Introduction

Geometry is the part of mathematics that gives algebra a picture.

In the earlier chapters, vectors were ordered lists of numbers and matrices were shape-aware transformations. Geometry asks what those numbers mean as positions, directions, distances, angles, regions, and boundaries.

That matters in machine learning because models rarely work with isolated numbers. They work with spaces:

• an image is a point in a high-dimensional input space
• an embedding is a point or direction in a learned space
• a classifier draws boundaries between regions
• a projection keeps one part of a representation and discards another
• similarity search depends on distance or angle

The Main Idea

The same vector can be read in two ways.

As a point, [3, 2] says where something is.

As a direction, [3, 2] says how to move.

Both readings are useful. Geometry teaches when to use each one.

Why This Chapter Comes Here

The vector chapter gave us coordinates, dot products, norms, projection, and embeddings. The matrix chapter gave us transformations.

Geometry connects those ideas:

coordinates
-> positions

dot products
-> angles and boundaries

norms
-> distance

matrices
-> transformations of space

When an equation has a picture, it is easier to notice what it is doing.

What This Chapter Covers

This chapter covers:

• points, directions, and coordinate systems
• length, distance, and circles
• angles and orthogonality
• projection and components
• lines, hyperplanes, and decision boundaries
• bases, coordinates, and subspaces
• embedding geometry
• high-dimensional intuition

The goal is not to become a geometer. The goal is to make ML equations easier to inspect.

Before Moving On

You should already be comfortable with vector addition, dot products, norms, matrix-vector products, and basic shape reasoning. This chapter uses those ideas but gives them a more visual meaning.