Journey into
Linear Algebra
Lecture 1
Welcome to
MATH 2660 👋
Meet the instructor
Minjae Park
Assistant Professor
Department of Mathematics & Statistics
E-mail: minjaep@auburn.edu
Office: Parker Hall 342
Office hours:
- TBD after HW 0
- By appointment
Meet each other!
Please introduce yourself to at least two classmates:
- Your name
- Your major and year
- Where you are from
- One thing you did this winter
Meet linear algebra
Linear algebra is the study of
linear transformations
between
vector spaces
Let’s unpack what each of these words means.
Linear transformation
- A transformation is another word for a map or function, emphasizing geometric changes.
- A linear function is one where changes in the output are proportional to changes in the input.
- A basic example is the function
\[ f : \mathbb{R} \to \mathbb{R}, \qquad f(x) = ax \] for some \(a \in \mathbb{R}\). - Despite its simplicity, this type of function appears everywhere in real-world applications.
Recall from calculus
- Most functions we encounter are differentiable.
- By definition, a differentiable function can be approximated by a linear function near each point.
- In practice, this allows us to simplify complicated situations by linearizing them.
Vector spaces
- A (real) vector is an ordered collection of real numbers, like \(\vec{v} = (1.5, -2, 0, \pi)\).
- The key feature of vectors is that we can perform certain operations on them, e.g. vector addition or scalar multiplication.
- A vector space is a collection of vectors where these operations are always valid and stay within the space.
Analogy: natural numbers
- Recall the natural numbers \(\mathbb N\): \(1, 2, 3, 4, \dots\).
- You can add two natural numbers, and the result is still a natural number.
- Subtraction is also an operation.
- However, subtracting one natural number from another does not always give a natural number.
- We say that the set of natural numbers is closed under addition, but not under subtraction.
Matrices
- A linear transformation between vector spaces can be represented by a matrix.
- Like vectors, a (real) matrix is an array of real numbers.
- By reading a matrix, you understand how the associated linear map transforms a vector.
- There are well-defined operations between matrices, and between matrices and vectors.
Meet linear algebra
Linear algebra is the study of
matrices
such as \[ M = \begin{pmatrix} 1.2 & -0.7 \\ 3.5 & 0 \\ -2.1 & 4.8 \end{pmatrix}. \]
The beauty of linear algebra
- Linear algebra trains your eye to see structure and relationships hidden inside collections of numbers.
- Vectors stop being “just lists,” and matrices stop being “just tables.”
- You begin to see the geometry and meaning behind the data.
A showcase: digital images
- Ordered collections and arrays of numbers naturally arise in real-world data.
- Throughout this course, we’ll explore many such examples.
- A digital image, for instance, is simply an array of numbers—one for each pixel.
Image compression
- By viewing an image as a matrix, we can extract its essential features using SVD (Singular Value Decomposition). Here is a demo.
- It may look like magic at first—but by the end of this course, you’ll understand how it works.
- This idea is a foundation for optimization, data analysis, and machine learning.
Course overview
Homepage
- All course materials
- Links to Canvas, Cengage & WebAssign, iClicker, Gradescope, etc.
Attendance and participation
Daily in lecture via iClicker
Tracked for credit, but not based on correctness, only participation.
Homework
Weekly (except quiz weeks) via WebAssign
More details will be announced later this week.
Quizzes
- Three quizzes during semester in-class: the format and policy will be announced later this week.
- Laptop is required to use iClicker
Grading
| Component | Weight | Details |
|---|---|---|
| Participation | 5% | In-class activities |
| Problem Sets | 15% | Weekly assignments (Lowest dropped) |
| Quizzes | 40% | In-class quizzes (Lowest dropped) |
| Final Exam | 40% | Comprehensive Final |
| Bonus | 0-10% | To be announced |
See course syllabus for how the final letter grade will be determined.
Participate 💻📱
Scan the QR code or go to join.iclicker.com/MBNJ.
Log in with our institution (Auburn - Mathematics & Statistics).
Wrap up
This week’s tasks
- Complete HW 0
- Read the syllabus