Geometry

What this chapter does

Geometry gives algebra a picture. This chapter turns coordinates into points, directions, distances, projections, boundaries, and neighborhoods so model representations feel inspectable instead of abstract.

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 geometry gives algebra a picture for machine learning.
02
Points, Directions, and Coordinate Systems
How coordinates name locations and movements in a space.
03
Length, Distance, and Spheres
Length for one vector, distance between points, and fixed-radius regions.
04
Angles and Orthogonality
Angles, zero dot products, and directions that do not overlap.
05
Projection and Components
How much of a vector lies along a chosen direction.
06
Lines and Hyperplanes
Boundaries written as dot products with a normal vector.
07
Bases and Coordinate Systems
Basis directions as the frame that gives coordinates meaning.
08
Subspaces
Smaller spaces that stay stable under addition and scaling.
09
Decision Boundaries
How classifiers divide representation space into regions.
10
Embedding Geometry
Neighborhoods, directions, and boundaries in learned vector spaces.
11
High-Dimensional Intuition
How to use low-dimensional pictures without over-trusting 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 4 accomplished and how geometry prepares decompositions.

Summary and Revision Notes

A compact review of geometric objects, formulas, and traps.

Exercises

Chapter-level practice for coordinates, distance, projection, and boundaries.

Hints

Low-spoiler nudges for the Chapter 4 exercises.

Solutions

Explained solutions for the Chapter 4 exercises.