (7.3, 7.4) Joe's Notes
Problem Session
4/25 PCA II (7.3) Joe's Notes Homework #15
(Not collected)

Final exams:
218D-1 (001):Thursday, 5/4, 9am–noon
218D-1 (002):Wednesday, 5/3, 2–5pm
Practice Final and Solutions