1. Introduction to Linear Algebra: Core Concepts

Don’t rush to memorize formulas when learning linear algebra. Instead, think of vectors as data, matrices as transformations, and systems of equations as constraints—this mindset will make subsequent machine learning concepts much easier to grasp.

Before diving into calculations, I always ask: What kind of object is being processed here? A single point? A batch of data? A set of constraints? Or a coordinate transformation?

Linear algebra is an indispensable mathematical foundation across data science, machine learning, and artificial intelligence. This article introduces its core concepts and lays a solid groundwork for what follows.

Vectors

In mathematics, a vector is a quantity possessing both magnitude and direction. In machine learning, vectors commonly represent data points—such as a collection of features. An $n$-dimensional vector can be written as:

There’s no need to get lost in notation right away. Start by placing vectors, matrices, transformations, and systems of equations side-by-side on a single conceptual map—this makes understanding features, dimensionality reduction, and neural network weights significantly more intuitive.

$$\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}$$

For example, when processing images, each pixel—or the entire image—can be represented as a vector. A $28 \times 28$ grayscale image, for instance, can be flattened into a 784-dimensional vector.

Vector Operations

Vectors support several fundamental operations, including addition and scalar multiplication:

Vector Addition:

$$\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ \vdots \\ u_n + v_n \end{pmatrix}$$

Scalar Multiplication:

$$c \cdot \mathbf{v} = \begin{pmatrix} c v_1 \\ c v_2 \\ \vdots \\ c v_n \end{pmatrix}$$

Case Study: Vector Representation of Image Data

Consider the following $3 \times 3$ color image, with pixel values:

[255, 0, 0]
[0, 255, 0]
[0, 0, 255]

This image can be represented as the vector
$\mathbf{v} = \begin{pmatrix} 255 \\ 0 \\ 0 \\ 0 \\ 255 \\ 0 \\ 0 \\ 0 \\ 255 \end{pmatrix}$.
In machine learning, such vectorized representations serve as the foundational input for classification, clustering, and other algorithmic tasks.

Matrices

A matrix is a rectangular array of numbers, typically used to represent linear transformations. An $m \times n$ matrix is written as:

When studying Introduction to Linear Algebra: Core Concepts, begin with a small, reproducible scenario you understand well—then map relevant concepts and practice steps onto it. After reading, re-express the material using your own example.

$$\mathbf{A} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$$

Matrix Operations

Key matrix operations include:

Matrix Addition:

$$\mathbf{A} + \mathbf{B} = \begin{pmatrix} a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\ a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \end{pmatrix}$$

Matrix Multiplication:

$$\mathbf{C} = \mathbf{A} \cdot \mathbf{B}$$

Each entry $c_{ij}$ in the resulting matrix $\mathbf{C}$ is computed as the dot product of row $i$ of $\mathbf{A}$ and column $j$ of $\mathbf{B}$.

Case Study: Image Transformation

Suppose the following $2 \times 2$ matrix represents a linear transformation (e.g., scaling) applied to an image:

$$\mathbf{M} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$$

Applying this transformation to a vector $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$ yields:

$$\mathbf{v'} = \mathbf{M} \cdot \mathbf{v} = \begin{pmatrix} 2x \\ 2y \end{pmatrix}$$

This means every coordinate in the image is scaled by a factor of two.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations and can be compactly expressed using matrices. For example, consider:

$$\begin{align*} 2x + 3y &= 8 \\ x - 4y &= -2 \end{align*}$$

This system can be rewritten in matrix form as:

$$\mathbf{A} \cdot \mathbf{x} = \mathbf{b}$$

where
$\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 1 & -4 \end{pmatrix}$,
$\mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}$, and
$\mathbf{b} = \begin{pmatrix} 8 \\ -2 \end{pmatrix}$.

Solving this system yields the values of $x$ and $y$. Such systems are essential tools in machine learning—especially in regression analysis and parameter estimation.

When reviewing Introduction to Linear Algebra: Core Concepts, place key ideas, procedural steps, and observable outcomes on the same page for efficient revision.

When practicing Introduction to Linear Algebra: Core Concepts, explicitly write down the input conditions, the operation performed, and the resulting output—this facilitates effective self-review later.

Summary

Vectors, matrices, and systems of linear equations form the bedrock of AI technologies. Whether representing data, applying transformations, or building and solving models, linear algebra sits at the heart of it all. In the next article, we’ll explore why linear algebra matters—and deepen your understanding of its pivotal role in artificial intelligence.