Summary and Revision Notes

Use these notes to review Chapter 4 quickly.

Key Ideas

Idea	Meaning
point	a location in a space
direction	a movement or change
coordinate system	a convention for naming points and directions
length	size of one vector
distance	length of the difference between two points
angle	relation between two directions
orthogonal	perpendicular; dot product is zero
projection	component of one vector along another direction
basis	directions used to describe vectors in a space
subspace	smaller space closed under addition and scaling
hyperplane	boundary written as $w \cdot x + b = 0$
embedding	learned vector representation of an object

Formulas To Remember

Euclidean length:

\|[x, y]\| = \sqrt{x^2 + y^2}

Distance:

d(a, b) = \|b - a\|

Dot product and angle:

u \cdot v = \|u\|\|v\|\cos(\theta)

Projection onto a nonzero vector:

\operatorname{proj}_b(a) = \frac{a \cdot b}{b \cdot b}b

Hyperplane:

w \cdot x + b = 0

Reading Guide

Situation	Ask
coordinates	point or direction?
distance	which metric defines close?
dot product	score, angle, or projection?
projection	scalar component or projected vector?
residual	what did the projection remove?
hyperplane	score positive, negative, or zero?
boundary score	raw score or normalized distance?
embedding	evidence of structure, or overclaim?
high dimension	does the picture need a formula or code check?

Common Traps

Treating a point and a direction as the same idea because the numbers match.
Forgetting that distance depends on the chosen metric.
Calling vectors orthogonal without checking the dot product.
Projecting onto a non-unit vector without dividing by $b \cdot b$ .
Confusing the boundary with the regions on either side.
Forgetting that a subspace must contain zero.
Treating nearby embeddings as complete explanations.
Trusting a two-dimensional picture too much in a high-dimensional setting.
Reading raw classifier score as distance without accounting for $\|w\|$ .
Forgetting that coordinates are weights on a chosen basis.

Mental Model

Geometry gives shape to computation.

When a formula uses vectors or matrices, ask whether it is measuring length, comparing directions, projecting onto a component, or separating space with a boundary.