MATH 135
Linear Algebra I
Summer Term 2026 |
Instructor: Staff |
Total sessions: 35 Sessions |
Office Hours: TBA |
Session Length: 145 Minutes |
Classroom: TBA |
Credits: 3 Units |
Class Length: 7 Weeks |
Language: English |
Course Overview:
This course provides an introduction to the fundamental concepts of linear algebra, focusing on vectors, matrices, and systems of linear equations. Students will develop both computational skills and geometric intuition, enabling them to work effectively in two-dimensional, three-dimensional, and higher-dimensional spaces. The course covers key topics such as vector spaces, linear independence, bases and dimension, matrix operations, linear transformations, determinants, and eigenvalues and eigenvectors. Emphasis is placed on understanding the connections between algebraic methods and geometric interpretations, as well as on clear and logical problem-solving. Applications, including Markov chains and other real-world models, are incorporated to demonstrate the importance of linear algebra in mathematics, science, and engineering.
Required Material:
David Poole, Linear Algebra: A Modern Introduction, Cengage Learning, 2025.
Learning Objectives:
1. Solve systems of linear equations
2. Work with vectors and matrices
3. Understand vector spaces and linear independence
4. Compute eigenvalues and eigenvectors
Course Outline:
Week 1:
Lecture 1-5: Complex Numbers & Vectors
Lecture 1: Complex Numbers Basics
o Definition and standard form
o Conjugate and modulus
Lecture 2: Operations with Complex Numbers
o Addition, subtraction, multiplication, division
o Polar and exponential form
Lecture 3: Introduction to Vectors
o Definition and notation
o Vector addition and scalar multiplication
Lecture 4: Norm & Dot Product
o Length (norm) of a vector
o Dot product and geometric meaning
Lecture 5: Vector Geometry
o Angle between vectors
o Orthogonality and projection
Assignment 1
More specific requirements (e.g., formatting requirements, deadlines, etc.) will be provided in the course.
Week 2:
Lecture 6-10: Geometry & Linear Systems
Lecture 6: Lines and Planes
o Parametric equations
o Standard form equations
Lecture 7: Euclidean Space
o Understanding ℝⁿ
o Geometric interpretation
Lecture 8: Systems of Linear Equations
o Writing systems in matrix form
o Types of solutions
Lecture 9: Gaussian Elimination
o Row operations
o Row echelon form
Lecture 10: Solving Linear Systems
o Back substitution
o Consistent vs inconsistent systems
Week 3:
Lecture 11-15: Matrix Algebra
Lecture 11: Matrix Basics
o Matrix notation and types
o Matrix addition and scalar multiplication
Lecture 12: Matrix Multiplication
o Definition and properties
o Applications
Lecture 13: Identity and Inverse Matrices
o Identity matrix
o Finding inverses
Lecture 14: Matrix Equations
o Solving AX = B
o Invertibility
Lecture 15: Applications of Matrices
o Systems modeling
o Real-world interpretation
Midterm Exam: multiple choice and problem-solving questions
(Contains topics in Lecture 1-15)
Week 4:
Lecture 16-20: Vector Spaces
Lecture 16: Span of Vectors
o Linear combinations
o Spanning sets
Lecture 17: Linear Independence
o Definition and tests
o Dependence vs independence
Lecture 18: Basis and Dimension
o Basis definition
o Dimension of a space
Lecture 19: Subspaces
o Definition and examples
o Testing subspaces
Lecture 20: Coordinate Systems
o Coordinate vectors
o Change of basis
Week 5:
Lecture 21-25: Fundamental Subspaces
Lecture 21: Column Space
o Definition and interpretation
o Basis of column space
Lecture 22: Row Space
o Row space properties
o Basis of row space
Lecture 23: Null Space
o Solving homogeneous systems
o Basis of null space
Lecture 24: Rank and Nullity
o Rank definition
o Rank-nullity theorem
Lecture 25: Applications of Subspaces
o Data interpretation
o Linear models
Assignment 2
More specific requirements (e.g., formatting requirements, deadlines, etc.) will be provided in the course.
Week 6:
Lecture 26-30: Linear Transformations & Determinants
Lecture 26: Linear Transformations
o Definition and examples
o Matrix representation
Lecture 27: Geometry of Transformations
o Rotations, reflections
o Projections
Lecture 28: Determinants Basics
o Definition and properties
o 2×2 and 3×3 determinants
Lecture 29: Cofactor Expansion
o Minor and cofactor
o Expansion method
Lecture 30: Applications of Determinants
o Invertibility test
o Geometric meaning
Week 7:
Lecture 31-35: Eigenvalues & Applications
Lecture 31: Eigenvalues and Eigenvectors
o Definition
o Characteristic equation
Lecture 32: Eigenspaces
o Finding eigenvectors
o Geometric meaning
Lecture 33: Diagonalization
o Conditions for diagonalization
o Computing diagonal form
Lecture 34: Similarity of Matrices
o Definition and properties
o Applications
Lecture 35: Markov Chains & Applications
o Transition matrices
o Long-term behavior
Final Exam: multiple choice and Problem-solving questions
(Contains topics in All Lectures)
Grading Assessment:
Assignment 1 |
15% |
Assignment 2 |
15% |
Midterm |
30% |
Final exam |
30% |
Seminar Participation |
10% |
Total |
100% |
Assignments:
Students are required to complete two assignments during the semester, each assignment will consist of a set of problems that align with the topics discussed in lectures, offering students the opportunity to apply the theories and techniques learned in a practical setting. The problems will focus on areas such as vectors, matrices, systems of linear equations, and related concepts. Students are expected to present clear, well-organized solutions with logical reasoning and proper mathematical notation. Step-by-step explanations should be included to demonstrate understanding. Assignments must be submitted as written documents. Late submissions may incur a penalty unless prior arrangements have been made.
Attendance:
Students are required to attend a weekly seminar led by TA to focus on the week's topic and deepen understanding. Seminar time assigned by TA. Seminar attendance counts toward the final grade.
Exams:
The examinations in this course consist of multiple choice and Problem-solving questions. The final exam is cumulative.
Final Evaluation:
Letter Grade |
Percentage (%) |
Letter Grade |
Percentage |
A+ |
≥95 |
C+ |
64-67 |
A |
89-94 |
C |
60-64 |
A- |
84-88 |
C- |
56-59 |
B+ |
79-83 |
D+ |
54-56 |
B |
73-78 |
D |
50-53 |
B- |
68-72 |
F |
≤50 |
General Policies:
Academic integrity
Academic integrity is the cornerstone of academia and requires students and researchers to maintain honesty, fairness, trust and responsibility in all academic activities. It includes not only avoiding dishonest behaviors such as plagiarism, cheating, and falsifying data, but also requires taking responsibility for one's own academic actions and ensuring that all work is done independently and accurately cites the research of others. Violations of academic integrity can result in severe academic penalties, such as zero grades, suspension or even expulsion, and can cause serious damage to an individual's reputation and future career. Upholding academic integrity is therefore essential to promoting a fair academic environment and facilitating the authentic dissemination of knowledge.
Accessible Resources Policy
The policy ensures that all students, especially those with disabilities, are able to participate equally in school learning and activities. The school provides a wide range of accessibility resources including, but not limited to, specialized classrooms, hearing aids, Braille textbooks, assistive technology, and flexible testing arrangements. Students are required to apply to the school in advance and provide appropriate medical or psychological evaluations so that an individualized support plan can be developed for them. This policy is designed to remove barriers in the academic environment and to ensure that every student has access to equitable learning opportunities.
Withdrawal Policy
Students may choose to withdraw from a course within a specified period of time, and may not be able to do so after the expiration date. When withdrawing from a course, students are required to fill out a withdrawal form with a reason, which will be reviewed and processed on a case-by-case basis. Withdrawal from a course may not affect the student's academic performance. If a student withdraws from a course with incomplete requirements, a “W” may be assigned instead of a grade, depending on the course.
