Vectors

What this chapter does

Vectors are small lists of numbers with meaning. This chapter begins with coordinates and simple arithmetic, then connects vectors to weighted sums, similarity, embeddings, and attention scores.

Lessons

Read these in order.

Start with the chapter introduction, then move through the topic lessons. The order is chosen so each page can reuse ideas from the pages before it.

01
Introduction
Why vectors come early in the Mathematics path and where they appear in ML.
02
What Is a Vector?
A first lesson on vectors as lists of numbers with meaning.
03
Vectors as Position and Direction
Two common readings of the same coordinates: location and movement.
04
Vector Addition
Combining movements and data by adding matching coordinates.
05
Subtraction and Scalar Multiplication
Difference vectors and stretching or flipping vectors with one number.
06
Dot Product
Multiplying matching entries, adding them, and reading the result as a score.
07
Norms and Distance
Vector length and distance as the length of a difference vector.
08
Angles and Cosine Similarity
How direction becomes a useful measure of similarity.
09
Projection
How much of one vector lies along another direction.
10
Vectors in Embeddings
Why learned representations are vectors and how vector operations compare them.

Review and practice

Close the chapter deliberately.

Use the conclusion and revision notes before the chapter exercises. Hints and solutions are collected here, while lesson-level exercises reveal their own help inline.

Conclusion

What Chapter 2 accomplished and how vectors prepare the matrix chapter.

Summary and Revision Notes

A compact review of vector operations, notation, and common traps.

Exercises

Chapter-level practice for vector arithmetic and interpretation.

Hints

Low-spoiler nudges for the Chapter 2 exercises.

Solutions

Explained solutions for the Chapter 2 exercises.