Math 22 Linear Algebra and Applications
(Dartmouth college, Summer 2026)

Main goals:
The major early motivation for the development of linear algebra was solving systems of linear equations, which often comes from optimization problems and are ubiquitous in many areas (from economics and engineering to chemistry and physics). We will first learn how to solve such systems, Ax = b, where A is a matrix of numbers, b is a vector of numbers and x is a vector of unknown variables.
Later, the subject evolved into the study of the more abstract concept of linear transformations, which are transformation of spaces of vectors by matrix multiplications. That is, if the vector x goes over all vectors in a certain space, what can we say about the set of vectors Ax? We will try to understand why linear transformations are important by emphasizing their geometric interpretations and some important applications (e.g., Markov chains and the Google page rank algorithm, the SVD decomposition and the Netflix recommending system).

Time and Place: MWF 11:30 AM - 12:35 AM. Kemeny 007, Dartmouth college.
Textbook: Linear Algebra and its applications, D. Lay, S. Lay, J. McDonald, 5th edition.
(the wonderful 3Blue1Brown series is also recommended);

Instructor: Stoyan Dimitrov (emailТoStoyan {at} gmail [dot] com; stoyan.dimitrov@dartmouth.edu).
Office Hours: Friday, 3:30pm - 5:30pm, Kemeny 320.
TA: Jack Reichert (28'), Sessions: T/Th/Sun 7pm-9pm, Kemeny 007.
Peet Tutoring: Hannah Hawkes (28'), Mon 7pm-8pm, Th 9pm - 10pm, drop-in sessions Sun 12pm-1pm.

Lecture Plan:

Week	Topic	Sources (Book Chapter)
1	Systems of Linear Equations. Row Reduction and Echelon Forms. Vector and Matrix Equations.	§1.1 - §1.4
2	Matrix Equations and their solutions. Linear Independence. Vector spaces. Linear transformation. Null and column space.	§1.5, §1.7, §4.1, §4.2
3	Kernel and Range, Injectivity and Surjectivity of Linear transformations. Inverse of a Matrix.	§1.9, §2.1, §2.2, §4.2
4	Inverse Matrix Theorem, Linear Independence. Basis. Coordinates, Dimension and Rank.	§2.3, §2.9, §4.3
5	Change of coordinates matrix and composition of linear transformations. Determinants.	§4.4, §4.7, §3.1, §3.2
6	Eigenvectors and Eigenspaces. Characteristic Equation. Diagonalization of a matrix.	§5.1, §5.2, §5.3, §5.4
7	Orthogonality, Projections, Gram-Schmidt Process and Least Squares.	§6.1, §6.2, §6.3, §6.4
8	Diagonalization of Symmetric Matrices. Applications: Markov Chains and Google's page rank algorithm.	§7.1, §4.9, §5.8
9	The SVD decomposition and its applications.	Link

Assignments and Grading scheme:

[42.5%] Final exam [TBD]
[32.5%] Midterm [Tuesday, 28 July, 5-7pm]
[21%] Quizzes [Every Monday except June 29th, 8 quizzes in total, the lowest is dropped, no make-ups]
[4%] Attendance [Each student needs to keep track of the number of missed lectures and write to my Dartmouth email in case of an absense]

HALL OF FAME (Top scorers in the class): TBD

Additional comments:

• This is the official site of the course. The Canvas page will be used only for announcements and to enter grades.
• There is no official written homework due to the advanced AI, but only weekly quizzes. However, since practice is important for learning Linear Algebra, you are encouraged to try solving the exercises after each chapter we cover in the textbook plus the practice problems there. Additional list of practice problems will be uploaded before the Midterm and the Final exam.
• A makeup Final exam or midterm will only be given in the rare case of a student who has a documented emergency, religious observance, scheduled extracurricular activity such as a game or performance [not practice], scheduled laboratory for another course, or similar commitment. You should also contact me at least 1 week in advance.
• Plagiarism and cheating in any form is strongly prohibited!

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