Journey into
Linear Algebra

Lecture 1

Minjae Park

Auburn University
MATH 2660 - Spring 2026

January 7, 2026

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

aub.ie/math2660

  • 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